Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3...
Transcript of Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3...
![Page 1: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/1.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard Floer homology
R. Lipshitz, P. Ozsvath and D. Thurston
May 13, 2009
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 2: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/2.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
1 Review of Heegaard Floer
2 Basic properties of bordered HF
3 Bordered Heegaard diagrams
4 The algebra
5 The cylindrical setting for Heegaard Floer
6 The module CFD
7 The module CFA
8 The pairing theorem
9 Four-dimensional information from bordered HF .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 3: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/3.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Classical Heegaard Floer theory assigns...
To Y 3 closed, oriented chain complexes CF (Y ), C F +(Y ), . . .well-defined up to homotopy equivalence.
To W 4 : Y 31 → Y 3
2 chain maps FW : CF (Y1)→ CF (Y2),. . .smooth, oriented well-defined up to chain homotopy.
Such that. . .
Theorem
If W1 : Y1 → Y2 and W2 : Y2 → Y3 then FW1∪Y2W2 = FW2 ◦ FW1 ,
. . .
(I’m omitting spinc -structures)
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 4: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/4.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Classical Heegaard Floer theory assigns...
To Y 3 closed, oriented chain complexes CF (Y ), C F +(Y ), . . .well-defined up to homotopy equivalence.
To W 4 : Y 31 → Y 3
2 chain maps FW : CF (Y1)→ CF (Y2),. . .smooth, oriented well-defined up to chain homotopy.
Such that. . .
Theorem
If W1 : Y1 → Y2 and W2 : Y2 → Y3 then FW1∪Y2W2 = FW2 ◦ FW1 ,
. . .
(I’m omitting spinc -structures)
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 5: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/5.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising HF . . .
HF contains lots of geometric content:
Detects smooth structures on 4-manifolds. (Ozsvath-Szabo)
Detects the genus of knots / Thurston norm of 3-manifolds.(Ozsvath-Szabo, Ni)
Detects fiberedness of knots / three-manifolds(Ozsvath-Szabo-Ghiggini-Ni, Juhasz,. . . )
Obstructs overtwistedness / Stein fillability of contactstructures (Ozsvath-Szabo)
Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 6: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/6.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising HF . . .
HF contains lots of geometric content:
Detects smooth structures on 4-manifolds. (Ozsvath-Szabo)
Detects the genus of knots / Thurston norm of 3-manifolds.(Ozsvath-Szabo, Ni)
Detects fiberedness of knots / three-manifolds(Ozsvath-Szabo-Ghiggini-Ni, Juhasz,. . . )
Obstructs overtwistedness / Stein fillability of contactstructures (Ozsvath-Szabo)
Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 7: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/7.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising HF . . .
HF contains lots of geometric content:
Detects smooth structures on 4-manifolds. (Ozsvath-Szabo)
Detects the genus of knots / Thurston norm of 3-manifolds.(Ozsvath-Szabo, Ni)
Detects fiberedness of knots / three-manifolds(Ozsvath-Szabo-Ghiggini-Ni, Juhasz,. . . )
Obstructs overtwistedness / Stein fillability of contactstructures (Ozsvath-Szabo)
Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 8: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/8.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising HF . . .
HF contains lots of geometric content:
Detects smooth structures on 4-manifolds. (Ozsvath-Szabo)
Detects the genus of knots / Thurston norm of 3-manifolds.(Ozsvath-Szabo, Ni)
Detects fiberedness of knots / three-manifolds(Ozsvath-Szabo-Ghiggini-Ni, Juhasz,. . . )
Obstructs overtwistedness / Stein fillability of contactstructures (Ozsvath-Szabo)
Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 9: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/9.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising HF . . .
HF contains lots of geometric content:
Detects smooth structures on 4-manifolds. (Ozsvath-Szabo)
Detects the genus of knots / Thurston norm of 3-manifolds.(Ozsvath-Szabo, Ni)
Detects fiberedness of knots / three-manifolds(Ozsvath-Szabo-Ghiggini-Ni, Juhasz,. . . )
Obstructs overtwistedness / Stein fillability of contactstructures (Ozsvath-Szabo)
Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 10: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/10.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising HF . . .
HF contains lots of geometric content:
Detects smooth structures on 4-manifolds. (Ozsvath-Szabo)
Detects the genus of knots / Thurston norm of 3-manifolds.(Ozsvath-Szabo, Ni)
Detects fiberedness of knots / three-manifolds(Ozsvath-Szabo-Ghiggini-Ni, Juhasz,. . . )
Obstructs overtwistedness / Stein fillability of contactstructures (Ozsvath-Szabo)
Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 11: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/11.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 12: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/12.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 13: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/13.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 14: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/14.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 15: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/15.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).
The cobordism maps FW are computable for most W(L-Manolescu-Wang).
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 16: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/16.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 17: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/17.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
But that’s it.
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 18: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/18.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
But that’s it.
The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 19: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/19.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
And yet. . .
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
All variants of HF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF (Y 3) is computable in general (Sarkar-Wang). So isC F−(Y )/U2C F−(Y ) (Ozsvath-Stipsicz-Szabo).The cobordism maps FW are computable for most W(L-Manolescu-Wang).
But that’s it.
The algorithms for HF and FW are inefficient and seem adhoc.
It’s like having only de Rham cohomology, except via nonlinearequations and without the Mayer-Vietoris theorem.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 20: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/20.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Floer homology
The rest of the talk is about joint work with Peter Ozsvathand Dylan Thurston.
Most of it can be found in “Bordered Heegaard Floerhomology: Invariance and pairing,” arXiv:0810.0687. (It’squite long.)
We also wrote an expository paper about some of the ideas,“Slicing planar grid diagrams: a gentle introduction tobordered Heegaard Floer homology,” arXiv:0810.0695,which we hope is easy to read.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 21: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/21.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The goals of bordered Floer homology
Theorem
(Ozsvath-Szabo) If Y = Y1#Y2 then
CF (Y ) ∼= CF (Y1)⊗F2 CF (Y2).
