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

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

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 2 k 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

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

More generally, given ( I , ρ ) where ρ = { ρ 1 , . . . , ρ ℓ } is a set of Reeb chords starting at I , with: i � = j implies ρ i and ρ j start and end on different pairs. { starting points of ρ i ’s } ⊂ I . specifies an algebra element a ( I , ρ ). From: These generate A ( Z ) over F 2 . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 14 / 50

That is, A ( Z ) is the subalgebra of the algebra of k-strand, upward-veering flattened braids on 4 k 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

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

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

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

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

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

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

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

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 F 2 , 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

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

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

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

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

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] = v 0 for some given v 0 . (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

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] = v 0 for some given v 0 . (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

It is easy to see that G ∼ = [Σ F , S 2 ]. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 22 / 50

It is easy to see that G ∼ = [Σ F , S 2 ]. It follows that G is a Z -central extension of H 1 ( F ), 0 → Z → G → H 1 ( F ) → 0 . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 22 / 50

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

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

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

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

The cylindrical setting for classical � CF: Fix an ordinary H.D. (Σ g , α , β , z ). (Here, α = { α 1 , . . . , α g } .) The chain complex � CF is generated over F 2 by g -tuples { x i ∈ α σ ( i ) ∩ β i } ⊂ α ∩ β . ( σ ∈ S g is a permutation.) (cf. T α ∩ T β ⊂ Sym g (Σ).) R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 24 / 50

The cylindrical setting for classical � CF: Fix an ordinary H.D. (Σ g , α , β , z ). (Here, α = { α 1 , . . . , α g } .) The chain complex � CF is generated over F 2 by g -tuples { x i ∈ α σ ( i ) ∩ β i } ⊂ α ∩ β . ( σ ∈ S g 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

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

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

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

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 ρ i × (1 , t i ). R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 26 / 50

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 ρ i × (1 , t i ). The asymptotics ρ i 1 , . . . , ρ i ℓ 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

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

Generators of � CFD... Fix a bordered Heegaard diagram (Σ g , α , β , z ) � CFD(Σ) is generated by g -tuples x = { x i } with: one x i on each β -circle one x i on each α -circle no two x i 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

Generators of � CFD... Fix a bordered Heegaard diagram (Σ g , α , β , z ) � CFD(Σ) is generated by g -tuples x = { x i } with: one x i on each β -circle one x i on each α -circle no two x i 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

...and associated idempotents. To x , associate the idempotent I ( x ), the α -arcs not occupied by x . x As a left A -module, � CFD = ⊕ x A I ( x ) . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 29 / 50

...and associated idempotents. To x , associate the idempotent I ( x ), the α -arcs not occupied by x . As a left A -module, � CFD = ⊕ x A I ( x ) . So, if I is a primitive idempotent, I x = 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

The differential on � CFD. � � d ( x ) = (# M ( x , y ; ρ 1 , . . . , ρ n )) a ( ρ 1 , I ( x )) · · · a ( ρ n , I n ) y . y ( ρ 1 ,...,ρ n ) where M ( x , y ; ρ 1 , . . . , ρ n ) consists of holomorphic curves asymptotic to x at −∞ y at + ∞ ρ 1 , . . . , ρ n at e ∞ . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 30 / 50

Example D1: a solid torus. 2 3 x z 1 a b d ( a ) = b + ρ 3 x d ( x ) = ρ 2 b d ( b ) = 0 . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 31 / 50

Example D2: same torus, different diagram. 2 3 x z 1 d ( x ) = ρ 2 ρ 3 x = ρ 23 x . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 32 / 50

� � Comparison of the two examples. First chain complex: a � � 1 � ρ 3 � � � � ρ 2 � � b x Second chain complex: ρ 23 � x x R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 33 / 50

� � Comparison of the two examples. First chain complex: a � � 1 � ρ 3 � � � � ρ 2 � � b x Second chain complex: ρ 23 � x x They’re homotopy equivalent! R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 33 / 50

� � Comparison of the two examples. First chain complex: a � � 1 � ρ 3 � � � � ρ 2 � � b x Second chain complex: ρ 23 � x 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

Generators and idempotents of � CFA. Fix a bordered Heegaard diagram (Σ g , α , β , z ) CFA(Σ) is generated by the same set as � � CFD: g -tuples x = { x i } with: one x i on each β -circle one x i on each α -circle no two x i on the same α -arc. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 34 / 50

Generators and idempotents of � CFA. Fix a bordered Heegaard diagram (Σ g , α , β , z ) CFA(Σ) is generated by the same set as � � CFD: g -tuples x = { x i } with: one x i on each β -circle one x i on each α -circle no two x i on the same α -arc. Over F 2 , � CFA = ⊕ x F 2 . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 34 / 50

Generators and idempotents of � CFA. Fix a bordered Heegaard diagram (Σ g , α , β , z ) CFA(Σ) is generated by the same set as � � CFD: g -tuples x = { x i } with: one x i on each β -circle one x i on each α -circle no two x i on the same α -arc. Over F 2 , � CFA = ⊕ x F 2 . This is much smaller than � CFD. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 34 / 50

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

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

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 � x if I = J ( x ) x I = 0 if I � = J ( x ) R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 36 / 50

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 � x if I = J ( x ) x I = 0 if I � = J ( x ) Given a set ρ of Reeb chords, define � x · a ( J ( x ) , ρ ) = (# M ( x , y ; ρ )) y 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

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

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 } ρ 1 = { s , x } , { r , y } ρ 1 = { s , y } , { r , x } ρ 3 = { r , y } , { s , x } ρ 3 = { s , y } , { r , x } ( ρ 1 ρ 3 ) = { s , y } . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

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 } ρ 1 = { s , x } , { r , y } ρ 1 = { s , y } , { r , x } ρ 3 = { r , y } , { s , x } ρ 3 = { s , y } , { r , x } ( ρ 1 ρ 3 ) = { s , y } . Example: { r , x } ρ 1 = { s , x } comes from this domain. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

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 } ρ 1 = { s , x } , { r , y } ρ 1 = { s , y } , { r , x } ρ 3 = { r , y } , { s , x } ρ 3 = { s , y } , { r , x } ( ρ 1 ρ 3 ) = { s , y } . Example: { r , x } ρ 3 = { r , y } comes from this domain. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

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 } ρ 1 = { s , x } , { r , y } ρ 1 = { s , y } , { r , x } ρ 3 = { r , y } , { s , x } ρ 3 = { s , y } , { r , x } ( ρ 1 ρ 3 ) = { s , y } . Example: { r , x } ( ρ 1 ρ 3 ) = { s , y } comes from this domain. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 37 / 50

Example A1: a solid torus. 2 1 x z 3 a b d ( a ) = b a ρ 1 = x a ρ 12 = b x ρ 2 = b . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

Example A1: a solid torus. 2 1 x z 3 a b d ( a ) = b a ρ 1 = x a ρ 12 = b x ρ 2 = b . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

Example A1: a solid torus. 2 1 x z 3 a b d ( a ) = b a ρ 1 = x a ρ 12 = b x ρ 2 = b . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

Example A1: a solid torus. 2 1 x z 3 a b d ( a ) = b a ρ 1 = x a ρ 12 = b x ρ 2 = b . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

Example A1: a solid torus. 2 1 x z 3 a b d ( a ) = b a ρ 1 = x a ρ 12 = b x ρ 2 = b . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 38 / 50

Why associativity should hold... ( x · ρ i ) · ρ j counts curves with ρ i and ρ j infinitely far apart. x · ( ρ i · ρ j ) counts curves with ρ i and ρ j at the same height. These are ends of a 1-dimensional moduli space, with height between ρ i and ρ j varying. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 39 / 50

The local model again. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 40 / 50

...and why it doesn’t. But this moduli space might have other ends: broken flows with ρ 1 and ρ 2 at a fixed nonzero height. R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 41 / 50

...and why it doesn’t. But this moduli space might have other ends: broken flows with ρ 1 and ρ 2 at a fixed nonzero height. These moduli spaces – M ( x , y ; ( ρ 1 , ρ 2 )) – measure failure of associativity. So... R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 41 / 50

Higher A ∞ -operations Define � m n +1 ( x , a ( ρ 1 ) , . . . , a ( ρ n )) = (# M ( x , y ; ( ρ 1 , . . . , ρ n ))) y y where M ( x , y ; ( ρ 1 , . . . , ρ n )) consists of holomorphic curves asymptotic to x at −∞ . y at + ∞ . ρ 1 all at one height at e ∞ , ρ 2 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

Example A2: same torus, different diagram. 2 1 x z 3 m 3 ( x , ρ 2 , ρ 1 ) = x m 4 ( x , ρ 2 , ρ 12 , ρ 1 ) = x m 5 ( x , ρ 2 , ρ 12 , ρ 12 , ρ 1 ) = x . . . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 43 / 50

� � Comparison of the two examples. First chain complex: a � � � � � � 1+ ρ 12 � m 2 ( · ,ρ 1 ) � � � � � � � � m 2 ( · ,ρ 2 ) � � b x Second chain complex: m 3 ( · ,ρ 2 ,ρ 1 )+ m 4 ( · ,ρ 2 ,ρ 12 ,ρ 1 )+ ... � x x R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 44 / 50

� � Comparison of the two examples. First chain complex: a � � � � � � 1+ ρ 12 � m 2 ( · ,ρ 1 ) � � � � � � � � m 2 ( · ,ρ 2 ) � � b x Second chain complex: m 3 ( · ,ρ 2 ,ρ 1 )+ m 4 ( · ,ρ 2 ,ρ 12 ,ρ 1 )+ ... � x 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

� � Comparison of the two examples. First chain complex: a � � � � � � 1+ ρ 12 � m 2 ( · ,ρ 1 ) � � � � � � � � m 2 ( · ,ρ 2 ) � � b x Second chain complex: m 3 ( · ,ρ 2 ,ρ 1 )+ m 4 ( · ,ρ 2 ,ρ 12 ,ρ 1 )+ ... � x x They’re A ∞ homotopy equivalent (exercise). Suggestive remark: (1 + ρ 12 ) − 1 “=” 1 + ρ 12 + ρ 12 , ρ 12 + . . . ρ 2 (1 + ρ 12 ) − 1 ρ 1 “=” ρ 2 , ρ 1 + ρ 2 , ρ 12 , ρ 1 + . . . . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 44 / 50

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

The pairing theorem Recall: Theorem If ∂ Y 1 = F = − ∂ Y 2 then CF( Y 1 ∪ ∂ Y 2 ) ≃ � � ⊗ A ( F ) � CFA( Y 1 ) � CFD( Y 2 ) . R. Lipshitz, P. Ozsv´ ath and D. Thurston ()Towards bordered Heegaard Floer homology June 10, 2008 46 / 50