Towards bordered Heegaard Floer homology R. Lipshitz, P. Ozsv ath - - PowerPoint PPT Presentation

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Towards bordered Heegaard Floer homology R. Lipshitz, P. Ozsv ath and D. Thurston June 10, 2008 R. Lipshitz, P. Ozsv ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 1 / 50 1 Review of Heegaard Floer 2 Basic

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SLIDE 1

Towards bordered Heegaard Floer homology

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston June 10, 2008

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 1 / 50

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SLIDE 2

1 Review of Heegaard Floer 2 Basic properties of bordered HF 3 Bordered Heegaard diagrams 4 The algebra 5 Gradings 6 The cylindrical setting for Heegaard Floer 7 The module

CFD

8 The module

CFA

9 The pairing theorem 10 Knot complements

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 2 / 50

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Review of Heegaard Floer

It’s like HM or ECH with different names.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 3 / 50

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SLIDE 4

Review of Heegaard Floer

It’s like HM or ECH with different names. We’ll focus on HF = H∗( CF), the mapping cone of U : CF+ → CF+.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 3 / 50

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Review of Heegaard Floer

It’s like HM or ECH with different names. We’ll focus on HF = H∗( CF), the mapping cone of U : CF+ → CF+. Conjecturally, HF+ =

  • HM = ECH∗.
  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 3 / 50

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SLIDE 6

Roughly, bordered HF assigns...

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

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 4 / 50

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SLIDE 7

Roughly, bordered HF assigns...

To a surface F, a (dg) algebra A(F). To a 3-manifold Y with boundary F, a

right A-module CFA(Y ) left A-module CFD(Y )

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 4 / 50

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Roughly, bordered HF assigns...

To a surface F, a (dg) algebra A(F). To a 3-manifold Y with boundary F, a

right A-module CFA(Y ) left A-module CFD(Y )

such that If Y = Y1 ∪F Y2 then

  • CF(Y ) =

CFA(Y1) ⊗A(F) CFD(Y2).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 4 / 50

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Precisely, bordered HF assigns...

To which is a Marked a connected, closed, A differential graded surface

  • riented surface,

algebra A(F) F + a handle decompos. of F + a small disk in F

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 5 / 50

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Precisely, bordered HF assigns...

To which is a Marked a connected, closed, A differential graded surface

  • riented surface,

algebra A(F) F + a handle decompos. of F + a small disk in F Bordered Y 3, a compact, oriented ∂Y 3 = F 3-manifold with connected boundary,

  • rientation-preserving

homeomorphism F → ∂Y

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 5 / 50

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Precisely, bordered HF assigns...

To which is a Marked a connected, closed, A differential graded surface

  • riented surface,

algebra A(F) F + a handle decompos. of F + a small disk in F Bordered Y 3, compact, oriented Right A∞-module ∂Y 3 = F 3-manifold with

  • CFA(Y ) over A(F),

connected boundary, Left dg-module

  • rientation-preserving
  • CFD(Y ) over A(−F),

homeomorphism F → ∂Y well-defined up to homotopy.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 5 / 50

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Satisfying the pairing theorem:

Theorem

If ∂Y1 = F = −∂Y2 then

  • CF(Y1 ∪∂ Y2) ≃

CFA(Y1) ⊗A(F) CFD(Y2).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 6 / 50

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Further structure (in progress):

To an φ ∈ MCG(F), bimodules CFDA(φ), CFAD(φ).

  • CFA(φ(Y )) ≃

CFA(Y ) ⊗A(F) CFDA(φ)

  • CFD(φ(Y )) ≃

CFAD(φ) ⊗A(−F) CFD(Y ).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 7 / 50

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Further structure (in progress):

To an φ ∈ MCG(F), bimodules CFDA(φ), CFAD(φ).

  • CFA(φ(Y )) ≃

CFA(Y ) ⊗A(F) CFDA(φ)

  • CFD(φ(Y )) ≃

CFAD(φ) ⊗A(−F) CFD(Y ). To F, bimodules CFDD and CFAAa, such that

  • CFD(Y ) ≃

CFA(Y ) ⊗A(F) CFDD

  • CFA(Y ) ≃

CFAA ⊗A(−F) CFD(Y ).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 7 / 50

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Further structure (in progress):

To an φ ∈ MCG(F), bimodules CFDA(φ), CFAD(φ).