(cf. homology: C F multiplicative rather than additive.)
Bordered Floer theory extends this more general decompositions of3-manifolds along surfaces.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 22: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/22.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The goals of bordered Floer homology
Theorem
(Ozsvath-Szabo) If Y = Y1#Y2 then
CF (Y ) ∼= CF (Y1)⊗F2 CF (Y2).
(cf. homology: C F multiplicative rather than additive.)
Bordered Floer theory extends this more general decompositions of3-manifolds along surfaces.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 23: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/23.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Roughly, bordered HF assigns...
To a surface F , a (dg) algebra A(F ).
To a 3-manifold Y with boundary F , a
right A(F )-module CFA(Y )
left A(−F )-module CFD(Y )
such that
If Y = Y1 ∪F Y2 then
CF (Y ) = CFA(Y1)⊗A(F ) CFD(Y2).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 24: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/24.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Roughly, bordered HF assigns...
To a surface F , a (dg) algebra A(F ).
To a 3-manifold Y with boundary F , a
right A(F )-module CFA(Y )
left A(−F )-module CFD(Y )
such that
If Y = Y1 ∪F Y2 then
CF (Y ) = CFA(Y1)⊗A(F ) CFD(Y2).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 25: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/25.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Roughly, bordered HF assigns...
To a surface F , a (dg) algebra A(F ).
To a 3-manifold Y with boundary F , a
right A(F )-module CFA(Y )
left A(−F )-module CFD(Y )
such that
If Y = Y1 ∪F Y2 then
CF (Y ) = CFA(Y1)⊗A(F ) CFD(Y2).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 26: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/26.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Precisely, bordered HF assigns...
To which is aMarked a connected, closed, A differential gradedsurface oriented surface, algebra A(F )F + a handle decompos. of F
+ a small disk in F
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 27: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/27.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Precisely, bordered HF assigns...
To which is aMarked a connected, closed, A differential gradedsurface oriented surface, algebra A(F )F + a handle decompos. of F
+ a small disk in F
Bordered Y 3, a compact, oriented∂Y 3 = F 3-manifold with
connected boundary,orientation-preservinghomeomorphism F → ∂Y
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 28: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/28.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Precisely, bordered HF assigns...
To which is aMarked a connected, closed, A differential gradedsurface oriented surface, algebra A(F )F + a handle decompos. of F
+ a small disk in F
Bordered Y 3, compact, oriented Right A∞-module
∂Y 3 = F 3-manifold with CFA(Y ) over A(F ),connected boundary, Left dg -module
orientation-preserving CFD(Y ) over A(−F ),homeomorphism F → ∂Y well-defined up to
homotopy equiv.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 29: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/29.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Satisfying the pairing theorem:
Theorem
If ∂Y1 = F = −∂Y2 then
CF (Y1 ∪∂ Y2) ' CFA(Y1)⊗A(F )CFD(Y2).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 30: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/30.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Further structure (in progress):
To an φ ∈ MCG(F ), bimodules CFDA(φ), CFDA(φ).
CFA(φ(Y )) ' CFA(Y ) ⊗A(F ) CFDA(φ)
CFD(φ(Y )) ' CFDA(φ) ⊗A(−F ) CFD(Y )
(inducing an action of MCG0(F ) on Db(A(F )-Mod)).
To F , bimodules CFDD and CFAA, such that
CFD(Y ) ' CFA(Y ) ⊗A(F ) CFDD
CFA(Y ) ' CFAA ⊗A(−F ) CFD(Y ).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 31: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/31.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Further structure (in progress):
To an φ ∈ MCG(F ), bimodules CFDA(φ), CFDA(φ).
CFA(φ(Y )) ' CFA(Y ) ⊗A(F ) CFDA(φ)
CFD(φ(Y )) ' CFDA(φ) ⊗A(−F ) CFD(Y )
(inducing an action of MCG0(F ) on Db(A(F )-Mod)).
To F , bimodules CFDD and CFAA, such that
CFD(Y ) ' CFA(Y ) ⊗A(F ) CFDD
CFA(Y ) ' CFAA ⊗A(−F ) CFD(Y ).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 32: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/32.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising bordered HF
It’s not tautologous.
It provides information about classical HF . For instance:
Theorem
Suppose CFK−(K ) ' CFK−(K ′). Let KC (resp. K ′C ) be thesatellite of K (resp. K ′) with companion C . ThenHFK−(KC ) ∼= HFK−(K ′C ).
It’s good for computations:
CFDA(φ) for generators φ of MCG0 can be computedexplicitly.
This leads to computations of CF (Y ) for any Y , by factoring.In fact, you can compute FW for any W 4.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 33: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/33.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising bordered HF
It’s not tautologous.
It provides information about classical HF . For instance:
Theorem
Suppose CFK−(K ) ' CFK−(K ′). Let KC (resp. K ′C ) be thesatellite of K (resp. K ′) with companion C . ThenHFK−(KC ) ∼= HFK−(K ′C ).
It’s good for computations:
CFDA(φ) for generators φ of MCG0 can be computedexplicitly.
This leads to computations of CF (Y ) for any Y , by factoring.In fact, you can compute FW for any W 4.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 34: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/34.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising bordered HF
It’s not tautologous.
It provides information about classical HF . For instance:
Theorem
Suppose CFK−(K ) ' CFK−(K ′). Let KC (resp. K ′C ) be thesatellite of K (resp. K ′) with companion C . ThenHFK−(KC ) ∼= HFK−(K ′C ).
It’s good for computations:
CFDA(φ) for generators φ of MCG0 can be computedexplicitly.
This leads to computations of CF (Y ) for any Y , by factoring.In fact, you can compute FW for any W 4.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 35: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/35.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising bordered HF
It’s not tautologous.
