R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
Transcript of R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
1/141
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
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
2/141
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 CFD7 The module CFA8 The pairing theorem
9 Four-dimensional information from bordered HF.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
3/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Classical Heegaard Floer theory assigns...
To Y3 closed, oriented chain complexes CF(Y), C F+(Y), . . .well-defined up to homotopy equivalence.
To W4 : Y3
1 Y3
2chain maps FW : CF(Y1) CF(Y2),...smooth, oriented well-defined up to chain homotopy.
Such that. . .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
4/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Classical Heegaard Floer theory assigns...
To Y3 closed, oriented chain complexes CF(Y), C F+(Y), . . .well-defined up to homotopy equivalence.
To W4 : Y3
1 Y3
2chain 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 FW1Y2W2 = FW2 FW1 ,. . .
(Im omitting spinc-structures)
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
B k d P i B d d di h l b C li d i l HF CFD CFA P i i 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
5/141
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)
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
B k d P ti B d d di Th l b C li d i l HF CFD CFA P i i 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
6/141
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)
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Backgro nd Properties Bordered diagrams The algebra C lindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
7/141
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,. . . )
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
8/141
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)
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
9/141
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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
10/141
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
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
11/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
12/141
g p g g y g
And yet...
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
13/141
g g g y g
And yet...
Heegaard Floer homology remains poorly understood:
The only known definition involves (nonlinear) partialdifferential equations.
Much of it is not yet algorithmically computable.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
14/141
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 ofHF for knots in S3 (Manolescu-Ozsvath-Sarkar).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
15/141
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 ofHF for knots in S3 (Manolescu-Ozsvath-Sarkar).HF(Y3) is computable in general (Sarkar-Wang). So isC F(Y)/U2C F(Y) (Ozsvath-Stipsicz-Szabo).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
16/141
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 ofHF for knots in S3 (Manolescu-Ozsvath-Sarkar).HF(Y3) 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).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
17/141
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 ofHF for knots in S3 (Manolescu-Ozsvath-Sarkar).HF(Y3) 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 thats it.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
18/141
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 ofHF for knots in S3 (Manolescu-Ozsvath-Sarkar).HF(Y3) 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 thats it.The algorithms for HF and FW are inefficient and seem adhoc.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
19/141
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 ofHF for knots in S3 (Manolescu-Ozsvath-Sarkar).
HF(Y3) 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 thats it.
The algorithms for HF and FW are inefficient and seem adhoc.
Its 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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
20/141
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 Floer
homology: Invariance and pairing, arXiv:0810.0687. (Itsquite long.)
We also wrote an expository paper about some of the ideas,Slicing planar grid diagrams: a gentle introduction to
bordered Heegaard Floer homology,arXiv:0810.0695
,which we hope is easy to read.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
21/141
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
22/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
23/141
Roughly, bordered HF assigns...
To a surface F, a (dg) algebra A(F).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
24/141
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)
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
25/141
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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
26/141
Precisely, bordered HF assigns...
To which is aMarked a connected, closed, A differential graded
surface 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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
27/141
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 Y3, a compact, orientedY3 = F 3-manifold with
connected boundary,orientation-preservinghomeomorphism F Y
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
28/141
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 Y3, compact, oriented Right A-module
Y3 = F 3-manifold with
CFA(Y) over A(F),
connected boundary, Left dg-moduleorientation-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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
29/141
Satisfying the pairing theorem:
Theorem
IfY1 = F = Y2 thenCF(Y1 Y2) CFA(Y1)A(F)CFD(Y2).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
30/141
Further structure (in progress):
To an MCG(F), bimodules CFDA(), CFDA().
CFA((Y)) CFA(Y) A(F)CFDA()CFD((Y)) CFDA() A(F) CFD(Y)
(inducing an action of MCG0(F) on Db(A(F)-Mod)).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
31/141
Further structure (in progress):
To an MCG(F), bimodules CFDA(), CFDA().
CFA((Y)) CFA(Y) A(F)CFDA()CFD((Y)) CFDA() 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) CFDDCFA(Y) CFAA A(F) CFD(Y).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
32/141
Advertising bordered HF
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
33/141
Advertising bordered HF
Its not tautologous.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Ad i i b d d HF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
34/141
Advertising bordered HF
Its 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
(KC).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Ad i i b d d HF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
35/141
Advertising bordered HF
Its 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
(KC).