  • CFA(φ(Y )) ≃

CFA(Y ) ⊗A(F) CFDA(φ)

  • CFD(φ(Y )) ≃

CFAD(φ) ⊗A(−F) CFD(Y ). To F, bimodules CFDD and CFAAa, such that

  • CFD(Y ) ≃

CFA(Y ) ⊗A(F) CFDD

  • CFA(Y ) ≃

CFAA ⊗A(−F) CFD(Y ).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 7 / 50

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Bordered Heegaard diagrams

Let (Σg, αc

1, . . . , αc g−k, β1, . . . , βg) be a Heegaard diagram for a Y 3

with bdy.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 8 / 50

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Bordered Heegaard diagrams

Let (Σg, αc

1, . . . , αc g−k, β1, . . . , βg) be a Heegaard diagram for a Y 3

with bdy. Let Σ′ be result of surgering along αc

1, . . . , αc g−k.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 8 / 50

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Bordered Heegaard diagrams

Let (Σg, αc

1, . . . , αc g−k, β1, . . . , βg) be a Heegaard diagram for a Y 3

with bdy. Let Σ′ be result of surgering along αc

1, . . . , αc g−k.

Let αa

1, . . . , αa 2k be circles in Σ′ \ (new disks intersecting in one point

p, giving a basis for π1(Σ′).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 8 / 50

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Bordered Heegaard diagrams

Let (Σg, αc

1, . . . , αc g−k, β1, . . . , βg) be a Heegaard diagram for a Y 3

with bdy. Let Σ′ be result of surgering along αc

1, . . . , αc g−k.

Let αa

1, . . . , αa 2k be circles in Σ′ \ (new disks intersecting in one point

p, giving a basis for π1(Σ′). These give circles αa

1, . . . , αa 2k in Σ.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 8 / 50

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Let Σ = Σ \ Dǫ(p). Σ, αc

1, . . . , αc g−k, αa 1, . . . , αa 2k, β1, . . . , βg) is a bordered Heegaard

diagram for Y .

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 9 / 50

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Let Σ = Σ \ Dǫ(p). Σ, αc

1, . . . , αc g−k, αa 1, . . . , αa 2k, β1, . . . , βg) is a bordered Heegaard

diagram for Y . Fix also z ∈ Σ near p.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 9 / 50

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A small circle near p looks like:

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 10 / 50

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A small circle near p looks like: This is called a pointed matched circle Z.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 10 / 50

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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 of ∂Y .

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 10 / 50

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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 of ∂Y . We will associate a dg algebra A(Z) to Z.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 10 / 50

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Where the algebra comes from.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 11 / 50

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Where the algebra comes from.

Decomposing ordinary (Σ, α, β) into bordered H.D.’s (Σ1, α1, β1) ∪ (Σ2, α2, β2), would want to consider holomorphic curves crossing ∂Σ1 = ∂Σ2.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 11 / 50

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Where the algebra comes from.

Decomposing ordinary (Σ, α, β) into bordered H.D.’s (Σ1, α1, β1) ∪ (Σ2, α2, β2), would want to consider holomorphic curves crossing ∂Σ1 = ∂Σ2. This suggests the algebra should have to do with Reeb chords in ∂Σ1 relative to α ∩ ∂Σ1.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 11 / 50

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Where the algebra comes from.

Decomposing ordinary (Σ, α, β) into bordered H.D.’s (Σ1, α1, β1) ∪ (Σ2, α2, β2), would want to consider holomorphic curves crossing ∂Σ1 = ∂Σ2. This suggests the algebra should have to do with Reeb chords in ∂Σ1 relative to α ∩ ∂Σ1. Analyzing some simple models, in terms of planar grid diagrams, suggested the product and relations in the algebra.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 11 / 50

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So...

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

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 12 / 50

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So...

Let Z be a pointed matched circle, for a genus k surface. Primitive idempotents of A(Z) correspond to k-element subsets I of the 2k pairs in Z. We draw them like this:

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 12 / 50

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A pair (I, ρ), where ρ is a Reeb chord in Z \ z starting at I specifies an algebra element a(I, ρ). We draw them like this:

From:

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 13 / 50

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More generally, given (I, ρ) where ρ = {ρ1, . . . , ρℓ} is a set of Reeb chords starting at I, with: i = j implies ρi and ρj start and end on different pairs. {starting points of ρi’s} ⊂ I. specifies an algebra element a(I, ρ).