It provides information about classical HF . For instance:
Theorem
Suppose CFK−(K ) ' CFK−(K ′). Let KC (resp. K ′C ) be thesatellite of K (resp. K ′) with companion C . ThenHFK−(KC ) ∼= HFK−(K ′C ).
It’s good for computations:
CFDA(φ) for generators φ of MCG0 can be computedexplicitly.
This leads to computations of CF (Y ) for any Y , by factoring.In fact, you can compute FW for any W 4.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 36: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/36.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising bordered HF
It’s not tautologous.
It provides information about classical HF . For instance:
Theorem
Suppose CFK−(K ) ' CFK−(K ′). Let KC (resp. K ′C ) be thesatellite of K (resp. K ′) with companion C . ThenHFK−(KC ) ∼= HFK−(K ′C ).
It’s good for computations:
CFDA(φ) for generators φ of MCG0 can be computedexplicitly.
This leads to computations of CF (Y ) for any Y , by factoring.In fact, you can compute FW for any W 4.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 37: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/37.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising bordered HF
It’s not tautologous.
It provides information about classical HF . For instance:
Theorem
Suppose CFK−(K ) ' CFK−(K ′). Let KC (resp. K ′C ) be thesatellite of K (resp. K ′) with companion C . ThenHFK−(KC ) ∼= HFK−(K ′C ).
It’s good for computations:
CFDA(φ) for generators φ of MCG0 can be computedexplicitly.
This leads to computations of CF (Y ) for any Y , by factoring.
In fact, you can compute FW for any W 4.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 38: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/38.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Advertising bordered HF
It’s not tautologous.
It provides information about classical HF . For instance:
Theorem
Suppose CFK−(K ) ' CFK−(K ′). Let KC (resp. K ′C ) be thesatellite of K (resp. K ′) with companion C . ThenHFK−(KC ) ∼= HFK−(K ′C ).
It’s good for computations:
CFDA(φ) for generators φ of MCG0 can be computedexplicitly.
This leads to computations of CF (Y ) for any Y , by factoring.In fact, you can compute FW for any W 4.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 39: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/39.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
Let (Σg , αc1, . . . , α
cg−k , β1, . . . , βg ) be a Heegaard diagram for
a Y 3 with bdy.
Let Σ′ be result of surgering along αc1, . . . , α
cg−k .
Let αa1, . . . , α
a2k be circles in Σ′ \ (new disks intersecting in
one point p, giving a basis for π1(Σ′).These give circles αa
1, . . . , αa2k in Σ.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 40: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/40.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
Let (Σg , αc1, . . . , α
cg−k , β1, . . . , βg ) be a Heegaard diagram for
a Y 3 with bdy.Let Σ′ be result of surgering along αc
1, . . . , αcg−k .
Let αa1, . . . , α
a2k be circles in Σ′ \ (new disks intersecting in
one point p, giving a basis for π1(Σ′).These give circles αa
1, . . . , αa2k in Σ.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 41: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/41.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
Let (Σg , αc1, . . . , α
cg−k , β1, . . . , βg ) be a Heegaard diagram for
a Y 3 with bdy.Let Σ′ be result of surgering along αc
1, . . . , αcg−k .
Let αa1, . . . , α
a2k be circles in Σ′ \ (new disks intersecting in
one point p, giving a basis for π1(Σ′).
These give circles αa1, . . . , α
a2k in Σ.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 42: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/42.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
Let (Σg , αc1, . . . , α
cg−k , β1, . . . , βg ) be a Heegaard diagram for
a Y 3 with bdy.Let Σ′ be result of surgering along αc
1, . . . , αcg−k .
Let αa1, . . . , α
a2k be circles in Σ′ \ (new disks intersecting in
one point p, giving a basis for π1(Σ′).These give circles αa
1, . . . , αa2k in Σ.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 43: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/43.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Let Σ = Σ \ Dε(p).
Σ, αc1, . . . , α
cg−k , α
a1, . . . , α
a2k , β1, . . . , βg ) is a bordered
Heegaard diagram for Y .
Fix also z ∈ Σ near p.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 44: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/44.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Let Σ = Σ \ Dε(p).
Σ, αc1, . . . , α
cg−k , α
a1, . . . , α
a2k , β1, . . . , βg ) is a bordered
Heegaard diagram for Y .
Fix also z ∈ Σ near p.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 45: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/45.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A small circle near p looks like:
This is called a pointed matched circle Z.This corresponds to a handle decomposition of ∂Y .We will associate a dg algebra A(Z) to Z.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 46: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/46.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A small circle near p looks like:This is called a pointed matched circle Z.
This corresponds to a handle decomposition of ∂Y .We will associate a dg algebra A(Z) to Z.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 47: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/47.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A small circle near p looks like:This is called a pointed matched circle Z.This corresponds to a handle decomposition of ∂Y .
We will associate a dg algebra A(Z) to Z.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 48: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/48.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A small circle near p looks like:This is called a pointed matched circle Z.This corresponds to a handle decomposition of ∂Y .We will associate a dg algebra A(Z) to Z.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 49: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/49.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Where the algebra comes from.
Decomposing ordinary (Σ,α,β) into bordered H.D.’s(Σ1,α1,β1) ∪ (Σ2,α2,β2), would want to considerholomorphic curves crossing ∂Σ1 = ∂Σ2.
This suggests the algebra should have to do with Reeb chordsin ∂Σ1 relative to α ∩ ∂Σ1.Analyzing some simple models, in terms of planar griddiagrams, suggested the product and relations in the algebra.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 50: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/50.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Where the algebra comes from.
Decomposing ordinary (Σ,α,β) into bordered H.D.’s(Σ1,α1,β1) ∪ (Σ2,α2,β2), would want to considerholomorphic curves crossing ∂Σ1 = ∂Σ2.This suggests the algebra should have to do with Reeb chordsin ∂Σ1 relative to α ∩ ∂Σ1.
