Non-product smooth actions onCartesian products of manifolds

Krzysztof Pawałowski (UAM Poznań Poland)

38th Symposium on Transformation GroupsNovember 18-20, 2011, Kobe, Japan

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

38th Symposium on Transformation GroupsNovember 18-20, 2011, Kobe, Japan

Non-product smooth actions onCartesian products of manifolds

Krzysztof Pawałowski (UAM Poznań Poland)

Joint work with Marek Kaluba and Wojciech PolitarczykGraduate students of Adam Mickiewicz University

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial

smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random,

it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold,

it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

By an asymmetric smooth manifold we mean a smooth manifoldwithout a non-trivial smooth finite group action.

A popular phrase about smooth manifolds reads as follows.

“If you choose a smooth manifold at random, it is asymmetric.”

Now, we wish to show that this phrase can be complemented bythe following comment.

“If you choose an asymmetric smooth manifold, it is symmetricup to cartesian product operation with sphere factors.”

Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds,

such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product

thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought

the product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem A

Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group.

For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3,

there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm

for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm.

One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1

and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1,

that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thoughtthe product contains an asymmetric manifold as a factor.

Theorem ALet G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

M =Mm × Sn1 × · · · × Snk .

Every fixed point set connected component is homeomorphicto the connected sum Mm#Σm for an appropriate homologym-sphere Σm. One may always assume that k = 1 and whenG = S1, that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds,

such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product

andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem B

For any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4

for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4.

For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product

and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M.

In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,

n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

We construct smooth group actions on Cartesian products ofsymplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Theorem BFor any compact Lie group G , there exists a smooth action ofG on a Cartesian product of symplectic manifolds of the form

M = CP2 × CPn1 × · · · × CPnk

such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP2#Σ4 for an arbitraryhomology 4-sphere Σ4. For an appropriate Σ4, the action of Gon M is non-product and non-symplectic with respect to anypossible symplectic structure on M. In the case where G = S1,n1 = · · · = nk = 1 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk ,

where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm

and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k .

Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n

insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined

by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n,

as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows.

Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R),

where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.

Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem A, consider the equivariant connected sum

(Mm × Sn1 × · · · × Snk ) # Sm+n # Sm+n # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on Xm and linearlyon Sni for i = 1, . . . , k . Moreover, G acts smoothly on Sm+n insuch a way that F (G , Sm+n) is homeomorphic to Σm.

The linear action of G on the product is determined by

Tx(Sm+n) = Rm ⊕ V n

at x ∈ Σm ⊂ Sm+n, as follows. Consider any decomposition

V n ∼= V n11 ⊕ · · · ⊕ V nkk .

For i = 1, . . . , k , let Sni = S(Vi ⊕ R), where G acts trivially on R.Then G acts linearly on Sni with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk ,

where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2

and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k.

Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined

by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4,

where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n.

Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

To prove Theorem B, consider the equivariant connected sum

(CP2 × CPn1 × · · · × CPnk ) # S2n+4 # S2n+4 # · · ·

for n = n1 + · · ·+ nk , where G acts trivially on CP2 and linearlyon CPni for i = 1, . . . , k. Moreover, G acts smoothly on S2n+4

in such a way that F (G , S2n+4) is diffeomorphic to Σ4.

The linear action of G on the product is determined by

Tx(S2n+4) = V 2n ⊕ R4

at x ∈ Σ4 ⊂ S2n+4, where V 2n is the realification of a complexG -module W n. Let

W n ∼= W n11 ⊕ · · · ⊕W nk

k

be the decomposition of W n into irreducible summands W nii .

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k ,

set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C)

and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni ,

the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi

and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k , set

S2ni+1 = S(Wi ⊕ C) and CPni = S(Wi ⊕ C)/S1,

where G acts trivially on C.

If dim Wi > 1, G acts on CPni with exactly one fixed point.

If dim Wi = 1, G acts on CP1 = S2 with exactly two fixed points.

At any fixed point x ∈ CPni , the tangent G -module Tx(CPni ) isthe realification of Wi and thus at any fixed point, the tangentG -module to the product is isomorphic to V 2n ⊕ R4.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)

Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group.

Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold.

Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F .

Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of G

and a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0.

Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D

such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F ,

the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x

is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex withXG = F . Let ξ be a G -vector bundle over X such that

ξ|F ∼= τF ⊕ ε⊕ ν

for a product vector bundle ε over F with the trivial action of Gand a G-vector bundle ν over F with dim νG = 0. Then there isa smooth action of G on a disk D such that the fixed point setis diffeomorphic to F and νF⊂D ∼= ν as G -vector bundles.

We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

Theorem

Let G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group.

Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.

Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n

such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof.

There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m.

In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order,

such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order,

such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

TheoremLet G be a compact Lie group. Let ∆m be a homology m-disk.Then there is a smooth action of G on a disk Dm+n such thatthe fixed point set F (G ,Dm+n) is diffeomorphic to ∆m.