From:

These generate A(Z) over F2.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 14 / 50

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That is, A(Z) is the subalgebra of the algebra of k-strand, upward-veering flattened braids on 4k positions where: no two start or end on the same pair

Not allowed.

Algebra elements are fixed by “horizontal line swapping”.

= +

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 15 / 50

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Multiplication...

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

=

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 16 / 50

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Multiplication...

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

= =

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 16 / 50

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Multiplication...

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

= = = 0

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 16 / 50

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Double crossings

We impose the relation (double crossing) = 0. e.g.,

= =0

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 17 / 50

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The differential

There is a differential d by d(a) =

  • smooth one crossing of a.

e.g.,

d

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 18 / 50

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Algebra – summary

The algebra is generated by the Reeb chords in Z, with certain

  • relations. e.g.,
  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 19 / 50

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Algebra – summary

The algebra is generated by the Reeb chords in Z, with certain

  • relations. e.g.,

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

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 19 / 50

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SLIDE 42

Algebra – summary

The algebra is generated by the Reeb chords in Z, with certain

  • relations. e.g.,

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

The algebra is finite-dimensional over F2, and has a nice description in terms of flattened braids.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 19 / 50

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Gradings

One can prove there is no Z-grading on A.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 20 / 50

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Gradings

One can prove there is no Z-grading on A. This bothered us.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 20 / 50

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Gradings

One can prove there is no Z-grading on A. This bothered us. Tim Perutz suggested we think about the geometric grading on HM.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 20 / 50

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Gradings

One can prove there is no Z-grading on A. This bothered us. Tim Perutz suggested we think about the geometric grading on HM. It was a good suggestion.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 20 / 50

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HM(Y ) is graded by homotopy classes of nonvanishing vector fields on Y . So A(F) should be graded by homotopy classes of nonvanishing vector fields v on F × [0, 1] such that v|F×∂[0,1] = v0 for some given v0. (Think of F × [0, 1] as a collar of ∂Y .)

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 21 / 50

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HM(Y ) is graded by homotopy classes of nonvanishing vector fields on Y . So A(F) should be graded by homotopy classes of nonvanishing vector fields v on F × [0, 1] such that v|F×∂[0,1] = v0 for some given v0. (Think of F × [0, 1] as a collar of ∂Y .) This is a group G under concatenation in [0, 1].

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 21 / 50

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It is easy to see that G ∼ = [ΣF, S2].

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 22 / 50

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It is easy to see that G ∼ = [ΣF, S2]. It follows that G is a Z-central extension of H1(F), 0 → Z → G → H1(F) → 0.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 22 / 50

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G is not commutative, but has a central element λ.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 23 / 50

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G is not commutative, but has a central element λ. There is a map gr : {gens. of A(F)} → G such that: gr(a · b) = gr(a) · gr(b) gr(d(a)) = λ · gr(a).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 23 / 50

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G is not commutative, but has a central element λ. There is a map gr : {gens. of A(F)} → G such that: gr(a · b) = gr(a) · gr(b) gr(d(a)) = λ · gr(a). The modules CFD and CFA are graded by G-sets.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 23 / 50

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G is not commutative, but has a central element λ. There is a map gr : {gens. of A(F)} → G such that: gr(a · b) = gr(a) · gr(b) gr(d(a)) = λ · gr(a). The modules CFD and CFA are graded by G-sets. Note: in the end, we define these gradings combinatorially, not geometrically.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 23 / 50

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The cylindrical setting for classical CF:

Fix an ordinary H.D. (Σg, α, β, z). (Here, α = {α1, . . . , αg}.) The chain complex CF is generated over F2 by g-tuples {xi ∈ ασ(i) ∩ βi} ⊂ α ∩ β. (σ ∈ Sg is a permutation.) (cf. Tα ∩ Tβ ⊂ Symg(Σ).)