Analyzing some simple models, in terms of planar griddiagrams, suggested the product and relations in the algebra.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 51: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/51.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Where the algebra comes from.
Decomposing ordinary (Σ,α,β) into bordered H.D.’s(Σ1,α1,β1) ∪ (Σ2,α2,β2), would want to considerholomorphic curves crossing ∂Σ1 = ∂Σ2.This suggests the algebra should have to do with Reeb chordsin ∂Σ1 relative to α ∩ ∂Σ1.Analyzing some simple models, in terms of planar griddiagrams, suggested the product and relations in the algebra.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 52: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/52.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So...
Let Z be a pointed matched circle, for a genus k surface.
Primitive idempotents of A(Z) correspond to k-elementsubsets I of the 2k pairs in Z.
We draw them like this:
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 53: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/53.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So...
Let Z be a pointed matched circle, for a genus k surface.
Primitive idempotents of A(Z) correspond to k-elementsubsets I of the 2k pairs in Z.
We draw them like this:
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 54: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/54.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A pair (I , ρ), where ρ is a Reeb chord in Z \ z starting at Ispecifies an algebra element a(I , ρ).
We draw them like this:
From:
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 55: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/55.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
More generally, given (I ,ρ) where ρ = {ρ1, . . . , ρ`} is a set ofReeb chords starting at I , with:
i 6= j implies ρi and ρj start and end on different pairs.
{starting points of ρi ’s} ⊂ I .
specifies an algebra element a(I ,ρ).
From:
These generate A(Z) over F2.R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 56: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/56.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
That is, A(Z) is the subalgebra of the algebra of k-strand,upward-veering flattened braids on 4k positions where:
no two start or end on the same pair
Not allowed.
Algebra elements are fixed by “horizontal line swapping”.
= +
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 57: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/57.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Multiplication...
...is concatenation if sensible, and zero otherwise.
=
=
=0
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Multiplication...
...is concatenation if sensible, and zero otherwise.
=
=
=0
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Multiplication...
...is concatenation if sensible, and zero otherwise.
=
=
=0
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Double crossings
We impose the relation
(double crossing) = 0.
e.g.,
= =0
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The differential
There is a differential d by
d(a) =∑
smooth one crossing of a.
e.g.,
d
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why?
Where do all of these relations (and differential) come from?
Studying degenerations of holomorphic curves.
They can all be deduced from some simple examples.See arXiv:0810.0695.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why?
Where do all of these relations (and differential) come from?
Studying degenerations of holomorphic curves.
They can all be deduced from some simple examples.See arXiv:0810.0695.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 64: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/64.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why?
Where do all of these relations (and differential) come from?
Studying degenerations of holomorphic curves.
They can all be deduced from some simple examples.See arXiv:0810.0695.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 65: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/65.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Algebra – summary
The algebra is generated by the Reeb chords in Z, withcertain relations. e.g.,
Multiplying consecutive Reeb chords concatenates them.Far apart Reeb chords commute.
The algebra is finite-dimensional over F2, and has a nicedescription in terms of flattened braids.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 66: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/66.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Algebra – summary
The algebra is generated by the Reeb chords in Z, withcertain relations. e.g.,
Multiplying consecutive Reeb chords concatenates them.Far apart Reeb chords commute.
The algebra is finite-dimensional over F2, and has a nicedescription in terms of flattened braids.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 67: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/67.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Algebra – summary
The algebra is generated by the Reeb chords in Z, withcertain relations. e.g.,
Multiplying consecutive Reeb chords concatenates them.Far apart Reeb chords commute.
The algebra is finite-dimensional over F2, and has a nicedescription in terms of flattened braids.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 68: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/68.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The cylindrical setting for classical CF :
Fix an ordinary H.D. (Σg ,α,β, z). (Here, α = {α1, . . . , αg}.)The chain complex CF is generated over F2 by g -tuples{xi ∈ ασ(i) ∩ βi} ⊂ α ∩ β. (σ ∈ Sg is a permutation.)(cf. Tα ∩ Tβ ⊂ Symg (Σ).)
Generators: {u, x}, {v , x}.
The differential counts embedded holomorphic maps
(S , ∂S)→ (Σ× [0, 1]× R, (α× 1× R) ∪ (β × 0× R))
asymptotic to x× [0, 1] at −∞ and y × [0, 1] at +∞.
For CF , curves may not intersect {z} × [0, 1]× R.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The cylindrical setting for classical CF :
Fix an ordinary H.D. (Σg ,α,β, z). (Here, α = {α1, . . . , αg}.)The chain complex CF is generated over F2 by g -tuples{xi ∈ ασ(i) ∩ βi} ⊂ α ∩ β. (σ ∈ Sg is a permutation.)
The differential counts embedded holomorphic maps
(S , ∂S)→ (Σ× [0, 1]× R, (α× 1× R) ∪ (β × 0× R))
asymptotic to x× [0, 1] at −∞ and y × [0, 1] at +∞.
For CF , curves may not intersect {z} × [0, 1]× R.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example of CF
Generators: {u, x}, {v , x}.
∂{u, x} = {v , x}+ {v , x} = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example of CF
Generators: {u, x}, {v , x}.
∂{u, x} = {v, x}+ {v , x} = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 72: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/72.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example of CF
Generators: {u, x}, {v , x}.
∂{u, x} = {v , x}{v, x} = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 73: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/73.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
For (Σ,α,β, z) a bordered Heegaard diagram, view ∂Σ as acylindrical end, p.
Maps
u : (S , ∂S)→ (Σ× [0, 1]× R, (α× 1× R) ∪ (β × 0× R))
have asymptotics at +∞, −∞ and the puncture p, i.e., east∞.
