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

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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/


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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.



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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. . .



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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)


B k d P i B d d di h l b C li d i l HF CFD CFA P i i 4D


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Advertising HF. . .

HF contains lots of geometric content:

Detects smooth structures on 4-manifolds. (Ozsvath-Szabo)


B k d P ti B d d di Th l b C li d i l HF CFD CFA P i i 4D


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Advertising HF. . .



Detects the genus of knots / Thurston norm of 3-manifolds.

(Ozsvath-Szabo, Ni)


Backgro nd Properties Bordered diagrams The algebra C lindrical HF CFD CFA Pairing 4D


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Advertising HF. . .




(Ozsvath-Szabo, Ni)Detects fiberedness of knots / three-manifolds(Ozsvath-Szabo-Ghiggini-Ni, Juhasz,. . . )




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Advertising HF. . .





Obstructs overtwistedness / Stein fillability of contactstructures (Ozsvath-Szabo)




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Advertising HF. . .






Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )




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Advertising HF. . .






Bounds the slice genus / minimal genus representatives ofhomology classes in 4-manifolds (Ozsvath-Szabo, . . . )



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g p g g y g

And yet...

Heegaard Floer homology remains poorly understood:

The only known definition involves (nonlinear) partialdifferential equations.




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g g g y g

And yet...



Much of it is not yet algorithmically computable.




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And yet...




All variants ofHF for knots in S3 (Manolescu-Ozsvath-Sarkar).




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And yet...




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).




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And yet...




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).




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And yet...





But thats it.




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And yet...





But thats it.The algorithms for HF and FW are inefficient and seem adhoc.




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And yet...




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.




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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.



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Roughly, bordered HF assigns...

To a surface F, a (dg) algebra A(F).




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To a 3-manifold Y with boundary F, a

right A(F)-module

CFA(Y)

left A(F)-module CFD(Y)




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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).




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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




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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




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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.




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Satisfying the pairing theorem:

Theorem

IfY1 = F = Y2 thenCF(Y1 Y2) CFA(Y1)A(F)CFD(Y2).




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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)).




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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).




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Advertising bordered HF




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Its not tautologous.



Ad i i b d d HF


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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).



Ad i i b d d HF


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Theorem


C


(KC).

Its good for computations:



Ad i i b d d HF


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Theorem


C


(KC).


CFDA() for generators of MCG0 can be computedexplicitly.



Ad ti i b d d HF


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Its not tautologous.It provides information about classical HF. For instance:

Theorem


C


(KC).



This leads to computations of CF(Y) for any Y, by factoring.



Ad ti i b d d HF


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Its not tautologous.It provides information about classical HF. For instance:

Theorem


C


(KC).



This leads to computations of CF(Y) for any Y, by factoring.In fact, you can compute FW for any W

4.



Bordered Heegaard diagrams


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Let (g, c1, . . . ,

cgk, 1, . . . , g) be a Heegaard diagram for

a Y3 with bdy.





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Let (g, c1, . . . ,


a Y3 with bdy.Let be result of surgering along c1, . . . ,

cgk.





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Let (g, c1, . . . ,



cgk.

Let a1, . . . , a2k be circles in

\ (new disks intersecting inone point p, giving a basis for 1(

).





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Let (g, 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 .




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Let = \ D(p).

, c1, . . . , cgk,

a1, . . . ,

a2k, 1, . . . , g) is a bordered

Heegaard diagram for Y .




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Let = \ D(p).

, c1, . . . , cgk,

a1, . . . ,

a2k, 1, . . . , g) is a bordered

Heegaard diagram for Y .Fix also z near p.



A small circle near p looks like:


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pThis is called a pointed matched circle Z.




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A small circle near p looks like:This is called a pointed matched circle Z.This corresponds to a handle decomposition ofY.




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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.



Where the algebra comes from.


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g

Decomposing ordinary (,,) into bordered H.D.s

(1,1,1) (2,2,2), would want to considerholomorphic curves crossing 1 = 2.





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g


(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.





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g


(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.



So...


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Let Z be a pointed matched circle, for a genus k surface.