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 24 / 50

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The cylindrical setting for classical CF:

Fix an ordinary H.D. (Σg, α, β, z). (Here, α = {α1, . . . , αg}.) The chain complex CF is generated over F2 by g-tuples {xi ∈ ασ(i) ∩ βi} ⊂ α ∩ β. (σ ∈ Sg is a permutation.) The differential counts embedded holomorphic maps (S, ∂S) → (Σ × [0, 1] × R, (α × 1 × R) ∪ (β × 0 × R)) asymptotic to x × [0, 1] at −∞ and y × [0, 1] at +∞. For CF, curves may not intersect {z} × [0, 1] × R.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 24 / 50

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SLIDE 57

A useless schematic of a curve in Σ × [0, 1] × R.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 25 / 50

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SLIDE 58

For (Σ, α, β, z) a bordered Heegaard diagram, view ∂Σ as a cylindrical end, p.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 26 / 50

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SLIDE 59

For (Σ, α, β, z) a bordered Heegaard diagram, view ∂Σ as a cylindrical end, p. Maps u : (S, ∂S) → (Σ × [0, 1] × R, (α × 1 × R) ∪ (β × 0 × R)) have asymptotics at +∞, −∞ and the puncture p, i.e., east ∞.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 26 / 50

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SLIDE 60

For (Σ, α, β, z) a bordered Heegaard diagram, view ∂Σ as a cylindrical end, p. Maps u : (S, ∂S) → (Σ × [0, 1] × R, (α × 1 × R) ∪ (β × 0 × R)) have asymptotics at +∞, −∞ and the puncture p, i.e., east ∞. The e∞ asymptotics are Reeb chords ρi × (1, ti).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 26 / 50

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SLIDE 61

For (Σ, α, β, z) a bordered Heegaard diagram, view ∂Σ as a cylindrical end, p. Maps u : (S, ∂S) → (Σ × [0, 1] × R, (α × 1 × R) ∪ (β × 0 × R)) have asymptotics at +∞, −∞ and the puncture p, i.e., east ∞. The e∞ asymptotics are Reeb chords ρi × (1, ti). The asymptotics ρi1, . . . , ρiℓ of u inherit a partial order, by R-coordinate.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 26 / 50

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SLIDE 62

Another useless schematic of a curve in Σ × [0, 1] × R.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 27 / 50

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SLIDE 63

Generators of CFD...

Fix a bordered Heegaard diagram (Σg, α, β, z)

  • CFD(Σ) is generated by g-tuples x = {xi} with:
  • ne xi on each β-circle
  • ne xi on each α-circle

no two xi on the same α-arc.

x x

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 28 / 50

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SLIDE 64

Generators of CFD...

Fix a bordered Heegaard diagram (Σg, α, β, z)

  • CFD(Σ) is generated by g-tuples x = {xi} with:
  • ne xi on each β-circle
  • ne xi on each α-circle

no two xi on the same α-arc.

y y

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 28 / 50

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SLIDE 65

...and associated idempotents.

To x, associate the idempotent I(x), the α-arcs not occupied by x.

x

As a left A-module,

  • CFD = ⊕xAI(x).
  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 29 / 50

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SLIDE 66

...and associated idempotents.

To x, associate the idempotent I(x), the α-arcs not occupied by x. As a left A-module,

  • CFD = ⊕xAI(x).

So, if I is a primitive idempotent, Ix = 0 if I = I(x) and I(x)x = x.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 29 / 50

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The differential on CFD.

d(x) =

  • y
  • (ρ1,...,ρn)

(#M(x, y; ρ1, . . . , ρn)) a(ρ1, I(x)) · · · a(ρn, In)y. where M(x, y; ρ1, . . . , ρn) consists of holomorphic curves asymptotic to x at −∞ y at +∞ ρ1, . . . , ρn at e∞.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 30 / 50

slide-68
SLIDE 68

Example D1: a solid torus.

z 1 2 3 x a b

d(a) = b + ρ3x d(x) = ρ2b d(b) = 0.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 31 / 50

slide-69
SLIDE 69

Example D2: same torus, different diagram.

z 1 2 3 x

d(x) = ρ2ρ3x = ρ23x.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 32 / 50

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SLIDE 70

Comparison of the two examples.

First chain complex: a

ρ3

  • 1
  • x

ρ2

b

Second chain complex: x

ρ23 x

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 33 / 50

slide-71
SLIDE 71

Comparison of the two examples.

First chain complex: a

ρ3

  • 1
  • x

ρ2

b

Second chain complex: x

ρ23 x

They’re homotopy equivalent!