The e∞ asymptotics are Reeb chords ρi × (1, ti ).
The asymptotics ρi1 , . . . , ρi` of u inherit a partial order, byR-coordinate.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
For (Σ,α,β, z) a bordered Heegaard diagram, view ∂Σ as acylindrical end, p.
Maps
u : (S , ∂S)→ (Σ× [0, 1]× R, (α× 1× R) ∪ (β × 0× R))
have asymptotics at +∞, −∞ and the puncture p, i.e., east∞.
The e∞ asymptotics are Reeb chords ρi × (1, ti ).
The asymptotics ρi1 , . . . , ρi` of u inherit a partial order, byR-coordinate.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 75: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/75.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
For (Σ,α,β, z) a bordered Heegaard diagram, view ∂Σ as acylindrical end, p.
Maps
u : (S , ∂S)→ (Σ× [0, 1]× R, (α× 1× R) ∪ (β × 0× R))
have asymptotics at +∞, −∞ and the puncture p, i.e., east∞.
The e∞ asymptotics are Reeb chords ρi × (1, ti ).
The asymptotics ρi1 , . . . , ρi` of u inherit a partial order, byR-coordinate.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 76: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/76.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
For (Σ,α,β, z) a bordered Heegaard diagram, view ∂Σ as acylindrical end, p.
Maps
u : (S , ∂S)→ (Σ× [0, 1]× R, (α× 1× R) ∪ (β × 0× R))
have asymptotics at +∞, −∞ and the puncture p, i.e., east∞.
The e∞ asymptotics are Reeb chords ρi × (1, ti ).
The asymptotics ρi1 , . . . , ρi` of u inherit a partial order, byR-coordinate.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 77: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/77.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators of CFD...
Fix a bordered Heegaard diagram (Σg ,α,β, z)
CFD(Σ) is generated by g -tuples x = {xi} with:
one xi on each β-circle
one xi on each α-circle
no two xi on the same α-arc.
xx
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators of CFD...
Fix a bordered Heegaard diagram (Σg ,α,β, z)
CFD(Σ) is generated by g -tuples x = {xi} with:
one xi on each β-circle
one xi on each α-circle
no two xi on the same α-arc.
y
y
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and associated idempotents.
To x, associate the idempotent I (x), the α-arcs not occupiedby x.
x
As a left A-module,
CFD = ⊕xAI (x).
So, if I is a primitive idempotent, I x = 0 if I 6= I (x) andI (x)x = x.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and associated idempotents.
To x, associate the idempotent I (x), the α-arcs not occupiedby x.
As a left A-module,
CFD = ⊕xAI (x).
So, if I is a primitive idempotent, I x = 0 if I 6= I (x) andI (x)x = x.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The differential on CFD.
d(x) =∑
y
∑(ρ1,...,ρn)
(#M(x, y; ρ1, . . . , ρn)) a(ρ1, I (x)) · · · a(ρn, In)y.
where M(x, y; ρ1, . . . , ρn) consists of holomorphic curvesasymptotic to
x at −∞y at +∞ρ1, . . . , ρn at e∞.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
d(b) = a + ρ3x
d(x) = ρ2a
d(a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
d(b) = a + ρ3x
d(x) = ρ2a
d(a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
d(b) = a + ρ3x
d(x) = ρ2a
d(a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
d(b) = a + ρ3x
d(x) = ρ2a
d(a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 86: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/86.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D2: same torus, different diagram.
d(x) = ρ2ρ3x = ρ23x.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D2: same torus, different diagram.
d(x) = ρ2ρ3x = ρ23x.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:a
ρ3
��
1
��???
????
?
xρ2 // b
Second chain complex:
xρ23 // x
They’re homotopy equivalent. In fact:
Theorem
If (Σ,α,β, z) and (Σ,α′, β′, z ′) are pointed bordered Heegaard
diagrams for the same bordered Y 3 then CFD(Σ) is homotopy
equivalent to CFD(Σ′).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:a
ρ3
��
1
��???
????
?
xρ2 // b
Second chain complex:
xρ23 // x
They’re homotopy equivalent. In fact:
Theorem
If (Σ,α,β, z) and (Σ,α′, β′, z ′) are pointed bordered Heegaard
diagrams for the same bordered Y 3 then CFD(Σ) is homotopy
equivalent to CFD(Σ′).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators and idempotents of CFA.
Fix a bordered Heegaard diagram (Σg ,α,β, z)
CFA(Σ) is generated by the same set as CFD: g -tuples x = {xi}with:
one xi on each β-circle
one xi on each α-circle
no two xi on the same α-arc.
Over F2,CFA = ⊕xF2.
This is much smaller than CFD.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators and idempotents of CFA.
Fix a bordered Heegaard diagram (Σg ,α,β, z)
CFA(Σ) is generated by the same set as CFD: g -tuples x = {xi}with:
one xi on each β-circle
one xi on each α-circle
no two xi on the same α-arc.
Over F2,CFA = ⊕xF2.
This is much smaller than CFD.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 92: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/92.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators and idempotents of CFA.
Fix a bordered Heegaard diagram (Σg ,α,β, z)
CFA(Σ) is generated by the same set as CFD: g -tuples x = {xi}with:
one xi on each β-circle
one xi on each α-circle
no two xi on the same α-arc.
Over F2,CFA = ⊕xF2.
This is much smaller than CFD.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 93: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/93.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The differential on CFA...
...counts only holomorphic curves contained in a compact subset ofΣ, i.e., with no asymptotics at e∞.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The module structure on CFA
To x, associate the idempotent J(x), the α-arcs occupied by
x (opposite from CFD).