Proof. There exists a finite contractible G -CW complex X suchthat XG = ∆m. In fact, if:

G is connected, set X = G ∗∆m with the joint action of G .

G/G0 is of prime power order, such an X exists by the workof L. Jones (1971).

G/G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic,

∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X

foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n,

one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G

suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m,

the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology disks

As ∆m is Z-acyclic, ∆m is stably parallelizable.

By making use of the product bundle X × (Rm ⊕ V n) over X foran appropriate n-dimensional real G -module V n, one can thickenup X to the standard disk Dm+n with a smooth action of G suchthat F (G ,Dm+n) is diffeomorphic to ∆m. �

We may always assume that n is even and at a chosen pointx ∈ ∆m, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere

such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks,

and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

By a homology m-sphere we mean a closed smooth manifold Σm

of dimension m ­ 0, with the homology H∗(Σm;Z) = H∗(Sm;Z).

Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Any homology 4-sphere bounds a contractible 5-disk.

For any homology m-sphere with m ­ 5, there is a uniquehomotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

There exist homology 3-spheres which bound homology4-disks, and some of the homology 4-disks are contractible.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3,

there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4,

there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5,

there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm,

the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Let G be a compact Lie group.

Corollary

There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ3, there exists a smooth actionof G on a sphere Sn+3 such that F is diffeomorphic to Σ3.

For any homology 4-spheres Σ4, there exists a smooth actionof G on a sphere Sn+4 such that F is diffeomorphic to Σ4.

For any homology m-spheres Σm with m ­ 5, there existsa smooth action of G on a sphere Sm+n such that F ishomeomorphic to Σm.

We may always assume that n is even and at a chosen pointx ∈ Σm, the normal G -module is the realification of a complexG -module.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X ,

let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)

Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds

suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0.

Then the connected sum X#Yis not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X , let b+2 (X ) be the number of positive

entries in a diagonalization of the intersection form of X over Q

H2(X ,Q)× H2(X ,Q)→ Q

(a, b) 7→ 〈a ∪ b, [X ]〉.

Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)Let X and Y be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and b+2 (Y ) > 0. Then the connected sum X#Y

is not a symplectic manifold.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)

Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds

suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0

and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.

Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1,

there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M.

Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃

= X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃.

As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0,

it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994)

that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic

and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)Let X and M be two closed oriented smooth 4-manifolds suchthat b+

2 (X ) > 0 and π1(M) has a subgroup of finite index k > 1.Then the connected sum X#M is not a symplectic manifold.

As π1(M) has a subgroup of finite index k > 1, there existsa k-sheeted covering M̃ → M. Then the k-sheeted coveringE → X#M has the form

E = kX#M̃ = X#((k − 1)X#M̃

).

Set Y = (k − 1)X#M̃. As b+2 (Y ) ­ b+

2 (X ) > 0, it follows bythe result of Taubes (1994) that E is not symplectic and thus,X#M is not symplectic too.

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)There exists a homology 4-sphere Σ4 with π1(Σ4) ∼= SL(2, 5).

By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)applied for X = CP2 and M = Σ4, Sato’s homology 4-sphere,the connected sum CP2#Σ4 is not a symplectic manifold. �

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)There exists a homology 4-sphere Σ4 with π1(Σ4) ∼= SL(2, 5).

By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)applied for X = CP2 and M = Σ4, Sato’s homology 4-sphere,the connected sum CP2#Σ4 is not a symplectic manifold. �

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)

There exists a homology 4-sphere Σ4 with π1(Σ4) ∼= SL(2, 5).

By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)applied for X = CP2 and M = Σ4, Sato’s homology 4-sphere,the connected sum CP2#Σ4 is not a symplectic manifold. �

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)There exists a homology 4-sphere Σ4 with π1(Σ4) ∼= SL(2, 5).

By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)applied for X = CP2 and M = Σ4, Sato’s homology 4-sphere,the connected sum CP2#Σ4 is not a symplectic manifold. �

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)There exists a homology 4-sphere Σ4 with π1(Σ4) ∼= SL(2, 5).

By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)applied for X = CP2

and M = Σ4, Sato’s homology 4-sphere,the connected sum CP2#Σ4 is not a symplectic manifold. �

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)There exists a homology 4-sphere Σ4 with π1(Σ4) ∼= SL(2, 5).

By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)applied for X = CP2 and M = Σ4, Sato’s homology 4-sphere,

the connected sum CP2#Σ4 is not a symplectic manifold. �

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)There exists a homology 4-sphere Σ4 with π1(Σ4) ∼= SL(2, 5).

By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)applied for X = CP2 and M = Σ4, Sato’s homology 4-sphere,the connected sum CP2#Σ4 is not a symplectic manifold. �

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds

ARIGATO GOSAIMASTA!

Krzysztof Pawałowski (UAM Poznań Poland) Non-product smooth actions on Cartesian products of manifolds