Differential cohomology in geometry and analysis · Differential cohomology in geometry and...
Transcript of Differential cohomology in geometry and analysis · Differential cohomology in geometry and...
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Differential cohomology in geometry and analysis
U. Bunke
Universität Regensburg
19.9.2008/ DMV-Tagung 2008
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Theorem
e =1
1− e−z −1z
.
this power series starts with
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Theorem
e =1
1− e−z −1z
.
this power series starts with
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 2 / 20
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Theorem
e =1
1− e−z −1z
.
this power series starts with
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 2 / 20
![Page 5: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/5.jpg)
Theorem
e =1
1− e−z −1z
.
this power series starts with
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) 19.9.2008/ DMV-Tagung 2008 2 / 20
![Page 6: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/6.jpg)
1 Differentiable cohomology
2 e-invariant
3 Application of differentiable K-theory
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Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
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Notation
X , Y , . . . - smooth compact manifoldspossibly with boundary.
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
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Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
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Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Examples: HZ, HQ, MU, , K
Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
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Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
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Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 4 / 20
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Notation
X , Y , . . . - smooth compact manifolds
h - generalized cohomology theory, h∗ := h(∗)Ω(X , h∗) := Ω(X )⊗Z h∗ smooth differential forms with coefficientsin h∗Ωd=0(X , h∗) ⊆ Ω(X , h∗) - closed forms
HdR(X , h∗) - cohomology of (Ω(X , h∗), d)
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Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
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Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
h(X )
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Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLL
h(X )
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Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLLHdR(X , h∗)
h(X )
OO
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Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ω(X , h∗)/im(d)
a&&NNNNNNNNNNN
d // Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLLHdR(X , h∗)
h(X )
OO
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Definition
A smooth extension of h is a contravariant functor h from compactsmooth manifolds to graded abelian groups which fits into the diagram
Ω(X , h∗)/h(X ) s
a&&MMMMMMMMMMM
d // Ωd=0(X , h∗)
h(X )
R99rrrrrrrrrr
I %% %%LLLLLLLLLLHdR(X , h∗)
h(X )
OO
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Flat theory
Definition
We define hflat(X ) := ker(R : h(X ) → Ω(X , h∗)).
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Flat theory
Definition
We define hflat(X ) := ker(R : h(X ) → Ω(X , h∗)).
Ω(X , h∗)/im(d)
a&&NNNNNNNNNNN
d // Ωd=0(X , h∗)
HdR(X , h∗)
OO
h(X )
R99rrrrrrrrrr
I %% %%KKKKKKKKKKHdR(X , h∗)
hflat(X )
+
88qqqqqqqqqqq
β// h(X )
OO
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Why?
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
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Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
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Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
![Page 25: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/25.jpg)
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
![Page 26: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/26.jpg)
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
![Page 27: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/27.jpg)
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
Other applications:
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
![Page 28: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/28.jpg)
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
Other applications:
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
![Page 29: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/29.jpg)
Why?
Differentiable cohomology provides a conceptual way to refinesecondary torsion invariants to R/Z-cohomology classes.