For I a primitive idempotent, define
xI =
{x if I = J(x)0 if I 6= J(x)
Given a set ρ of Reeb chords, define
x · a(J(x),ρ) =∑
y
(#M(x, y;ρ)) y
where M(x, y;ρ) consists of holomorphic curves asymptoticto
x at −∞.y at +∞.ρ at e∞, all at the same height.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 95: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/95.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The module structure on CFA
To x, associate the idempotent J(x), the α-arcs occupied by
x (opposite from CFD).
For I a primitive idempotent, define
xI =
{x if I = J(x)0 if I 6= J(x)
Given a set ρ of Reeb chords, define
x · a(J(x),ρ) =∑
y
(#M(x, y;ρ)) y
where M(x, y;ρ) consists of holomorphic curves asymptoticto
x at −∞.y at +∞.ρ at e∞, all at the same height.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 96: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/96.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The module structure on CFA
To x, associate the idempotent J(x), the α-arcs occupied by
x (opposite from CFD).
For I a primitive idempotent, define
xI =
{x if I = J(x)0 if I 6= J(x)
Given a set ρ of Reeb chords, define
x · a(J(x),ρ) =∑
y
(#M(x, y;ρ)) y
where M(x, y;ρ) consists of holomorphic curves asymptoticto
x at −∞.y at +∞.ρ at e∞, all at the same height.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 97: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/97.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A local example of the module structure on CFA.
Consider the following piece of a Heegaard diagram, withgenerators {r , x}, {s, x}, {r , y}, {s, y}.
The nonzero products are: {r , x}ρ1 = {s, x},{r , y}ρ1 = {s, y}, {r , x}ρ3 = {r , y}, {s, x}ρ3 = {s, y},{r , x}(ρ1ρ3) = {s, y}.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A local example of the module structure on CFA.
Consider the following piece of a Heegaard diagram, withgenerators {r , x}, {s, x}, {r , y}, {s, y}.The nonzero products are: {r , x}ρ1 = {s, x},{r , y}ρ1 = {s, y}, {r , x}ρ3 = {r , y}, {s, x}ρ3 = {s, y},{r , x}(ρ1ρ3) = {s, y}.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 99: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/99.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A local example of the module structure on CFA.
Consider the following piece of a Heegaard diagram, withgenerators {r , x}, {s, x}, {r , y}, {s, y}.The nonzero products are: {r , x}ρ1 = {s, x},{r , y}ρ1 = {s, y}, {r , x}ρ3 = {r , y}, {s, x}ρ3 = {s, y},{r , x}(ρ1ρ3) = {s, y}.Example: {r , x}ρ1 = {s, x} comes from this domain.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 100: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/100.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A local example of the module structure on CFA.
Consider the following piece of a Heegaard diagram, withgenerators {r , x}, {s, x}, {r , y}, {s, y}.The nonzero products are: {r , x}ρ1 = {s, x},{r , y}ρ1 = {s, y}, {r , x}ρ3 = {r , y}, {s, x}ρ3 = {s, y},{r , x}(ρ1ρ3) = {s, y}.Example: {r , x}ρ3 = {r , y} comes from this domain.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 101: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/101.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A local example of the module structure on CFA.
Consider the following piece of a Heegaard diagram, withgenerators {r , x}, {s, x}, {r , y}, {s, y}.The nonzero products are: {r , x}ρ1 = {s, x},{r , y}ρ1 = {s, y}, {r , x}ρ3 = {r , y}, {s, x}ρ3 = {s, y},{r , x}(ρ1ρ3) = {s, y}.Example: {r , x}(ρ1ρ3) = {s, y} comes from this domain.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 102: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/102.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
d(u) = v
uρ2 = t
uρ23 = v
tρ3 = v .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
d(u) = v
uρ2 = t
aρ23 = v
tρ3 = v .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 104: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/104.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
d(u) = v
uρ2 = t
aρ23 = v
tρ3 = v .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 105: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/105.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
d(u) = v
uρ2 = t
aρ23 = v
tρ3 = v .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 106: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/106.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
d(u) = v
uρ2 = t
aρ23 = v
tρ3 = v .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 107: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/107.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why associativity should hold...
(x · ρi ) · ρj counts curves with ρi and ρj infinitely far apart.
x · (ρi · ρj) counts curves with ρi and ρj at the same height.
These are ends of a 1-dimensional moduli space, with heightbetween ρi and ρj varying.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The local model again.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and why it doesn’t.
But this moduli space might have other ends: broken flowswith ρ1 and ρ2 at a fixed nonzero height.
These moduli spaces – M(x, y; (ρ1, ρ2)) – measure failure ofassociativity. So...
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 110: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/110.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and why it doesn’t.
But this moduli space might have other ends: broken flowswith ρ1 and ρ2 at a fixed nonzero height.
These moduli spaces – M(x, y; (ρ1, ρ2)) – measure failure ofassociativity. So...
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 111: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/111.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Higher A∞-operations
Define
mn+1(x, a(ρ1), . . . , a(ρn)) =∑
y
(#M(x, y; (ρ1, . . . ,ρn))) y
where M(x, y; (ρ1, . . . ,ρn)) consists of holomorphic curvesasymptotic to
x at −∞.
y at +∞.
ρ1 all at one height at e∞, ρ2 at some other (higher) heightat e∞, and so on.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A2: same torus, different diagram.
m3(x , ρ3, ρ2) = x
m4(x , ρ3, ρ23, ρ2) = x
m5(x , ρ3, ρ23, ρ23, ρ2) = x
...
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A2: same torus, different diagram.
m3(x, ρ3, ρ2) = x
m4(x , ρ3, ρ23, ρ2) = x
m5(x , ρ3, ρ23, ρ23, ρ2) = x
...
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
u
m2(·,ρ2)
��
1+ρ23
��@@@
@@@@
@@@@
@@@@
@
xm2(·,ρ3) // v
Second chain complex:
xm3(·,ρ3,ρ2)+m4(·,ρ3,ρ23,ρ2)+... // x
They’re A∞ homotopy equivalent (exercise).Suggestive remark:
(1 + ρ23)−1“=”1 + ρ23 + ρ23, ρ23 + . . .