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 33 / 50

slide-72
SLIDE 72

Comparison of the two examples.

First chain complex: a

ρ3

  • 1
  • x

ρ2

b

Second chain complex: x

ρ23 x

They’re homotopy equivalent!A relief, since

Theorem

If (Σ, α, β, z) and (Σ, α′, β′, z′) are pointed bordered Heegaard diagrams for the same bordered Y 3 then CFD(Σ) is homotopy equivalent to

  • CFD(Σ′).
  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 33 / 50

slide-73
SLIDE 73

Generators and idempotents of CFA.

Fix a bordered Heegaard diagram (Σg, α, β, z)

  • CFA(Σ) is generated by the same set as

CFD: g-tuples x = {xi} with:

  • ne xi on each β-circle
  • ne xi on each α-circle

no two xi on the same α-arc.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 34 / 50

slide-74
SLIDE 74

Generators and idempotents of CFA.

Fix a bordered Heegaard diagram (Σg, α, β, z)

  • CFA(Σ) is generated by the same set as

CFD: g-tuples x = {xi} with:

  • ne xi on each β-circle
  • ne xi on each α-circle

no two xi on the same α-arc. Over F2,

  • CFA = ⊕xF2.
  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 34 / 50

slide-75
SLIDE 75

Generators and idempotents of CFA.

Fix a bordered Heegaard diagram (Σg, α, β, z)

  • CFA(Σ) is generated by the same set as

CFD: g-tuples x = {xi} with:

  • ne xi on each β-circle
  • ne xi on each α-circle

no two xi on the same α-arc. Over F2,

  • CFA = ⊕xF2.

This is much smaller than CFD.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 34 / 50

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SLIDE 76

The differential on CFA...

...counts only holomorphic curves contained in a compact subset of Σ, i.e., with no asymptotics at e∞.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 35 / 50

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SLIDE 77

The module structure on CFA

To x, associate the idempotent J(x), the α-arcs occupied by x (opposite from CFD).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 36 / 50

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SLIDE 78

The module structure on CFA

To x, associate the idempotent J(x), the α-arcs occupied by x (opposite from CFD). For I a primitive idempotent, define xI = x if I = J(x) if I = J(x)

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 36 / 50

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SLIDE 79

The module structure on CFA

To x, associate the idempotent J(x), the α-arcs occupied by x (opposite from CFD). For I a primitive idempotent, define xI = x if I = J(x) if I = J(x) Given a set ρ of Reeb chords, define x · a(J(x), ρ) =

  • y

(#M(x, y; ρ)) y where M(x, y; ρ) consists of holomorphic curves asymptotic to

x at −∞. y at +∞. ρ at e∞, all at the same height.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 36 / 50

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SLIDE 80

A local example of the module structure on CFA.

Consider the following piece of a Heegaard diagram, with generators {r, x}, {s, x}, {r, y}, {s, y}.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

slide-81
SLIDE 81

A local example of the module structure on CFA.

Consider the following piece of a Heegaard diagram, with generators {r, x}, {s, x}, {r, y}, {s, y}. The nonzero products are: {r, x}ρ1 = {s, x}, {r, y}ρ1 = {s, y}, {r, x}ρ3 = {r, y}, {s, x}ρ3 = {s, y}, {r, x}(ρ1ρ3) = {s, y}.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

slide-82
SLIDE 82

A local example of the module structure on CFA.

Consider the following piece of a Heegaard diagram, with generators {r, x}, {s, x}, {r, y}, {s, y}. The nonzero products are: {r, x}ρ1 = {s, x}, {r, y}ρ1 = {s, y}, {r, x}ρ3 = {r, y}, {s, x}ρ3 = {s, y}, {r, x}(ρ1ρ3) = {s, y}. Example: {r, x}ρ1 = {s, x} comes from this domain.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

slide-83
SLIDE 83

A local example of the module structure on CFA.

Consider the following piece of a Heegaard diagram, with generators {r, x}, {s, x}, {r, y}, {s, y}. The nonzero products are: {r, x}ρ1 = {s, x}, {r, y}ρ1 = {s, y}, {r, x}ρ3 = {r, y}, {s, x}ρ3 = {s, y}, {r, x}(ρ1ρ3) = {s, y}. Example: {r, x}ρ3 = {r, y} comes from this domain.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

slide-84
SLIDE 84

A local example of the module structure on CFA.