Chern-Simons invariants
Characteristic classes for flat vector bundles
Invariants of elements in stable homotopy groups
Other applications:
topological terms in σ-models
Configuration spaces of field theories with differential form fieldstrength
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 7 / 20
![Page 30: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/30.jpg)
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
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Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 35: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/35.jpg)
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 36: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/36.jpg)
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 37: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/37.jpg)
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 38: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/38.jpg)
Natural questions
Existence of h?First example: HZ -Cheeger-Simons characters 1985K - Freed, Hopkins-Freed (geometric model, not complete) ∼ 2000,B. -Schick (analytic model) 2003S, MU and other bordism theories, B.-Schick (geometric model) ∼2004general h - Hopkins-Singer (homotopy theoretic model), 2005Landweber exact theories B.-Schick, based on MU, 2007
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Uniqueness of h?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 44: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/44.jpg)
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 45: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/45.jpg)
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 46: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/46.jpg)
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 47: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/47.jpg)
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 48: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/48.jpg)
Natural questions
Uniqueness of h?ordinary cohomology theory (like HZ ) is unique- Wiethaup,Simons-Sullivan, 2006-7Differentiable extensions are not unique in general. K -theory is anexample.K is unique, if it is in addition compatible with suspension(Wiethaup).No clear picture in general.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 49: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/49.jpg)
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 50: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/50.jpg)
Natural questions
What is hflat(X )?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatThis is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
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Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 57: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/57.jpg)
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 58: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/58.jpg)
Natural questions
What is hflat(X )?Consequence of axioms:
X 7→ hflat(X ) := ker(R)
is homotopy invariant.natural candidate:fibre sequence of cohomology theories
→ h → hR → hR/Z →
In most examplesh−1
R/Z(X ) ∼= hflat(X )
so thatβ : hflat(X ) → h(X )
is Bockstein.This is clear by construction for the Hopkins-Singer example.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 59: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/59.jpg)
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 60: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/60.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 61: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/61.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 62: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/62.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 63: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/63.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 64: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/64.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 65: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/65.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 66: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/66.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 67: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/67.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 68: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/68.jpg)
Natural questions
Cup product?Assume that h is multiplicative.Require hat I and R are homomorphisms of rings.Then we call h a multiplicative extension.HZ is multiplicative - Cheeger-SimonsK and bordism theories like S, MU have multiplicative extensions(B.-Schick)Extensions constructed using Landweber exactness from MU aremultiplicative.The general case is not clear.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 69: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/69.jpg)
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 70: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/70.jpg)
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 71: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/71.jpg)
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 72: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/72.jpg)
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 73: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/73.jpg)
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 74: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/74.jpg)
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).
Ω(X , h∗)/im(d)
RX/Y A(of )∧...
a // h(X )
f!
I //
R**
h(X )
f!
Ω(X , h∗)RX/Y A(of )∧...
Ω(Y , h∗)/im(d)
a// h(Y )
I//
R
44h(Y ) Ω(Y , h∗)
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 75: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/75.jpg)
Natural questions
Orientation and integration?f : X → Y proper submersion, h-multiplicative.A h-orientation of f is a choice h-Thom class of the stable normalbundle of f .In this case have integration f! : h(X ) → h(Y ).Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 76: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/76.jpg)
Natural questions
Orientation and integration?Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 77: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/77.jpg)
Natural questions
Orientation and integration?Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 78: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/78.jpg)
Natural questions
Orientation and integration?Want a good concept of h-orientation and integrationf! : h(X ) → h(Y )
No additional structures needed for cHZ (Brylinski, Dupont-Ljungman,Gomi)The concept is developed for bordism theories S, MU (geometricconstruction).Landweber exact theories admit integration for MU-oriented maps(B.-Schick).Theory for K is developed and based on local index theory(B.-Schick).
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 79: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/79.jpg)
Natural questions
Existence of h?
Uniqueness of h?
What is hflat(X )?
Cup product?
Orientation and integration?
Riemann-Roch?
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 80: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/80.jpg)
Natural questions
Riemann-Roch? Example: K.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 81: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/81.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 82: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/82.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(Td(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 83: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/83.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 84: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/84.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 85: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/85.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 86: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/86.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 87: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/87.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 88: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/88.jpg)
Natural questions
Riemann-Roch? Example: K.f : X → Y proper submersion,Let f be K-oriented.Differentiable Riemann-Roch states that
K(X )
f!
ch // HQ(X )
f!(cTd(T v f )∪... )
K(Y )
ch // HQ(Y )
commutes.
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 8 / 20
![Page 89: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/89.jpg)
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
![Page 90: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/90.jpg)
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
![Page 91: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/91.jpg)
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
![Page 92: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/92.jpg)
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
![Page 93: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/93.jpg)
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
![Page 94: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/94.jpg)
How?
There are various methods to construct smooth extensions:
Sheaf theory (Deligne cohomology) for HZDifferential characters HZ (Cheeger-Simons), K (Maghfoul, 2008).
Homotopy theory (Hopkins-Singer, 2005)
Cycles and relations (e.g. B. -Schick)
U. Bunke (Universität Regensburg) Differentiable cohomology 19.9.2008/ DMV-Tagung 2008 9 / 20
![Page 95: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/95.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 96: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/96.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 97: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/97.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 98: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/98.jpg)
Stably framed manifolds
M manifold
TM stably framedTM ⊕ Rk
M is trivialized for some large k .
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 99: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/99.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 100: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/100.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordantThere exists a manifold W such that TW is stably framed and∂W ∼= M t −M ′
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 101: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/101.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 102: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/102.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 103: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/103.jpg)
Stably framed manifolds
M manifold
TM stably framed
consider M ∼ M ′ if M and M ′ are framed bordant
[M] ∈ Ωfr - class of M in the group of framed bordism classes
Question: Is [M] trivial?
Pontrjagin-Thom: Ωfr ∼= πS. This group is complicated and onlypartially known.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 10 / 20
![Page 104: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/104.jpg)
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
![Page 105: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/105.jpg)
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
![Page 106: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/106.jpg)
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
![Page 107: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/107.jpg)
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
![Page 108: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/108.jpg)
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
![Page 109: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/109.jpg)
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
![Page 110: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/110.jpg)
Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
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Primary Invariants
Construct homomorphisms ε : Ωfr → A to known groups A and studyimage ε([M]) ∈ A.Ωfr are coefficients of a generalized homology theory represented byspectrum S.[M] corresponds to homotopy class f : Σdim(M)S → SIdea: map to simpler homology theories.Choices: HZ, K, MUuse unit ε : S → KProblem: The primary invariant vanishes for dim(M) > 0 since Ωfr
>0 istorsion (Serre) and K∗ is free.
∗ −2 −1 0 1 2 3 4 5 6 7Ωfr∗ 0 0 Z Z
2Z2
Z24 0 0 Z
2Z
240K∗ Z 0 Z 0 Z 0 Z 0 Z 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 11 / 20
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Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f // S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
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Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
0%%
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
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Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
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Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
Problem: Lift f ∈ Kdim(M)+1 is not unique!
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
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Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
Problem: Lift f ∈ Kdim(M)+1 is not unique!Problem: K∗ is as complicated as πS
∗
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
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Secondary invariants
Vanishing of primary invariant implies existence of lift in
Σ−1K
Σdim(M)S
f //
f99
0%%
S
ε
K
Problem: Lift f ∈ Kdim(M)+1 is not unique!Problem: K∗ is as complicated as πS
∗
Need simpler targets!
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 12 / 20
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R/Z-invariants
KR/Z,∗ is known
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R/Z-invariants
KR/Z,∗ is known
KR/Z,ev∼= R/Z , KR/Z,odd
∼= 0
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
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R/Z-invariants
KR/Z,∗ is known
KR/Z,ev∼= R/Z , KR/Z,odd
∼= 0
Use KR/Z as target!
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R/Z-invariants
RationalizationS // SR
take homotopy cofibre
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R/Z-invariants
Σ−1SR/Zδ // S // SR .
Relate with K -theory.
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R/Z-invariants
Σ−1K
0
""Σ−1SR/Z
δ // S //
ε
SR
K
.
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R/Z-invariants
Σ−1K
0
""
u
yyt tt
tt
Σ−1SR/Zδ // S //
ε
SR
K
.
observe existence and uniqueness of u!
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R/Z-invariants
Σ−1K
0
""
u
yyss
ss
s
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
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R/Z-invariants
Σdim(M)S
f
Σ−1K
0
$$
u
xxqq
q
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
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R/Z-invariants
Σdim(M)S
f
e
Σ−1K
0
$$
u
xxqq
q
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
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R/Z-invariants
Σdim(M)S
f
e
Σ−1K
0
$$
u
xxqq
q
Σ−1SR/Z
εR/Z
δ // S
ε
// SR
εR
Σ−1KR/Z // K // KR
.
Observe that e ∈ KR/Z,dim(M)+1 is well-defined.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 13 / 20
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A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
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A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Assume that primary invariant vanishes :
Σ−1K
ΣkB+
f //
0$$
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
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A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Have liftΣ−1K
ΣkB+
f //
f::
0$$
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
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A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Secondary invariant:
Σ−1K
εR/Zu// Σ−1KR/Z
ΣkB+
e
##
f //
f;;
0$$
S
ε
K
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
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A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
Secondary invariant:
Σ−1K
εR/Zu// Σ−1KR/Z
ΣkB+
e
##
f //
f;;
0$$
S
ε
K
e ∈K−k−1
R/Z (B)
S−k−1R (B)
is well-defined.
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
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A family version
Consider space B and stable cohomotopy class
f : ΣkB+ → S
e ∈K−k−1
R/Z (B)
S−k−1R (B)
is well-defined.
Note that for k ≥ 0 we have
e ∈ K−k−1R/Z (B) .
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 14 / 20
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Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
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Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
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Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
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Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundle
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
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Special Examples
Of particular interest is the following special case.
π : W → B - locally trivial fibre bundle
framing of vertical bundle T vπ := ker(dπ)
π : W → B with framing represents class
ΣkB+f→ S , k = dim(B)− dim(W )
more special case: π : W → B a G-principal bundlebasis of Lie(G) induces trivialization of T vπ by fundamental vectorfields
U. Bunke (Universität Regensburg) e-invariant 19.9.2008/ DMV-Tagung 2008 15 / 20
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A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
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A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
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A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
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A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
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A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
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A secondary index theorem
π : W → B, T vπ framed, f ∈ [ΣkB+, S], e ∈ K−k−1R/Z (B)
Theorem (B.-Schick)
π has canonical K-orientation.
Defineπ!(1) ∈ K(B) .
Note that π!(1) is flat.Define
ean := π!(1) ∈ Kflat ,−k (B) ∼= K−k−1R/Z (B)
Theorem (Secondary index theorem)
ean = e
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 16 / 20
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Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
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Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 17 / 20
![Page 148: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/148.jpg)
Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 17 / 20
![Page 149: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/149.jpg)
Bordism formula
Assume that q : V → B is a zero bordism of π : W → Bas K -oriented bundle.Can extend the smooth K -orientation of π over to q.
Theorem (bordism formula)
π!(1) = a(
∫V/B
Td)
This integral can often be evaluated!
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 17 / 20
![Page 150: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/150.jpg)
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
![Page 151: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/151.jpg)
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
![Page 152: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/152.jpg)
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
![Page 153: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/153.jpg)
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
![Page 154: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/154.jpg)
Principal bundles
π : W → B a G-principal bundleT ⊆ G maximal toruschoose U(1) ⊆ Tlet U(1) act on D2 in the standard way, ∂D2 ∼= U(1).q : V := W ×U(1) D2 → B is K -oriented zero bordism of π : W → B
TheoremIf rank(G) ≥ 2, then e = 0.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 18 / 20
![Page 155: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/155.jpg)
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
![Page 156: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/156.jpg)
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
![Page 157: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/157.jpg)
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
![Page 158: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/158.jpg)
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
![Page 159: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/159.jpg)
rank one case: U(1)
universal bundle W → BU(1)use Chern character
ch : KR → HRper
in order to identify
KR/Z(BU(1)) ∼=H(BU(1); R)
im(ch)
H(BU(1); R) ∼= R[[z]] , |z| = 2
Theorem ∫V/B
Td =1
1− e−z −1z
.
this power series starts with12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 19 / 20
![Page 160: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/160.jpg)
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
![Page 161: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/161.jpg)
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
![Page 162: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/162.jpg)
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
![Page 163: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/163.jpg)
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
![Page 164: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/164.jpg)
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
![Page 165: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/165.jpg)
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20
![Page 166: Differential cohomology in geometry and analysis · Differential cohomology in geometry and analysis U. Bunke Universität Regensburg 19.9.2008/ DMV-Tagung 2008 U. Bunke (Universität](https://reader036.fdocuments.in/reader036/viewer/2022062921/5f03a0387e708231d409fa65/html5/thumbnails/166.jpg)
Remarks
12
+1
12z − 1
720z3 +
130240
z5 − 11209600
z7 +1
47900160z9 − . . . .
first term 12 is Adams e invariant of S1
second term 112 shows that transfer in stable homotopy along the Hopf bundle
produces an element of order at least 12 in πS3∼= Z
24pairing e with higher dimensional primitive classes in K∗(BU(1)) reproducesresults by Miller and Knapp (late 70ties) on the transfer for U(1)-bundles
for higher rank, e.g. U(1)n must replace K by more complicated cohomologytheory, e.g. elliptic cohomologythis is work in progress with N. Naumann.
U. Bunke (Universität Regensburg) Application of differentiable K-theory 19.9.2008/ DMV-Tagung 2008 20 / 20