So...


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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:




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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:



More generally, given (I,) where = {1, . . . , } is a set ofR b h d i I i h


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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


That is, A(Z) is the subalgebra of the algebra of k-strand,


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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.



Multiplication...


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...is concatenation if sensible, and zero otherwise.



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Multiplication...


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...is concatenation if sensible, and zero otherwise.



Double crossings


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We impose the relation

(double crossing) = 0.

e.g.,



The differential


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There is a differential d by

d(a) = smooth one crossing of a.e.g.,



Why?


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Where do all of these relations (and differential) come from?



Why?


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Studying degenerations of holomorphic curves.



Why?


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Studying degenerations of holomorphic curves.

They can all be deduced from some simple examples.See arXiv:0810.0695.



Algebra summary


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The algebra is generated by the Reeb chords in Z, withcertain relations. e.g.,



Algebra summary


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Multiplying consecutive Reeb chords concatenates them.Far apart Reeb chords commute.



Algebra summary


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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.



The cylindrical setting for classical CF:


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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}.



The cylindrical setting for classical CF:


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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.



Example ofCF


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{u, x} = {v, x} + {v, x} = 0.



Example ofCF


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{u, x} = {v, x}+ {v, x} = 0.



Example ofCF


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{u, x} = {v, x}{v, x} = 0.




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For (,,, z) a bordered Heegaard diagram, view as a

cylindrical end, p.




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cylindrical end, p.Maps

u: (S, S) ( [0, 1] R, ( 1 R) ( 0 R))

have asymptotics at +, and the puncture p, i.e., east.




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u: (S, S) ( [0, 1] R, ( 1 R) ( 0 R))


The e asymptotics are Reeb chordsi (1, ti).




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u: (S, S) ( [0, 1] R, ( 1 R) ( 0 R))


The e asymptotics are Reeb chordsi (1, ti).

The asymptotics i1 , . . . , i of u inherit a partial order, by

R-coordinate.



Generators ofCFD...

Fi b d d H d di ( )


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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



Generators ofCFD...

Fi b d d H d di ( )


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Fix a bordered Heegaard diagram (g,,, z)

CFD() is generated by g-tuples x = {xi} with:one xi on each -circle


no two xi on the same -arc.

y

y



...and associated idempotents.

T i t th id t t I ( ) th t i d


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To x, associate the idempotent I(x), the -arcs not occupiedby x.

x

As a left A-module,

CFD = xAI(x).



...and associated idempotents.


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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.



The differential on CFD.


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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.


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d(b) = a + 3xd(x) = 2a

d(a) = 0.




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d(b) = a + 3xd(x) = 2a

d(a) = 0.




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d(b) = a + 3xd(x) = 2a

d(a) = 0.




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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.


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d(x) = 23x = 23x.


Example D2: same torus, different diagram.


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d(x) = 23x = 23x.


Comparison of the two examples.

First chain complex:


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a3

1

????????

x2

// b

Second chain complex:

x 23 // x





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a3

1

????????

x2

// b


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().


Generators and idempotents ofCFA.


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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 -circleno two xi on the same -arc.

R Lipshitz P Ozsvath and D Thurston Bordered Heegaard Floer homology


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Generators and idempotents ofCFA.


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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 -circleno two xi on the same -arc.

Over F2,

CFA = xF2.

This is much smaller than CFD.


The differential on CFA...


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...counts only holomorphic curves contained in a compact subset of

, i.e., with no asymptotics at e.


The module structure on CFATo x, associate the idempotent J(x), the -arcs occupied by


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p ( ) p y

x (opposite from

CFD).




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( )

x (opposite from

CFD).For I a primitive idempotent, define

xI =

x if I = J(x)0 if I = J(x)




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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.


A local example of the module structure on CFA.Consider the following piece of a Heegaard diagram, with


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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


{ } { } { } { }


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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


{ } { } { } { }


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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


{ } { } { } { }


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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.




t { } { } { } { }


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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.



Example A1: a solid torus.


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d(u) = v

u2 = tu23 = v

t3 = v.