Consider the following piece of a Heegaard diagram, with generators {r, x}, {s, x}, {r, y}, {s, y}. The nonzero products are: {r, x}ρ1 = {s, x}, {r, y}ρ1 = {s, y}, {r, x}ρ3 = {r, y}, {s, x}ρ3 = {s, y}, {r, x}(ρ1ρ3) = {s, y}. Example: {r, x}(ρ1ρ3) = {s, y} comes from this domain.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

slide-85
SLIDE 85

Example A1: a solid torus.

z 1 2 3 x a b

d(a) = b aρ1 = x aρ12 = b xρ2 = b.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

slide-86
SLIDE 86

Example A1: a solid torus.

z 1 2 3 x a b

d(a) = b aρ1 = x aρ12 = b xρ2 = b.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

slide-87
SLIDE 87

Example A1: a solid torus.

z 1 2 3 x a b

d(a) = b aρ1 = x aρ12 = b xρ2 = b.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

slide-88
SLIDE 88

Example A1: a solid torus.

z 1 2 3 x a b

d(a) = b aρ1 = x aρ12 = b xρ2 = b.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

slide-89
SLIDE 89

Example A1: a solid torus.

z 1 2 3 x a b

d(a) = b aρ1 = x aρ12 = b xρ2 = b.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

slide-90
SLIDE 90

Why associativity should hold...

(x · ρi) · ρj counts curves with ρi and ρj infinitely far apart. x · (ρi · ρj) counts curves with ρi and ρj at the same height. These are ends of a 1-dimensional moduli space, with height between ρi and ρj varying.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 39 / 50

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SLIDE 91

The local model again.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 40 / 50

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SLIDE 92

...and why it doesn’t.

But this moduli space might have other ends: broken flows with ρ1 and ρ2 at a fixed nonzero height.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 41 / 50

slide-93
SLIDE 93

...and why it doesn’t.

But this moduli space might have other ends: broken flows with ρ1 and ρ2 at a fixed nonzero height. These moduli spaces – M(x, y; (ρ1, ρ2)) – measure failure of

  • associativity. So...
  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 41 / 50

slide-94
SLIDE 94

Higher A∞-operations

Define mn+1(x, a(ρ1), . . . , a(ρn)) =

  • y

(#M(x, y; (ρ1, . . . , ρn))) y where M(x, y; (ρ1, . . . , ρn)) consists of holomorphic curves asymptotic to x at −∞. y at +∞. ρ1 all at one height at e∞, ρ2 at some other (higher) height at e∞, and so on.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 42 / 50

slide-95
SLIDE 95

Example A2: same torus, different diagram.

z 1 2 3 x

m3(x, ρ2, ρ1) = x m4(x, ρ2, ρ12, ρ1) = x m5(x, ρ2, ρ12, ρ12, ρ1) = x . . .

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 43 / 50

slide-96
SLIDE 96

Comparison of the two examples.

First chain complex: a

m2(·,ρ1)

  • 1+ρ12
  • x

m2(·,ρ2)

b

Second chain complex: x

m3(·,ρ2,ρ1)+m4(·,ρ2,ρ12,ρ1)+...

x

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 44 / 50

slide-97
SLIDE 97

Comparison of the two examples.

First chain complex: a

m2(·,ρ1)

  • 1+ρ12
  • x

m2(·,ρ2)

b

Second chain complex: x

m3(·,ρ2,ρ1)+m4(·,ρ2,ρ12,ρ1)+...

x

They’re A∞ homotopy equivalent (exercise).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 44 / 50

slide-98
SLIDE 98

Comparison of the two examples.

First chain complex: a

m2(·,ρ1)

  • 1+ρ12
  • x

m2(·,ρ2)

b

Second chain complex: x

m3(·,ρ2,ρ1)+m4(·,ρ2,ρ12,ρ1)+...

x

They’re A∞ homotopy equivalent (exercise). Suggestive remark: (1 + ρ12)−1“=”1 + ρ12 + ρ12, ρ12 + . . . ρ2(1 + ρ12)−1ρ1“=”ρ2, ρ1 + ρ2, ρ12, ρ1 + . . . .