ρ3(1 + ρ23)−1ρ2“=”ρ3, ρ2 + ρ3, ρ23, ρ2 + . . . .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
u
m2(·,ρ2)
��
1+ρ23
��@@@
@@@@
@@@@
@@@@
@
xm2(·,ρ3) // v
Second chain complex:
xm3(·,ρ3,ρ2)+m4(·,ρ3,ρ23,ρ2)+... // x
They’re A∞ homotopy equivalent (exercise).
Suggestive remark:
(1 + ρ23)−1“=”1 + ρ23 + ρ23, ρ23 + . . .
ρ3(1 + ρ23)−1ρ2“=”ρ3, ρ2 + ρ3, ρ23, ρ2 + . . . .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 116: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/116.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
u
m2(·,ρ2)
��
1+ρ23
��@@@
@@@@
@@@@
@@@@
@
xm2(·,ρ3) // v
Second chain complex:
xm3(·,ρ3,ρ2)+m4(·,ρ3,ρ23,ρ2)+... // x
They’re A∞ homotopy equivalent (exercise).Suggestive remark:
(1 + ρ23)−1“=”1 + ρ23 + ρ23, ρ23 + . . .
ρ3(1 + ρ23)−1ρ2“=”ρ3, ρ2 + ρ3, ρ23, ρ2 + . . . .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
In general:
Theorem
If (Σ,α,β, z) and (Σ,α′, β′, z ′) are pointed bordered Heegaard
diagrams for the same bordered Y 3 then CFA(Σ) is A∞-homotopy
equivalent to CFA(Σ′).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The pairing theorem
Recall:
Theorem
If ∂Y1 = F = −∂Y2 then
CF (Y1 ∪∂ Y2) ' CFA(Y1)⊗A(F )CFD(Y2).
We’ll illustrate this with three examples.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)
// v
a
ρ3
��
1
��???
????
xρ2 // b
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x , t ⊗ a, t ⊗ b.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)
// v
a
ρ3
��
1
��???
????
xρ2 // b
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x , t ⊗ a, t ⊗ b.
d(t ⊗ b) = t ⊗ a + t ⊗ ρ3x = t ⊗ a + tρ3 ⊗ x = t ⊗ a + v ⊗ x
d(u ⊗ x) = v ⊗ x + u ⊗ ρ2a = v ⊗ x + uρ2 ⊗ a = v ⊗ x + t ⊗ a
d(v ⊗ x) = v ⊗ ρ2a = vρ2 ⊗ a = 0
d(t ⊗ a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 121: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/121.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)
// v
a
ρ3
��
1
��???
????
xρ2 // b
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x , t ⊗ a, t ⊗ b.
d(t ⊗ b) = t⊗ a + t ⊗ ρ3x = t ⊗ a + tρ3 ⊗ x = t ⊗ a + v ⊗ x
d(u ⊗ x) = v ⊗ x + u ⊗ ρ2a = v ⊗ x + uρ2 ⊗ a = v ⊗ x + t ⊗ a
d(v ⊗ x) = v ⊗ ρ2a = vρ2 ⊗ a = 0
d(t ⊗ a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 122: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/122.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)
// v
a
ρ3
��
1
��???
????
xρ2 // b
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x , t ⊗ a, t ⊗ b.
d(t ⊗ b) = t ⊗ a + t⊗ ρ3x = t ⊗ a + tρ3 ⊗ x = t ⊗ a + v ⊗ x
d(u ⊗ x) = v ⊗ x + u ⊗ ρ2a = v ⊗ x + uρ2 ⊗ a = v ⊗ x + t ⊗ a
d(v ⊗ x) = v ⊗ ρ2a = vρ2 ⊗ a = 0
d(t ⊗ a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 123: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/123.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)
// v
a
ρ3
��
1
��???
????
xρ2 // b
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x , t ⊗ a, t ⊗ b.
d(t ⊗ b) = t ⊗ a + t ⊗ ρ3x = t ⊗ a + tρ3 ⊗ x = t ⊗ a + v ⊗ x
d(u ⊗ x) = v ⊗ x + u ⊗ ρ2a = v ⊗ x + uρ2 ⊗ a = v ⊗ x + t ⊗ a
d(v ⊗ x) = v ⊗ ρ2a = vρ2 ⊗ a = 0
d(t ⊗ a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 124: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/124.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)
// v
a
ρ3
��
1
��???
????
xρ2 // b
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x , t ⊗ a, t ⊗ b.
d(t ⊗ b) = t ⊗ a + t ⊗ ρ3x = t ⊗ a + tρ3 ⊗ x = t ⊗ a + v ⊗ x
d(u ⊗ x) = v ⊗ x + u ⊗ ρ2a = v ⊗ x + uρ2 ⊗ a = v ⊗ x + t ⊗ a
d(v ⊗ x) = v ⊗ ρ2a = vρ2 ⊗ a = 0
d(t ⊗ a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 125: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/125.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)
// v
a
ρ3
��
1
��???
????
xρ2 // b
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x , t ⊗ a, t ⊗ b.
d(t ⊗ b) = t ⊗ a + t ⊗ ρ3x = t ⊗ a + tρ3 ⊗ x = t ⊗ a + v ⊗ x
d(u ⊗ x) = v ⊗ x + u ⊗ ρ2a = v ⊗ x + uρ2 ⊗ a = v ⊗ x + t ⊗ a
d(v ⊗ x) = v ⊗ ρ2a = vρ2 ⊗ a = 0
d(t ⊗ a) = 0.