Its good for computations:
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Ad i i b d d HF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
36/141
Advertising bordered HF
Its 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
(KC).
Its good for computations:
CFDA() for generators of MCG0 can be computedexplicitly.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Ad ti i b d d HF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
37/141
Advertising bordered HF
Its 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
(KC).
Its good for computations:
CFDA() for generators of MCG0 can be computedexplicitly.
This leads to computations of CF(Y) for any Y, by factoring.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Ad ti i b d d HF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
38/141
Advertising bordered HF
Its 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
(KC).
Its good for computations:
CFDA() for generators 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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
39/141
Bordered Heegaard diagrams
Let (g, c1, . . . ,
cgk, 1, . . . , g) be a Heegaard diagram for
a Y3 with bdy.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
40/141
Bordered Heegaard diagrams
Let (g, c1, . . . ,
cgk, 1, . . . , g) be a Heegaard diagram for
a Y3 with bdy.Let be result of surgering along c1, . . . ,
cgk.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
41/141
Bordered Heegaard diagrams
Let (g, c1, . . . ,
cgk, 1, . . . , g) be a Heegaard diagram for
a Y3 with bdy.Let be result of surgering along c1, . . . ,
cgk.
Let a1, . . . , a2k be circles in
\ (new disks intersecting inone point p, giving a basis for 1(
).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Bordered Heegaard diagrams
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
42/141
Bordered Heegaard diagrams
Let (g, c1, . . . ,
cgk, 1, . . . , g) be a Heegaard diagram for
a Y3 with bdy.Let be result of surgering along c1, . . . ,
cgk.
Let a1, . . . , a2k be circles in
\ (new disks intersecting inone point p, giving a basis for 1(
).These give circles a1, . . . ,
a2k in .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
43/141
Let = \ D(p).
, c1, . . . , cgk,
a1, . . . ,
a2k, 1, . . . , g) is a bordered
Heegaard diagram for Y .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
44/141
Let = \ D(p).
, c1, . . . , cgk,
a1, . . . ,
a2k, 1, . . . , g) is a bordered
Heegaard diagram for Y .Fix also z near p.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A small circle near p looks like:
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
45/141
A small circle near p looks like:
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
A small circle near p looks like:
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
46/141
pThis is called a pointed matched circle Z.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
47/141
A small circle near p looks like:This is called a pointed matched circle Z.This corresponds to a handle decomposition ofY.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
48/141
A small circle near p looks like:This is called a pointed matched circle Z.This corresponds to a handle decomposition ofY.We will associate a dg algebra A(Z) to Z.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Where the algebra comes from.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
49/141
g
Decomposing ordinary (,,) into bordered H.D.s
(1,1,1) (2,2,2), would want to considerholomorphic curves crossing 1 = 2.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Where the algebra comes from.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
50/141
g
Decomposing ordinary (,,) into bordered H.D.s
(1,1,1) (2,2,2), would want to considerholomorphic curves crossing 1 = 2.This suggests the algebra should have to do with Reeb chordsin 1 relative to 1.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Where the algebra comes from.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
51/141
g
Decomposing ordinary (,,) into bordered H.D.s
(1,1,1) (2,2,2), would want to considerholomorphic curves crossing 1 = 2.This suggests the algebra should have to do with Reeb chordsin 1 relative to 1.Analyzing some simple models, in terms of planar grid
diagrams, suggested the product and relations in the algebra.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So...
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
52/141
Let Z be a pointed matched circle, for a genus k surface.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So...
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
53/141
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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
54/141
A pair (I, ), where is a Reeb chord in Z \ z starting at I
specifies an algebra element a(I, ).We draw them like this:
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
More generally, given (I,) where = {1, . . . , } is a set ofR b h d i I i h
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
55/141
Reeb chords starting at I, with:
i = j implies i and j start and end on different pairs.
{starting points of is} I.specifies an algebra element a(I,).
These generate A(Z) over F2.R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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,
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
56/141
upward-veering flattened braids on 4k positions where:
no two start or end on the same pair
Algebra elements are fixed by horizontal line swapping.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Multiplication...
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
57/141
...is concatenation if sensible, and zero otherwise.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
58/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Multiplication...
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
59/141
...is concatenation if sensible, and zero otherwise.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Double crossings
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
60/141
We impose the relation
(double crossing) = 0.
e.g.,
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The differential
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
61/141
There is a differential d by
d(a) = smooth one crossing of a.e.g.,
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why?
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
62/141
Where do all of these relations (and differential) come from?
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why?
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
63/141
Where do all of these relations (and differential) come from?
Studying degenerations of holomorphic curves.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why?
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
64/141
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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Algebra summary
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
65/141
The algebra is generated by the Reeb chords in Z, withcertain relations. e.g.,
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Algebra summary
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
66/141
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.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Algebra summary
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
67/141
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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The cylindrical setting for classical CF:
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
68/141
Fix an ordinary H.D. (g,,, z). (Here, = {1, . . . , g}.)
The chain complex CF is generated over F2 by g-tuples{xi (i) i} . ( Sg is a permutation.)(cf. T T Sym
g().)
Generators: {u, x}, {v, x}.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The cylindrical setting for classical CF:
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
69/141
Fix an ordinary H.D. (g,,, z). (Here, = {1, . . . , g}.)
The chain complex CF is generated over F2 by g-tuples{xi (i) i} . ( Sg is a permutation.)
The differential counts embedded holomorphic maps
(S, S) ( [0, 1] R, ( 1 R) ( 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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example ofCF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
70/141
Generators: {u, x}, {v, x}.
{u, x} = {v, x} + {v, x} = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example ofCF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
71/141
Generators: {u, x}, {v, x}.
{u, x} = {v, x}+ {v, x} = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example ofCF
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
72/141
Generators: {u, x}, {v, x}.
{u, x} = {v, x}{v, x} = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
73/141
For (,,, z) a bordered Heegaard diagram, view as a
cylindrical end, p.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
74/141
For (,,, z) a bordered Heegaard diagram, view as a
cylindrical end, p.Maps
u: (S, S) ( [0, 1] R, ( 1 R) ( 0 R))
have asymptotics at +, and the puncture p, i.e., east.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
75/141
For (,,, z) a bordered Heegaard diagram, view as a
cylindrical end, p.Maps
u: (S, S) ( [0, 1] R, ( 1 R) ( 0 R))
have asymptotics at +, and the puncture p, i.e., east.
The e asymptotics are Reeb chordsi (1, ti).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
76/141
For (,,, z) a bordered Heegaard diagram, view as a
cylindrical end, p.Maps
u: (S, S) ( [0, 1] R, ( 1 R) ( 0 R))
have asymptotics at +, and the puncture p, i.e., east.
The e asymptotics are Reeb chordsi (1, ti).
The asymptotics i1 , . . . , i of u inherit a partial order, by
R-coordinate.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators ofCFD...
Fi b d d H d di ( )
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
77/141
Fix a bordered Heegaard diagram (g,,, z)
CFD() is generated by g-tuples x = {xi} with:one xi on each -circle
one xi on each -circle
no two xi on the same -arc.
x
x
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators ofCFD...
Fi b d d H d di ( )
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
78/141
Fix a bordered Heegaard diagram (g,,, z)
CFD() is generated by g-tuples x = {xi} with:one xi on each -circle
one xi on each -circle
no two xi on the same -arc.
y
y
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and associated idempotents.
T i t th id t t I ( ) th t i d
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
79/141
To x, associate the idempotent I(x), the -arcs not occupiedby x.
x
As a left A-module,
CFD = xAI(x).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and associated idempotents.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
80/141
To x, associate the idempotent I(x), the -arcs not occupiedby x.
As a left A-module,
CFD = xAI(x).
So, if I is a primitive idempotent, Ix = 0 ifI = I(x) and
I(x)x = x.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The differential on CFD.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
81/141
d(x) =
y
(1,...,n)
(#M(x, y; 1, . . . , n)) a(1, I(x)) a(n, In)y.
where M(x, y; 1, . . . , n) consists of holomorphic curvesasymptotic to
x at
y at +
1, . . . , n at e.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
82/141
d(b) = a + 3xd(x) = 2a
d(a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
83/141
d(b) = a + 3xd(x) = 2a
d(a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
84/141
d(b) = a + 3xd(x) = 2a
d(a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D1: a solid torus.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
85/141
d(b) = a + 3xd(x) = 2a
d(a) = 0.
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D2: same torus, different diagram.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
86/141
d(x) = 23x = 23x.
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example D2: same torus, different diagram.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
87/141
d(x) = 23x = 23x.
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
88/141
First chain complex:
a3
1
????????
x2
// b
Second chain complex:
x 23 // x
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
89/141
First chain complex:
a3
1
????????
x2
// b
Second chain complex:
x 23 // x
Theyre homotopy equivalent. In fact:
Theorem
If (,,, z) and (,
,
, z
) are pointed bordered Heegaarddiagrams for the same bordered Y3 then CFD() is homotopy
equivalent toCFD().
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators and idempotents ofCFA.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
90/141
Fix a bordered Heegaard diagram (g,,, z)CFA() is generated by the same set as CFD: g-tuples x = {xi}with:
one xi on each -circle
one xi on each -circleno two xi on the same -arc.
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
91/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Generators and idempotents ofCFA.
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
92/141
Fix a bordered Heegaard diagram (g,,, z)CFA() is generated by the same set as CFD: g-tuples x = {xi}with:
one xi on each -circle
one xi on each -circleno two xi on the same -arc.
Over F2,
CFA = xF2.
This is much smaller than CFD.
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The differential on CFA...
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
93/141
...counts only holomorphic curves contained in a compact subset of
, i.e., with no asymptotics at e.
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The module structure on CFATo x, associate the idempotent J(x), the -arcs occupied by
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
94/141
p ( ) p y
x (opposite from
CFD).
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The module structure on CFATo x, associate the idempotent J(x), the -arcs occupied by
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
95/141
( )
x (opposite from
CFD).For I a primitive idempotent, define
xI =
x if I = J(x)0 if I = J(x)
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The module structure on CFATo x, associate the idempotent J(x), the -arcs occupied by
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
96/141
x (opposite from
CFD).For I a primitive idempotent, define
xI =
x if I = J(x)0 if I = J(x)
Given a set of Reeb chords, define
x a(J(x),) =
y
(#M(x, y;)) y
where M(x, y;) consists of holomorphic curves asymptotictox at .y at +. at e, all at the same height.
R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology 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, with
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
97/141
generators {r, x}, {s, x}, {r, y}, {s, y}.
R Li shit P O s ath and D Th rston Bordered Heegaard Floer homolog 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, with
{ } { } { } { }
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
98/141
generators {r, x}, {s, x}, {r, y}, {s, y}.The nonzero products are: {r, x}1 = {s, x},{r, y}1 = {s, y}, {r, x}3 = {r, y}, {s, x}3 = {s, y},{r, x}(13) = {s, y}.
R Li hit P O th d D Th t B d d H d Fl h l 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, with
{ } { } { } { }
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
99/141
generators {r, x}, {s, x}, {r, y}, {s, y}.The nonzero products are: {r, x}1 = {s, x},{r, y}1 = {s, y}, {r, x}3 = {r, y}, {s, x}3 = {s, y},{r, x}(13) = {s, y}.Example: {r, x}1 = {s, x} comes from this domain.
R Li hit P O th d D Th t B d d H d Fl h l 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, with
{ } { } { } { }
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
100/141
generators {r, x}, {s, x}, {r, y}, {s, y}.The nonzero products are: {r, x}1 = {s, x},{r, y}1 = {s, y}, {r, x}3 = {r, y}, {s, x}3 = {s, y},{r, x}(13) = {s, y}.Example: {r, x}3 = {r, y} comes from this domain.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
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, with
t { } { } { } { }
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
101/141
generators {r, x}, {s, x}, {r, y}, {s, y}.The nonzero products are: {r, x}1 = {s, x},{r, y}1 = {s, y}, {r, x}3 = {r, y}, {s, x}3 = {s, y},{r, x}(13) = {s, y}.Example: {r, x}(13) = {s, y} comes from this domain.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
102/141
d(u) = v
u2 = tu23 = v
t3 = v.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
103/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
104/141
d(u) = v
u2 = ta23 = v
t3 = v.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
105/141
d(u) = v
u2 = ta23 = v
t3 = v.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A1: a solid torus.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
106/141
d(u) = v
u2 = ta23 = v
t3 = v .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Why associativity should hold...
(x i) j counts curves with i and j infinitely far apart.
x ( ) co nts c r es ith and at the same height
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
107/141
x (i j) counts curves with i and j at the same height.
These are ends of a 1-dimensional moduli space, with heightbetween i and j varying.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The local model again.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
108/141
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and why it doesnt.
But this moduli space might have other ends: broken flowswith 1 and 2 at a fixed nonzero height.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
109/141
t 1 a d 2 at a ed o e o e g t
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
...and why it doesnt.
But this moduli space might have other ends: broken flowswith 1 and 2 at a fixed nonzero height.
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
110/141
1 2 g
These moduli spaces M(x, y; (1, 2)) measure failure ofassociativity. So...
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Higher A-operations
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
111/141
Define
mn+1(x, a(1), . . . , a(n)) =
y
(#M(x, y; (1, . . . ,n))) y
where M(x, y; (1
, . . . ,n)) consists of holomorphic curvesasymptotic to
x at .
y at +.
1 all at one height at e, 2 at some other (higher) heightat e, and so on.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
112/141
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Example A2: same torus, different diagram.
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
113/141
m3(x, 3, 2) = x
m4(x, 3, 23, 2) = xm5(x, 3, 23, 23, 2) = x
...
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
u@
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
114/141
m2(,2)
1+23
@@@@@@@@@@@@@@@@
xm2(,3)
// v
Second chain complex:
xm3(,3,2)+m4(,3,23,2)+...
// x
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
u@
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
115/141
m2(,2)
1+23
@@@@@@@@@@@@@@@
@
xm2(,3)
// v
Second chain complex:
xm3(,3,2)+m4(,3,23,2)+...
// x
Theyre A homotopy equivalent (exercise).
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Comparison of the two examples.
First chain complex:
u@
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
116/141
m2(,2)
1+23
@@@@@@@@@@@@@@@
@
xm2(,3)
// v
Second chain complex:
xm3(,3,2)+m4(,3,23,2)+...
// x
Theyre A homotopy equivalent (exercise).
Suggestive remark:(1 + 23)
1=1 + 23 + 23, 23 + . . .
3(1 + 23)12=3, 2 + 3, 23, 2 + . . . .
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
117/141
In general:
Theorem
If (,,, z) and (,, , z) are pointed bordered Heegaard
diagrams for the same bordered Y3 then CFA() is A-homotopyequivalent to CFA().
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The pairing theorem
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
118/141
Recall:
Theorem
IfY1 = F = Y2 then
CF(Y1 Y2) CFA(Y1)A(F)CFD(Y2).Well illustrate this with three examples.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
119/141
u
m2(,2)
1+m2(,23)
@@@@@@@
xm2(,3)
// v
a
3
1
???????
x2
// 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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(,2)1+m2(,23)@
@@@ a
1???
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
120/141
( )
@@@
xm2(,3)
// v
3
????
x2
//b
Generators of CFA(Y1) CFD(Y2): u x, v x, t a, t b.d(t b) = t a + t 3x = t a + t3 x = t a + v x
d(u x) = v x + u 2a = v x + u2 a = v x + t a
d(v x) = v 2a = v2 a = 0
d(t a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(,2)1+m2(,23)@
@@@ a
1???
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
121/141
@@@
xm2(,3)
// v
3
????
x2
//b
Generators of CFA(Y1) CFD(Y2): u x, v x, t a, t b.d(t b) = t a + t 3x = t a + t3 x = t a + v x
d(u x) = v x + u 2a = v x + u2 a = v x + t a
d(v x) = v 2a = v2 a = 0
d(t a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(,2)1+m2(,23)@
@@@ a
1????
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
122/141
@@@
xm2(,3)
// v
3
????
x2
//b
Generators of CFA(Y1) CFD(Y2): u x, v x, t a, t b.d(t b) = t a + t 3x = t a + t3 x = t a + v x
d(u x) = v x + u 2a = v x + u2 a = v x + t a
d(v x) = v 2a = v2 a = 0
d(t a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(,2)1+m2(,23)
@@
@@@@ a
1?
????
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
123/141
@@@
xm2(,3)
// v
3
???
x2
//b
Generators of CFA(Y1) CFD(Y2): u x, v x, t a, t b.d(t b) = t a + t 3x = t a + t3 x = t a + v x
d(u x) = v x + u 2a = v x + u2 a = v x + t a
d(v x) = v 2a = v2 a = 0
d(t a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(,2)1+m2(,23)
@@
@@@@ a
3
1?????
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
124/141
@@@
xm2(,3)
// v
3
???
x2
//b
Generators of CFA(Y1) CFD(Y2): u x, v x, t a, t b.d(t b) = t a + t 3x = t a + t3 x = t a + v x
d(u x) = v x + u 2a = v x + u2 a = v x + t a
d(v x) = v 2a = v2 a = 0
d(t a) = 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
u
m2(,2)
1+m2(,23)
@@
@@@@@ a
31
??
????
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
125/141
@@
xm2(,3)
// v
???
x2
// b
Generators of CFA(Y1) CFD(Y2): u x, v x, t a, t b.d(t b) = t a + t 3x = t a + t3 x = t a + v x
d(u x) = v x + u 2a = v x + u2 a = v x + t a
d(v x) = v 2a = v2 a = 0d(t a) = 0.
This simplifies to F2t a + u x F2t b = v x.R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
126/141
u
m2(,2)
1+m2(,23)
@@@@@@@
xm2(,3)
// v
x23
// x
Generators of
CFA(Y1) CFD(Y2): u x, v x.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
127/141
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
23x = v x + u
23 x = v x + v x = 0.
-
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
128/141
23 23d(v x) = v 23x = v23 x = 0.
The most interesting part is the interaction:
d
m2
u
v
x
x
23
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
a
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
129/141
xm3(,3,2)+...
// x
a
3
1
???????
x2
//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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
a
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
130/141
xm3(,3,2)+...
// x
a
3
1
???????
x2
//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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
a
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
131/141
xm3(,3,2)+...
// x
a
3
1
???????
x2
//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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(,3,2)+...
// x
a
3
1
???????
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
132/141
?
x2
// 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
3
2
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(,3,2)+...
// x
a
3
1
???????
2
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
133/141
?
x2
// 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
3
2
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
xm3(,3,2)+...
// x
a
3
1
??
?????
2
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
134/141
x
2// 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
3
2
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The surgery exact sequence
Theorem
(Ozsvath-Szab o) For K a knot in Y there is an exact sequence
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
135/141
HF(Y(K)) HF(Y1(K)) HF(Y0(K)) HF(Y(K))
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
The surgery exact sequence
Theorem
(Ozsvath-Szab o) For K a knot in Y there is an exact sequence
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
136/141
HF(Y(K)) HF(Y1(K)) HF(Y0(K)) HF(Y(K))
Proof via bordered Floer.
Define
H :
0z
12
3
r
r
H1 :
0z
12
3
a
a
b b H0 :
0z
12
3
n n
Theres a s.e.s.0 CFD(H) CFD(H1) CFD(H0) 0.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Is it the same sequence?
For
12
r
12
a
12
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
137/141
H :
0z
12
3
r
H1 :
0z
12
3
a
b b H0 :
0z
12
3
n n
the maps are
(r) = b
z
+ 2a
z
(a) = n
z
(b) = 2n
z
.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
Is it the same sequence?
For
12
r
12
a
12
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
138/141
H :
0z
12
3
r
H1 :
0z
12
3
a
b b H0 :
0z
12
3
n n
the maps are
(r) = b
z
+ 2a
z
(a) = n
z
(b) = 2n
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
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So...
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
139/141
The map in the surgery sequence is induced by a 2-handleattachment W.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So...
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
140/141
The map in the surgery sequence is induced by a 2-handleattachment W.
So, this map has a universal definition as a map between CFD
of solid tori.
R. Lipshitz, P. Ozsvath and D. Thurston Bordered Heegaard Floer homology
Background Properties Bordered diagrams The algebra Cylindrical HF CFD CFA Pairing 4D
So...
http://find/http://goback/ -
8/3/2019 R. Lipshitz, P. Ozsvath and D. Thurston- Bordered Heegaard Floer homology
141/141
The map in the surgery sequence is induced by a 2-handleattachment W.
So, this map has a universal definition as a map between CFD
of 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
http://find/http://goback/