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 44 / 50

slide-99
SLIDE 99

Again, that’s a relief, since:

Theorem

If (Σ, α, β, z) and (Σ, α′, β′, z′) are pointed bordered Heegaard diagrams for the same bordered Y 3 then CFA(Σ) is A∞-homotopy equivalent to

  • CFA(Σ′).
  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 45 / 50

slide-100
SLIDE 100

The pairing theorem

Recall:

Theorem

If ∂Y1 = F = −∂Y2 then

  • CF(Y1 ∪∂ Y2) ≃

CFA(Y1) ⊗A(F) CFD(Y2).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 46 / 50

slide-101
SLIDE 101

The pairing theorem

Recall:

Theorem

If ∂Y1 = F = −∂Y2 then

  • CF(Y1 ∪∂ Y2) ≃

CFA(Y1) ⊗A(F) CFD(Y2). At this point, one might wonder: Why the distinction between CFD and CFA? And why is the pairing theorem true?

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 46 / 50

slide-102
SLIDE 102

Consider this local picture Here, d(xA ⊗ xD) = xA ⊗ d(xD) = xA ⊗ γyD = xAγ ⊗ yD = yA ⊗ yD

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 47 / 50

slide-103
SLIDE 103

Consider this local picture Here, d(xA ⊗ xD) = xA ⊗ d(xD) = xA ⊗ γyD = xAγ ⊗ yD = yA ⊗ yD as desired.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 47 / 50

slide-104
SLIDE 104

Using “nice diagrams” (analogous to Sarkar-Wang), such rectangles are the only rigid curves crossing the boundary.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 48 / 50

slide-105
SLIDE 105

Using “nice diagrams” (analogous to Sarkar-Wang), such rectangles are the only rigid curves crossing the boundary. Any Heegaard diagram is equivalent to a nice one, so the pairing theorem follows from this simple case and invariance.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 48 / 50

slide-106
SLIDE 106

Using “nice diagrams” (analogous to Sarkar-Wang), such rectangles are the only rigid curves crossing the boundary. Any Heegaard diagram is equivalent to a nice one, so the pairing theorem follows from this simple case and invariance. This proof probably wouldn’t work for CF−. There is a more involved proof that should – and perhaps gives insight into the right definition

  • f CFD− and CFA−...

but we’ll omit it for lack of time.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 48 / 50

slide-107
SLIDE 107

Using “nice diagrams” (analogous to Sarkar-Wang), such rectangles are the only rigid curves crossing the boundary. Any Heegaard diagram is equivalent to a nice one, so the pairing theorem follows from this simple case and invariance. This proof probably wouldn’t work for CF−. There is a more involved proof that should – and perhaps gives insight into the right definition

  • f CFD− and CFA−...

but we’ll omit it for lack of time.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 48 / 50

slide-108
SLIDE 108

Computing CFD for knot complements.

For a knot K in S3, CFD and CFA are determined by CFK−(K).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 49 / 50

slide-109
SLIDE 109

Computing CFD for knot complements.

For a knot K in S3, CFD and CFA are determined by CFK−(K). The proof involves winding one of the α-curves like this

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 49 / 50

slide-110
SLIDE 110

Computing CFD for knot complements.

For a knot K in S3, CFD and CFA are determined by CFK−(K). The proof involves winding one of the α-curves like this ...and studying boundary degenerations when curves in a bordered H.D. are allowed to cross z.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 49 / 50

slide-111
SLIDE 111

Satellites

It is easy to compute HFK of satellites from these results.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 50 / 50

slide-112
SLIDE 112

Satellites

It is easy to compute HFK of satellites from these results. In particular, one can reprove results of Eaman Eftekhary and Matt Hedden.

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 50 / 50

slide-113
SLIDE 113

Satellites

It is easy to compute HFK of satellites from these results. In particular, one can reprove results of Eaman Eftekhary and Matt Hedden. More generally, these techniques imply HFK− of satellites of K is determined by CFK− of K. i.e.,

Theorem

Suppose K and K ′ are knots with CFK−(K) filtered homotopy equivalent to CFK−(K ′). Let KC (resp. K ′

C) be the satellite of K (resp. K ′) with

companion C. Then HFK−(KC) ∼ = HFK−(K ′

C).

  • R. Lipshitz, P. Ozsv´

ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 50 / 50