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d(u) = v

u2 = ta23 = v

t3 = v.





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d(u) = v

u2 = ta23 = v

t3 = v.





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d(u) = v

u2 = ta23 = v

t3 = v .



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


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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.



The local model again.


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...and why it doesnt.

But this moduli space might have other ends: broken flowswith 1 and 2 at a fixed nonzero height.


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t 1 a d 2 at a ed o e o e g t



...and why it doesnt.

But this moduli space might have other ends: broken flowswith 1 and 2 at a fixed nonzero height.


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1 2 g

These moduli spaces M(x, y; (1, 2)) measure failure ofassociativity. So...



Higher A-operations


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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.



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Example A2: same torus, different diagram.


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m3(x, 3, 2) = x

m4(x, 3, 23, 2) = xm5(x, 3, 23, 23, 2) = x

...





u@


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m2(,2)

1+23

@@@@@@@@@@@@@@@@

xm2(,3)

// v


xm3(,3,2)+m4(,3,23,2)+...

// x





u@


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m2(,2)

1+23

@@@@@@@@@@@@@@@

@

xm2(,3)

// v


xm3(,3,2)+m4(,3,23,2)+...

// x

Theyre A homotopy equivalent (exercise).





u@


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m2(,2)

1+23

@@@@@@@@@@@@@@@

@

xm2(,3)

// v


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 + . . . .




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In general:

Theorem

If (,,, z) and (,, , z) are pointed bordered Heegaard

diagrams for the same bordered Y3 then CFA() is A-homotopyequivalent to CFA().



The pairing theorem


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Recall:

Theorem

IfY1 = F = Y2 then

CF(Y1 Y2) CFA(Y1)A(F)CFD(Y2).Well illustrate this with three examples.




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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.



u

m2(,2)1+m2(,23)@

@@@ a

1???


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( )

@@@

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.



u

m2(,2)1+m2(,23)@

@@@ a

1???


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@@@

xm2(,3)

// v

3

????

x2

//b



d(v x) = v 2a = v2 a = 0

d(t a) = 0.



u

m2(,2)1+m2(,23)@

@@@ a

1????


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@@@

xm2(,3)

// v

3

????

x2

//b



d(v x) = v 2a = v2 a = 0

d(t a) = 0.



u

m2(,2)1+m2(,23)

@@

@@@@ a

1?

????


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@@@

xm2(,3)

// v

3

???

x2

//b



d(v x) = v 2a = v2 a = 0

d(t a) = 0.



u

m2(,2)1+m2(,23)

@@

@@@@ a

3

1?????


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@@@

xm2(,3)

// v

3

???

x2

//b



d(v x) = v 2a = v2 a = 0

d(t a) = 0.



u

m2(,2)

1+m2(,23)

@@

@@@@@ a

31

??

????


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@@

xm2(,3)

// v

???

x2

// b



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



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u

m2(,2)

1+m2(,23)

@@@@@@@

xm2(,3)

// v

x23

// x

Generators of

CFA(Y1) CFD(Y2): u x, v x.



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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.


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23 23d(v x) = v 23x = v23 x = 0.

The most interesting part is the interaction:

d

m2

u

v

x

x

23



a


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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.



a


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.



a


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.



xm3(,3,2)+...

// x

a

3

1

???????


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



xm3(,3,2)+...

// x

a

3

1

???????

2


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



xm3(,3,2)+...

// x

a

3

1

??

?????

2


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



The surgery exact sequence

Theorem

(Ozsvath-Szab o) For K a knot in Y there is an exact sequence


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HF(Y(K)) HF(Y1(K)) HF(Y0(K)) HF(Y(K))



The surgery exact sequence

Theorem

(Ozsvath-Szab o) For K a knot in Y there is an exact sequence


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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.



Is it the same sequence?

For

12

r

12

a

12


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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

.



Is it the same sequence?

For

12

r

12

a

12


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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.



So...


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The map in the surgery sequence is induced by a 2-handleattachment W.



So...


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So, this map has a universal definition as a map between CFD

of solid tori.



So...


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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

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