This simplifies to F2〈t ⊗ a + u ⊗ x〉 ⊕ F2〈t ⊗ b = v ⊗ x〉.R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 126: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/126.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)// v
xρ23 // x
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 127: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/127.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(·,ρ2)
��
1+m2(·,ρ23)
@@@
@@@@
xm2(·,ρ3)// v
xρ23 // x
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x .
d(u ⊗ x) = v ⊗ x + u ⊗ ρ23x = v ⊗ x + uρ23 ⊗ x = v ⊗ x + v ⊗ x = 0.
d(v ⊗ x) = v ⊗ ρ23x = vρ23 ⊗ x = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 128: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/128.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators of CFA(Y1)⊗ CFD(Y2): u ⊗ x , v ⊗ x .
d(u ⊗ x) = v ⊗ x + u ⊗ ρ23x = v ⊗ x + uρ23 ⊗ x = v ⊗ x + v ⊗ x = 0.
d(v ⊗ x) = v ⊗ ρ23x = vρ23 ⊗ x = 0.
The most interesting part is the interaction:
d
m2
u
v
x
x
ρ23
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 129: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/129.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(·,ρ3,ρ2)+...// x
a
ρ3
��
1
��???
????
xρ2 // b
〈t ⊗ a, t ⊗ b | d(t ⊗ a) = t ⊗ a + t ⊗ b = 0, d(t ⊗ a) = 0〉.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 130: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/130.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(·,ρ3,ρ2)+...// x
a
ρ3
��
1
��???
????
xρ2 // b
〈t ⊗ a, t ⊗ b | d(t ⊗ b) = t⊗ a + t ⊗ a = 0, d(t ⊗ a) = 0〉.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 131: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/131.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(·,ρ3,ρ2)+...// x
a
ρ3
��
1
��???
????
xρ2 // b
〈t ⊗ a, t ⊗ b | d(t ⊗ b) = t ⊗ a + t⊗ a = 0, d(t ⊗ a) = 0〉.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 132: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/132.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(·,ρ3,ρ2)+...// x
a
ρ3
��
1
��???
????
xρ2 // b
〈t ⊗ a, t ⊗ b | d(t ⊗ b) = t ⊗ a + t⊗ a = 0, d(t ⊗ a) = 0〉.
d
d
m3
t
t
b
x
a
ρ3
ρ2
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 133: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/133.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(·,ρ3,ρ2)+...// x
a
ρ3
��
1
��???
????
xρ2 // b
〈t ⊗ a, t ⊗ b | d(t ⊗ b) = t ⊗ a + t⊗ a = 0, d(t ⊗ a) = 0〉.
d
d
m3
t
t
b
x
a
ρ3
ρ2
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 134: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/134.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(·,ρ3,ρ2)+...// x
a
ρ3
��
1
��???
????
xρ2 // b
〈t ⊗ a, t ⊗ b | d(t ⊗ b) = t ⊗ a + t⊗ a = 0, d(t ⊗ a) = 0〉.
d
d
m3
t
t
b
x
a
ρ3
ρ2
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 135: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/135.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The surgery exact sequence
Theorem
(Ozsvath-Szabo) For K a knot in Y there is an exact sequence
→ HF (Y∞(K ))→ HF (Y−1(K ))→ HF (Y0(K ))→ HF (Y∞(K ))→
Proof via bordered Floer.
Define
H∞ :
0z
12
3
r
r
H−1 :
0z
12
3
a
a
b b H0 :
0z
12
3
n n
There’s a s.e.s.0→ CFD(H∞)→ CFD(H−1)→ CFD(H0)→ 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 136: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/136.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The surgery exact sequence
Theorem
(Ozsvath-Szabo) For K a knot in Y there is an exact sequence
→ HF (Y∞(K ))→ HF (Y−1(K ))→ HF (Y0(K ))→ HF (Y∞(K ))→
Proof via bordered Floer.
Define
H∞ :
0z
12
3
r
r
H−1 :
0z
12
3
a
a
b b H0 :
0z
12
3
n n
There’s a s.e.s.0→ CFD(H∞)→ CFD(H−1)→ CFD(H0)→ 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 137: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/137.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Is it the same sequence?
For
H∞ :
0z
12
3
r
r
H−1 :
0z
12
3
a
a
b b H0 :
0z
12
3
n n
the maps are
ϕ(r) = b
z
+ ρ2a
z
ψ(a) = n
z
ψ(b) = ρ2n
z
.
A version of the pairing theorem shows this gives the triangle mapon HF .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 138: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/138.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Is it the same sequence?
For
H∞ :
0z
12
3
r
r
H−1 :
0z
12
3
a
a
b b H0 :
0z
12
3
n n
the maps are
ϕ(r) = b
z
+ ρ2a
z
ψ(a) = n
z
ψ(b) = ρ2n
z
.
A version of the pairing theorem shows this gives the triangle mapon HF .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 139: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/139.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So. . .
The map in the surgery sequence is induced by a 2-handleattachment W .
So, this map has a universal definition as a map between CFDof solid tori.
More generally, the map for attaching handles along a link is
given by a concrete map between CFD of handlebodies.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 140: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/140.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So. . .
The map in the surgery sequence is induced by a 2-handleattachment W .
So, this map has a universal definition as a map between CFDof solid tori.
More generally, the map for attaching handles along a link is
given by a concrete map between CFD of handlebodies.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
![Page 141: Bordered Heegaard Floer homology - Columbia UniversityAll variants of HF for knots in S3 (Manolescu-Ozsv ath-Sarkar). HFc(Y3) is computable in general (Sarkar-Wang). So is CF (Y)=U2CF](https://reader030.fdocuments.in/reader030/viewer/2022040223/5e5ac6be77c4f779cb5dd226/html5/thumbnails/141.jpg)
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So. . .
The map in the surgery sequence is induced by a 2-handleattachment W .
So, this map has a universal definition as a map between CFDof solid tori.
More generally, the map for attaching handles along a link is
given by a concrete map between CFD of handlebodies.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology