robert lipshitz,peter s. ozsváth and dylan p. thurston- tour of bordered floer theory

8/3/2019 Robert Lipshitz,Peter S. Ozsvth and Dylan P. Thurston- Tour of bordered Floer theory

1/8

Tour of bordered Floer theoryRobert Lipshitza, Peter S. Ozsvthb, and Dylan P. Thurstonc,1

aDepartment of Mathematics, Columbia University, New York, NY 10027; bDepartment of Mathematics, Massachusetts Institute of Technology,Cambridge, MA 02139; and cDepartment of Mathematics, Barnard College, Columbia University, New York, NY 10027

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved February 2, 2011 (received for review December 18, 2010)

Heegaard Floer theory is a kind of topological quantum field theory(TQFT), assigning graded groups to closed, connected, oriented

3-manifolds and group homomorphisms to smooth, oriented

four-dimensional cobordisms. Bordered Heegaard Floer homology

is an extension of Heegaard Floer homology to 3-manifolds with

boundary, with extended-TQFT-type gluing properties. In this sur-

vey, we explain the formal structure and construction of bordered

Floer homology and sketch how it can be used to compute some

aspects of Heegaard Floer theory.

Heegaard Floer homology 4-manifolds

Heegaard Floer homology, introduced in a series of papers(13) of Szab and the second author has become a usefultool in three- and four-dimensional topology. The HeegaardFloer invariants contain subtle topological information,allowing one to detect the genera of knots and homology classes(4); detect fiberedness for knots (59) and 3-manifolds (1013);bound the slice genus (14) and unknotting number (15, 16); provetightness and obstruct Stein fillability of contact structures (5, 17);and more. It has been useful for resolving a number of conjec-tures, particularly related to questions about Dehn surgery(18, 19); see also ref. 20. It is either known or conjectured tobe equivalent to several other gauge-theoretic or holomorphiccurve invariants in low-dimensional topology, including mono-pole Floer homology (21), embedded contact homology (22),and the Lagrangian matching invariants of 3- and 4-manifolds(2325). Heegaard Floer homology is known to relate to Khova-nov homology (2628), and more relations with Khovanov

Rozansky-type homologies are conjectured (29).Heegaard Floer homology has several variants; the technically

simplest is dHF, which is sufficient for most of the three-dimen-sional applications discussed above. Bordered Heegaard Floer

homology, the focus of this paper, is an extension ofdHF to 3-mani-folds with boundary (30). This extension gives a conceptuallysatisfying way to compute essentially all aspects of the Heegaard

Floer package related to dHF. [There are also other algorithms forcomputing many parts of Heegaard Floer theory (3139).]

We will start with the formal structure of bordered HeegaardFloer homology. Most of the paper is then devoted to sketchingits definition. We conclude by explaining how bordered Floerhomology can be used for calculations of Heegaard Floerinvariants.

Formal Structure

Review of Heegaard Floer Theory. Heegaard Floer theory has manycomponents. Most basic among them, it associates:

To a closed, connected, oriented 3-manifold Y, an abelian groupdHFY and ZU modules HFY, HFY, and HFY.

These are the homologies of chain complexes cCFY,CFY, CFY, and CFY, respectively. The chain com-plexes (and their homology groups) decompose into spinc struc-tures, CFY

sspincYCFY ;s, where CFisany ofthe four

chain complexes. Each CFY ;s has a relative grading modulo

the divisibility of c1s (1). The chain complex cCFY is theU 0 specialization of CFY.

To a smooth, compact, oriented cobordism W from Y1 to Y2,maps FW: HFY1HFY2 induced by chain mapsfW: CFY1 CFY2.* These maps decompose accordingto spinc structures on W (3).

The maps FW satisfy a topological quantum field theory (TQFT)composition law:

If W0 is another cobordism, from Y2 to Y3, then FW0 FW FW0W (3).

The Heegaard Floer invariants are defined by counting pseu-doholomorphic curves in symmetric products of Heegaardsurfaces. The Heegaard Floer groups were conjectured to beequivalent to the monopole Floer homology groups (definedby counting solutions of the SeibergWitten equations), via

the correspondence: HFYHM^

Y, HFY dHMY,

HFYHMY, and similarly for the corresponding cobord-ism maps. A proof of this conjecture has recently been announcedby Kutluhan et al. (4042). Colin et al. have announced an inde-pendent proof for the U 0 specialization (43).

In particular, the Heegaard Floer package contains enoughinformation to detect exotic smooth structures on 4-manifolds(10, 44). For closed 4-manifolds, this information is contained

in HF and HF; the weaker invariant dHF is not useful for dis-tinguishing smooth structures on closed 4-manifolds.

The Structure of Bordered Floer Theory. Bordered Floer homology is

an extension of dHF to 3-manifolds with boundary, in a TQFT

form. Bordered Floer homology associates:

To a closed, oriented, connected surface F, together with someextra markings (see Definition 1), a differential graded (dg)algebra AF.

To a compact, oriented 3-manifold Y with connected bound-ary, together with a diffeomorphism : F Y marking theboundary, a module over AF. Actually, there are two differ-

ent invariants for Y: dCFDY, a left dg module over AF,and CFAY, a rightA module overAF, each well-definedup to quasi-isomorphism. We sometimes refer to a 3-manifoldY with Y F; we actually mean Y together with an identifi-cation ofY withF. We call these data a bordered 3-manifold.

More generally, to a 3-manifold Y with two boundary compo-nents LY and RY, diffeomorphisms L: FL LY andR: FR RY and a framed arc from LY to RY (compatible

with L and R in a suitable sense), a dg bimodule dCFDDYwith commuting left actions ofAFL and AFR; an Abimodule dCFDAY with a left action ofAFL and a right

action ofAFR; and anA bimodule dCFAAY with commut-

ing right actions of AFL and AFR. Each of dCFDDY,

Author contributions: R.L., P.S.O., and D.P.T. performed research; and R.L., P.S.O., andD.P.T. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

*For CF and CF, we mean the completions with respect to the formal variable U.

1To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1019060108 PNAS May 17, 2011 vol. 108 no. 20 80858092

M A T H E M A T I

C S

S P E C I A L F E A T U R E


2/8

dCFDAY, and dCFAAY is well-defined up to quasi-isomorph-ism. We call the data Y ; L; R; a strongly bordered 3-manifold with two boundary components.

To keep the sidedness straight, note that type D boundaries arealways on the left, and type A boundaries are always on the right;and for 0;1 F, the boundary component on the left side,f0g F, is oriented as F, whereas the one on the right side

is oriented as F.Gluing 3-manifolds corresponds to tensoring invariants; forany valid tensor product (necessarily matching D sides with Asides) there is a corresponding gluing. More concretely (30, 45):

Given 3-manifolds Y1 and Y2 with Y1 F Y2, there is aquasi-isomorphism

cCFY1FY2 dCFAY1 ~AF dCFDY2; [1]

where ~ denotes the derived tensor product. So,dHFY1FY2 TorAF dCFAY1; dCFDY2.

More generally, given 3-manifolds Y1 and Y2 withY1 F1 F2 and Y2 F2 F3, there are quasi-iso-morphisms of bimodules corresponding to any valid tensorproduct. For instance,

dCFDDY1F2 Y2dCFDAY1 ~AF2

dCFDDY2

dCFDAY2 ~AF2dCFDDY1:

Also, F1 or F3 may be S2 (or empty), in which case these state-

ments reduce to pairing theorems for a module and abimodule.

We refer to theorems of this kind as pairing theorems.There is also a self-pairing theorem. Let Y be a 3-manifold

with Y F Fand be a framed arc connecting correspond-ing points in the boundary components of Y. The self-pairingtheorem relates the Hochschild homology of the bimodule

dCFDAY with the knot Floer homology of a generalized openbook decomposition associated to Y and .

These invariants satisfy a number of duality properties(46); e.g.:

The algebra AF is the opposite algebra ofAF. (Thereare also more subtle duality properties of the algebras; seeRemark 4.)

The module dCFDY is dual [over AF] to dCFAY:

dCFDYMorAF dCFAY;AF

dCFAYMorAF dCFDY;AF:

The module dCFDDY is the one-sided dual of dCFAAY:

MorAF2dCFDDY;AF2 dCFAAY:

The symmetric statement also holds, as does the corresponding

statement for dCFDAY.

As a consequence of these dualities, one can give pairingtheorems using the Hom functor rather than the tensor product(46); e.g.:

Let Y1 and Y2 be 3-manifolds with Y1 Y2 F. Then

cCFY1 Y2 MorAF dCFDY1; dCFDY2:

Similar statements hold for dCFA and for bimodules. Given 3-manifolds Y1 and Y2 with Y1 F Y2,

cCFY1 F Y2MorAFAF dCFDD0;1 F;

dCFDY1 F2dCFDY2: [2]

Similarly, if Y2 has another boundary component F0, then

dCFDY1 F Y2MorAFAF dCFDD0;1 F;

dCFDY1 F2dCFDDY2: [3]

(If both Y1 and Y2 had two boundary components, then the

left-hand side would pick up a change of framing.)

Remark 1:Some of the duality properties discussed above can alsobe seen from the Fukaya-categorical perspective (47).

Remark 2:It is natural to expect that to a 4-manifold with cornersone would associate a map of bimodules, satisfying certain gluingaxioms. We have not done this; however, as discussed below, even

without this bordered Floer homology allows one to computethe maps ^FW associated to cobordisms W between closed3-manifolds.

The AlgebrasAs mentioned earlier, the bordered Floer algebras are associated

to surfaces together with some extra markings. We encode thesemarkings as pointed matched circlesZ, which we discuss next. Wethen introduce a simpler algebra, An, depending only on aninteger n, of which the bordered Floer algebras AZ aresubalgebras. The definition of AZ itself is given in the lastsubsection.

Pointed Matched Circles.

Definition 1: A pointed matched circle Z consists of an orientedcircle Z; 4k points a fa1;; a4kg in Z; a matching M of thepoints in a in pairs, which we view as a fixed-point free involution

M: a a; and a basepoint z Z \a. We require that performingsurgery on Z along the matched pairs of points yields a connected1-manifold. A pointed matched circle Z with jaj 4k specifies:

A closed surface FZ of genus k, as follows: Fill Z with a diskD. Attach a two-dimensional 1-handle to each pair of points ina matched by M. By hypothesis, the result has connectedboundary; fill that boundary with a second disk.

A distinguished disk in FZ: the disk D (say). A basepoint z in the boundary of the distinguished disk.

Remark 3: Matched circles can be seen as a special case of fatgraphs (48). They are also dual to the typical representation ofa genus g surface as a 4g-gon with sides glued together.

The Strands Algebra. We next define a differential algebraAn, de-pending only on an integer n; the algebra AZ associated to apointed matched circle with jaj 4k will be a subalgebra ofA4k.

The algebra An has an F2 basis consisting of all triplesS; T; , where S and T are subsets of n f1;; ng and :S T is a bijection such that for all s S, s s. Given sucha map , let Inv fs1; s2 S S s1 < s2; s2 < s1gand inv jInvj, so inv is the number ofinversions of.

The product S; T; U; V; in An is defined to be 0if U T or if U T, but inv inv inv. IfU T and inv inv inv, then let S; T; U; V; S; V; . In particular, the elements S; S; I(where I denotes the identity map) are the indecomposable idem-potents in An.

Given a generator S; T; An and an element s1; s2 Inv, let : S T be the map defined bys s ifs s1; s2; s1 s2; and s2 s1. De-fine a differential on An by

8086 www.pnas.org/cgi/doi/10.1073/pnas.1019060108 Lipshitz et al.


3/8

S; T; Invinvinv1

S; T; :

See Fig. 1 for a graphical representation.Given a generator S; T; An, define the weight of

S; T; to be the cardinality ofS. LetAn; i be the subalgebraof An generated by elements of weight i, so An ni0An;i.

The Algebra Associated to a Pointed Matched Circle. Fix a pointedmatched circle Z Z; a; M; z with jaj 4k. After cutting Zat z, the orientation of Z identifies a with 4k, so we can view

M as a matching of 4k.Call a basis element S;T; of A4k equitable if no two

elements of4k that are matched (with respect to M) both occurin S, and no two elements of4k that are matched both occur in T.

Given equitable basis elements x S; T; andy S0; T0; ofA4k, we say that x and y are related by hor-izontal strand swapping, and write x y, if there is a subset U Ssuch that S0 S \ U MU, jS\U jS\U, jU IU, andjMU IMU.

Given an equitable basis element x ofA4k, let ax yx y.See Fig. 2 for an example. Define AZ A4k to be the F2subspace with basis fax x is equitableg. It is straightforwardto verify that AZ is a differential subalgebra ofA4k. We callthe elements ax basic generators ofAZ. If x is not equitable,set ax 0 and extend a linearly to a map a: A4kAZ.(This is not an algebra homomorphism.)

Indecomposable idempotents ofAZ correspond to subsetsof the set of matched pairs in a. These generate a subalgebraIZ where all strands are horizontal. The algebra AZdecomposes as kikAZ; i, where AZ; i AZ A4k; k i.

As the figures suggest, we often think of elements ofAZ interms of sets ofchords in Z; a, i.e., arcs in Z with endpoints in a,

with orientations induced from Z. Given a chord in Z \ fzg; a,let (respectively, ) be the initial (respectively, terminal)endpoint of. Given a set fig of chords in Z \ fzg; a suchthat no two i have the same initial (respectively, terminal)endpoint, let fi g and

fi g; we can think of as amap :

. Let

a S0

S0

S0; S0; IS\ :

Example 1:The algebra associated to the unique pointed matchedcircle for S2 is F2. The algebra AT

2; 0 associated to the uniquepointed matched circle for T2, with 1 3; 2 4, is given by thepath algebra with relations:

The algebra associated to the pointed matched circle Z for a

genus 2 surface with matching 1 3; 2 4; 5 7; 6 8 hasPoincar polynomial (45, Sect. 4)

i

dimF2 HAZ; iTi T2 32T1 98 32T1 T2:

The algebra associated to the pointed matched circle Z0 for agenus 2 surface with matching 1 5; 2 6; 3 7; 4 8has Poincar polynomial

i

dimF2 HAZ0; iTi T2 32T1 70 32T1 T2:

The ranks in the genus two examples which are equal areexplained by the observations that for any pointed matched

circle, AZ; k F2; AZ; k 1 has no differential; thedimension of AZ; k 1 is independent of the matching;and the following:

Remark 4:The algebras AZ; i andAZ; i are Koszul dual.(Here, Z denotes the pointed matched circle obtained by rever-sing the orientation onZ.) Also, given a pointed matched circleZfor F, letZ denote the pointed matched circle corresponding tothe dual handle decomposition ofF. ThenAZ; i andAZ; iare Koszul dual. In particular, AZ; i is quasi-isomorphic toAZ; i (46).

Remark 5: In Zarevs bordered-sutured extension of the theory(49), the strands algebra An; k has a topological interpretation

as the algebra associated to a disk with boundary sutures.

Combinatorial Representations of Bordered 3-ManifoldsA bordered 3-manifold is a 3-manifold Y together with anorientation-preserving homeomorphism : FZ Y forsome pointed matched circle Z. Two bordered 3-manifoldsY1; 1: FZ1 Y1 and Y2; 2: FZ2 Y2 are calledequivalent if there is an orientation-preserving homeomorphism: Y1 Y2 such that 1 2 ; in particular, this implies thatZ1 Z2. Bordered Floer theory associates homotopy equiva-lence classes of modules to equivalence classes of bordered3-manifolds. Just as the bordered Floer algebras are associatedto combinatorial representations of surfaces, not directly to sur-faces, the bordered Floer modules are associated to combinator-ial representations of bordered 3-manifolds.

Fig. 1. The strands algebra. A product, two vanishing products, and a differential. In this notation, the restrictions on the number of inversions means thatelements with double crossings in the product or differential are set to 0.

Fig. 2. The algebra AZ. An example of the operation ax and a short-hand for the resulting element of AZ A4k.

Lipshitz et al. PNAS May 17, 2011 vol. 108 no. 20 8087

M A T H E M A T I C S



4/8

The Closed Case. Recall that a three-dimensional handlebody is aregular neighborhood of a connected graph in R3. Accordingto a classical result of Heegaard (50), every closed, orientable3-manifold can be obtained as a union of two such handlebodies,

H and H. Such a representation is called a Heegaard splitting.A Heegaard splitting along an orientable surface of genus gcanbe represented by a Heegaard diagram: a pair of g tuples ofpairwise disjoint, homologically linearly independent, embeddedcircles f1;; gg and f1;; gg in . These curves are

chosen so that each i (respectively, i) bounds a disk in thehandlebodyH (respectively, H). Any two Heegaard diagramsfor the same manifold Y are related by a sequence of moves,called Heegaard moves; see, for instance, ref. 51 or ref. 1, Sect. 2.1.

Representing 3-Manifolds with Boundary. The story extends easily to3-manifolds with boundary, using a slight generalization of han-dlebodies. Acompression body (with both boundaries connected)is the result of starting with a connected orientable surface 2times [0,1] and then attaching thickened disks (three-dimensional2-handles) along some number of homologically linearly indepen-dent, disjoint circles in 2 f0g. A compression body has twoboundary components, 1 and 2, with genera g1 g2. Up tohomeomorphism, a compression body is determined by itsboundary.

A Heegaard decomposition of a 3-manifold Y with nonempty,connected boundary is a decomposition Y H2H, where His a compression body and H is a handlebody. Let gbe the genusof2 andk the genus ofY. AHeegaard diagram for Y is gotten bychoosing g pairwise disjoint circles 1;;g in 2 and gk dis-

joint circles c1;;cgk in 2 so that

The circles 1;;g bound disks D1 ;;Dg in H such thatH \ D1Dg is topologically a ball, and

The circles c1;;cgk bound disks Dc1 ;;Dcgk in H such that

H \ Dc1Dc

gk is topologically the product of a surface

and an interval.

To specify a parametrization, or bordering, of Y, we need alittle more data. A bordered Heegaard diagram for Y is a tuple

H ; c1;;cgk;

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{c

a1;;a2k;

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{a

1;;gzfflfflfflfflffl}|fflfflfflfflffl{

; z;

where

is an oriented surface with a single boundary component; D2; c; is a Heegaard diagram for Y; a1;;

a2k are pairwise disjoint, embedded arcs in with bound-

ary on , and are disjoint from the ci ; \ c1

cgk

a1

a2k is a disk with 2gk holes;

z is a point in , disjoint from all of the ai .

Let a c.A bordered Heegaard diagramH specifies a pointed matched

circle ZH Z ; a a ; M; z, where two pointsin a are matched in M if they lie on the same ai . A borderedHeegaard diagram for Y also specifies an identification: FZ Y, well-defined up to isotopy.

There are moves, analogous to Heegaard moves, relatingany two bordered Heegaard diagrams for equivalent bordered3-manifolds.

The Modules and BimodulesAs discussed above, there are two invariants of a 3-manifold Y

with boundary FZ. dCFDY has a straightforward modulestructure but a differential that counts holomorphic curves,

whereas dCFAY uses holomorphic curves to define the modulestructure itself.

Fix a bordered Heegaard diagramH ; ; ; z for Y. LetSH be the set of g tuples x fxig

gi1 so that there is

exactly one point xi on each -circle and on each -circle, and

there is at most one xi on each -arc. The invariant dCFAY isa direct sum of copies of F2, one for each element ofSH,

whereas dCFDY is a direct sum of elementary projectiveAH modules, one for each element of SH. Let XHbe the F2-vector space generated bySH, which is also the vec-

tor space underlyingd

CFAY.Each generator x SH determines a spinc structuresx spincY; the construction (30) is an easy adaptation ofthe closed case (1, Sect. 2.6).

Before continuing to describe the bordered Floer modules, wedigress to briefly discuss the moduli spaces of holomorphic curves.

Moduli Spaces of Holomorphic Curves. Fix a bordered Heegaarddiagram H ; ; ; z. Let \ . Choose a symplecticform on giving it a cylindrical end and a complex structure

j compatible with , making into a punctured Riemannsurface. Let p denote the puncture in . We choose the ai sothat their intersections with (also denoted ai ) are cylindrical(R-invariant) in a neighborhood of p.

We consider curves

u: S; S 0;1 R; f1g R f0g R;

holomorphic with respect to an appropriate almost complexstructure J, satisfying conditions spelled out in ref. 30. The readermay wish to simply think of a product complex structure j jD,though these complex structures may not be general enough toachieve transversality.

Such holomorphic curves u have asymptotics in three places:

0;1 fg. We consider curves asymptotic to g-tuples ofstrips x 0;1 R at and y 0;1 R at ,

where x; y SH. fpg 0;1 R, which we denote e. We consider curves

asymptotic to chords

i in

;a

at a point 1

; ti 0;1 R. [These are chords for the coisotropic foliation of 0;1 R, whose leaves are the circles fs0; t0g.]We impose the condition that these chords i not cross z .

Topological maps of this form can be grouped into homologyclasses. Let 2x; y denote the set of homology classes of mapsasymptotic to x 0;1 at and y 0;1 at . Then 2x; x iscanonically isomorphic to H2Y ; Y; 2x; y is nonempty ifand only if sx sy; and if sx sy then 2x; y is anaffine copy of H2Y ; Y, under concatenation by elements of2x; x [or 2y; y] (30). [Again, these results are easy adapta-tions of the corresponding results in the closed case (1, Sect. 2).]Note that our usage of 2x; y differs from the usage in ref. 1,

where homology classes are allowed to crossz, but agrees with the

usage in ref. 30.Given generators x; y SH, a homology class B 2x; y,and a sequence ~ 1;;n of sets i fi;1;;i;mi g of Reeb

chords, let fMBx; y; ~ denote the moduli space of embeddedholomorphic curves u in the homology class B, asymptotic tox 0;1 at , y 0;1 at , and i;j 1; ti at e, for somesequence of heights t1 < < tn. There is an action of R onfMBx; y; ~ , by translation. LetMBx; y; ~ fM

Bx; y; ~ R.

The modules dCFDH and dCFAH will be defined usingcounts of zero-dimensional moduli spaces MBx; y; ~ . Provingthat these modules satisfy 2 0 and the A relations, respec-tively, involves studying the ends of one-dimensional modulispaces. These ends correspond to the following four kinds ofdegenerations:



5/8

1. Breaking into a two-story holomorphic building. That is, the Rcoordinate of some parts of the curve go to with respect toother parts, giving an element ofMB1x;y; ~1 M

B2x;y; ~2,where B is the concatenation B1 B2 and ~ is the concatena-tion ~1; ~2.

2. Degenerations in which a boundary branch point of the pro-jection u approaches e, in such a way that some chordi;j splits into a pair of chords a, b with i;j a b. Thisdegeneration results in a curve at e, a join curve, and anelement ofMBx; y; ~ 0, where ~ 0 is obtained by replacingthe chord i;j i ~ with two chords, a and b.

3. The difference in R coordinates ti1 ti between two conse-cutive sets of chords i and i1 in ~ going to 0. In the process,some boundary branch points of u may approach e, de-generating a split curve, along with an element ofMBx; y; ~ 0,

where ~ 0 1;;ii1;;n and ii1 is gotten fromii1 by gluing together any pairs of chords i;j;i1;where i;j ends at the starting point ofi1; (i.e.,

i;j

i1;).

4. Degenerations in which a pair of boundary branch points of u approach e, causing a pair of chords i;j and i; insome i whose endpoints

i;j and

i; are nested, say

i;j < i; <

i; <

i;j, to break apart and recombine into a

pair of chordsa i;j; i;

and b i;;

i;j. This gives an

odd shuffle curve ate andan element ofMBx; y; ~ 0, where ~ 0is obtained from ~ by replacing i;j and i; in i with a and b.

See Fig. 3 for examples of the first three kinds of degen-erations.

Remark 6:This analytic setup builds on the cylindrical reformula-tion of Heegaard Floer theory (52). It relates to the original for-mulation of Heegaard Floer theory, in terms of holomorphic disksin Symg, by thinking of a map D Symg as a multivaluedmap D and then taking the graph. See, for instance, ref. 52,Sect. 13. Some of the results were previously proved in ref. 53.

Type D Modules. Fix a bordered Heegaard diagram H and a sui-

table almost complex structure J. Let Z H be the orienta-tion reverse of the pointed matched circle given by H. Given agenerator x SH, let IDx denote the indecomposable idem-potent ofAZ;0 AZ corresponding to the set of-arcs thatare disjoint from x. This gives an action of the subring IZ of

AZ on XH. As a module, dCFDH AZ IZ XH:

Define a differential on dCFDH by

x ySH

B2 x;y1 ;;n

#MBx; y; f1g;; fng

af1gafng y;

with the convention that the number of elements in an infinite setis zero. Here denotes a chord in H;a with orientationreversed, so as to be a chord in Z. [To ensure finiteness of thesesums, we need to impose an additional condition on the Hee-gaard diagram H, called provincial admissibility (30).] Extend

to all of dCFDH by the Leibniz rule.

Theorem 2. (30) The map is a differential, i.e., 2 0.

Proof sketch:Let ax;y denote the coefficient ofyin x. The equa-tion 2x 0 simplifies to the equation that, for all y, dax;ywax;waw;y 0. As usual in Floer theory, we prove this by con-sidering the boundary of the one-dimensional moduli spaces ofcurves. Of the four types of degenerations, type 4 does not occur,because each i is a singleton set. Type 1 gives the terms of theform ax;waw;y. Type 3 with a split curve degenerating correspondsto dax;y. Type 3 with no split curves cancel in pairs against them-selves and type 2. See also ref. 54.

Theorem 3. (30) Up to homotopy equivalence, the module dCFDHis independent of the (provincially admissible) bordered Heegaarddiagram H representing the bordered 3-manifold Y.

The proof is similar to the closed case (1, Theorem 6.1). The-

orem 3 justifies writing dCFDY for the homotopy equivalenceclass of dCFDH for any bordered Heegaard diagram H for Y.

Remark 7: Because 2x;y is empty unless sx sy, dCFDYdecomposes as a direct sum over spinc structures on Y.

Type A Modules.Again, fix a bordered Heegaard diagramH and asuitable almost complex structureJ, but now letZ H. Given agenerator x SH, let IAx denote the indecomposable idem-potent in AZ;0 AZ corresponding to the set of -arcsintersecting x [opposite of IDx], again making XH into amodule over IZ AZ.

Define an A action ofAZ on XH by setting

mn1x;a1;;an ySH B2x;y

#MBx;y; 1;;n y;

and extending multilinearly. As for dCFD, to ensure finiteness ofthese sums, we need to assume thatH is provincially admissible.

Theorem 4. The operations mn1 satisfy the A module relation.

Proof sketch: Because AZ is a differential algebra, the Arelation for dCFAY takes the form

0 ijn2

mimjx; a1;; aj1;; an

n

1mn1x; a1;; a;; an

n1

1

mnx; a1;; aa1;; an: [4]

The first term in Eq. 4 corresponds to degenerations of type 1.The second term corresponds to degenerations of types 2 and 4,depending on whether one of the strands in the crossing beingresolved is horizontal (2) or not (4). The third term correspondsto degenerations of type 3. This proves the result.

Theorem 5. (30) Up to A homotopy equivalence, the A moduledCFAH is independent of the (provincially admissible) bordered

Heegaard diagramH representing the bordered 3-manifold Y.

Fig. 3. Degenerations of holomorphic curves. Degenerations of types 1, 2,and 3 are shown, in that order. The dots indicate branch points, which can bethought of as the ends of cuts. (This figure is drawn from ref. 30.)





6/8

Again, the proof is similar to the closed case (1, Theorem 6.1).

Remark 8:Like dCFDY, the module dCFAY breaks up as a sumover spinc structures on Y.

Remark 9:It is always possible to choose a Heegaard diagram H

for Y so that the higher products mn, n > 2, vanish on dCFAH,

so that dCFAH is an honest differential module. One way to doso is using an analogue of nice diagrams (31).

Bimodules. Next, suppose Y is a strongly bordered 3-manifold withtwo boundary components. By this we mean that we have a3-manifold Y with boundary decomposed as Y LYRY,homeomorphisms L: FZL LY and R: FZR RY,and a framed arc connecting the basepoints in FZL and

FZR and pointing into the preferred disks of FZL and

FZR. Associated to Y are bimodules AZL;AZRdCFDDY;

AZLdCFDAY

AZR; and dCFAAY

AZL;AZRdefined by treat-

ing, respectively, both LY and RY in type D fashion; LY in typeD fashion, and RY in type A fashion; and both LY and RY intype A fashion.

An important special case of 3-manifolds with two boundary

components is mapping cylinders. Given an isotopy class of maps: FZ1FZ2 taking the distinguished disk of FZ1 tothe distinguished disk of FZ2 and the basepoint of FZ1 tothe basepoint of FZ2called a strongly based mapping classthe mapping cylinder M of is a strongly bordered 3-manifold

with two boundary components. Let dCFDD dCFDDM,dCFDA dCFDAM, and dCFAA dCFAAM.

The set of strongly based mapping classes forms a groupoid, with objects the pointed matched circles representing genus gsurfaces and HomZ1;Z2 the strongly based mapping classes

FZ1FZ2. In particular, the automorphisms of a particularpointed matched circle Z form the (strongly based) mappingclass group.

Remark 10: The functors dCFDA give an action of thestrongly based mapping class group on the (derived) categoryof leftAZ modules; this action categorifies the standard actionon H1FZ; F2. This action is faithful (55).

GradingsLetZ Z; a; M; z denote the genus 1 pointed matched circlefrom Example 1. Consider the following elements ofAZ;1:

x af1; 2g; f2; 3g; 1 3; 2 2

y af1; 2g; f1; 4g; 1 1; 2 4:

A short computation shows that y x ddx y. It followsthat there is no Z grading on AZ;1 with homogeneous basicgenerators. A similar argument applies to AZ0; i for anyZ0,

as long as AZ0; i involves at least two moving strands.There is, however, a grading in a more complicated sense. Let

G be a group and G a distinguished central element. A grad-ing of a differential algebra A by G; is a decompositionA gGAg so that dAg A1g and Ag Ah Agh . TakingG Z and 1 recovers the usual notion of a Z grading ofhomological type. The corresponding notion for modules is agrading by a G set. A grading of a left differential A module

M by a left G set S is a decomposition M sSMs so thatAgMs Mgs and Ms M1s. Similarly, right modules aregraded by right G sets. IfMis graded by a G set S and Nis gradedbyT, then M A N is graded byS G T, which retains an actionof. These more general kinds of gradings have been consideredby, e.g., Nstsescu and Van Oystaeyen (56).

Given a surface F, let GF be the Z-central extension ofH1F, Z GF H1F, where 1 Z maps to GZ.Explicitly, GF 1

2Z H1F with

k1;1 k2;2 k1 k2 1 2;1 2:

Here, denotes the intersection pairing on H1F. It turns outthat AZ has a grading by GFZ; (30, Sect3.3). Simi-larly, given a 3-manifold Y bordered byFZ, one can construct

GFZ-set gradings on dCFAY and dCFDY (30).

Even in the closed case, the grading on Heegaard Floer homol-ogy has a somewhat nonstandard form: a partial relative cyclicgrading. That is, generators do not have well-defined gradings,but only well-defined grading differences grx; y; the grading dif-ference grx; y is defined only for generators representing thesame spinc structure; and grx; y is well-defined only modulothe divisibility ofc1sx. A partial relative cyclic grading is pre-cisely a grading by a Z set. This leads naturally to a graded versionof the pairing theorems, including Eq. 1 (30).

Deforming the Diagonal and the Pairing Theorems.The tensor product pairing theorems are the main motivation forthe definitions of the modules and bimodules. We will sketch theproof of the archetype, Eq. 1. Fix bordered Heegaard diagramsH1 and H2 for Y1 and Y2, respectively, with H1 H2. It iseasy to see that H H1 H2 is a Heegaard diagram forY1 Y2.

There are two sides to the proof, one algebraic and one ana-lytic. We start with the algebra. Typically, the A tensor product

M ~ NofA modulesMandNis defined using a chain complexwhose underlying vector space is M k T

A k N(where T

A is

the tensor algebra ofA, and kis the ground ring ofAfor us, thering of idempotents). This complex is typically infinite-dimen-

sional, and so is unlikely to align easily with cCF.

However, the differential module dCFDH2 has a specialform: It is given as AZ IZ XH2, so the differential is en-

coded by a map 1: XH2AZ IZ XH2. This allowsus to define a smaller model for the A tensor product. Let

n

: XH

2A

Z

n

XH

2 be the result of iterating 1

ntimes. For notational convenience, let M dCFAH1 and

X XH2. Define dCFAH1 dCFDH2 to be the F2-vectorspaceM IZ X, with differential (graphically depicted in Fig. 4)

i0

mi1 IX IM i: [5]

The sum in Eq. 5 is not a priori finite. To ensure that it is finite, we need to assume an additional boundedness condition on

either dCFAH1 or dCFDH2. These boundedness conditionscorrespond to an admissibility hypothesis for Hi, which in turnguarantees that H1H2 is (weakly) admissible.

Lemma 1. There is a canonical homotopy equivalence

dCFAH1 ~AZ dCFDH2 dCFAH1 AZ dCFDH2:

The proof is straightforward.We turn to the analytic side of the argument next. Because

of how the idempotents act on dCFAH1 and dCFDH2, thereis an obvious identification between generators xL xR ofdCFAH1 dCFDH2 and generators x of cCFH.

Let Z H1 H2 H. The differential on cCFH countsrigid J-holomorphic curves in 0;1 R. For a sequence ofalmost complex structures Jr with longer and longer necks at

Z, such curves degenerate to pairs of curves uL; uR for H1and H2, with matching asymptotics at e. More precisely, inthe limit as we stretch the neck, the moduli space degeneratesto a fibered product



7/8

1 ;;i

MxL; yL; 1;;ievLevR MxR; yR; 1;;i: [6]

Here, evL: Mx; y; 1;;i RiR is given by taking the

successive height differences (in the R coordinate) of the chords1;;i, and similarly for evR. Also, we are suppressing homologyclasses from the notation.

Because we are taking a fiber product over a large-dimensionalspace, the moduli spaces in Formula 6 are not conducive todefining invariants ofH1 and H2. To deal with this, we deformthe matching condition, considering instead the fiber products,

MxL; yL; 1;;iRevLevR MxR;yR;1;;i;

and sending R. In the limit, some of the chords on the leftcollide, whereas some of the chords on the right become infinitelyfar apart. The result exactly recaptures the definitions ofdCFAHL and dCFDHR and the algebra of Eq. 5 (30).

Remark 11: In ref. 30, we also give another proof of the pairingtheorem (Eq. 1), using the technique of nice diagrams (31).

Computing with Bordered Floer Homology

Computing cCF. Let Y be a closed 3-manifold. As discussed earlier,Y admits a Heegaard splitting into two handlebodies, glued bysome homeomorphism between their boundaries. Via the pair-

ing theorems (Eqs. 2 and 3), this reduces computing dHFY tocomputing dCFDHg for some particular bordered handlebody

Hg of each genus g and dCFDD for arbitrary in the strongly

based mapping class group. For an appropriate Hg, dCFDHg is

easy to compute. Moreover we do not need to compute dCFDDfor every mapping class, just for generators for the mapping classgroupoid. This groupoid has a natural set of generators: arcslides(compare refs. 57 and 58). It turns out that the type DD invariantsof arcslides are determined by a small amount of geometric input(essentially, the set of generators and a nondegeneracy conditionfor the differential) and the condition that 2 0 (59).

These techniques also allow one to compute all types of thebordered invariants for any bordered 3-manifold.

Cobordism Maps. Next, we discuss how to compute the map^fW: cCFY1 cCFY4 associated to a 4-dimensional cobordismW from Y1 to Y4. The cobordism W can be decomposed intothree cobordisms W1W2W3, where Wi: Yi Yi1 consists of

i-handle attachments and ^fW is a corresponding composition^fW3

^fW2 ^fW1 .

The maps ^fW1 and^fW3 are simple to describe:

Y2Y1#kS2 S1, whereas Y3Y4#S2 S1; cCFS2 S1 is(homotopy equivalent to) F2 F2 HS1; F2; and the invar-iant dHFY satisfies a Knneth theorem for connect sums, so

cCFY1#kS2 S1 cCFY1 F2 HT

k; F2

(with respect to appropriate Heegaard diagrams), whereTk S1k is the k-dimensional torus. Let be the top-dimen-sional generator ofHT

k; F2 and the bottom-dimensional gen-

erator of HT; F2. Then ^fW1 is x x , whereas

^fW3 takesx x and x 0 if gr > gr.

By contrast, ^fW2 is defined by counting holomorphic trianglesin a suitable Heegaard triple diagram. Two additional propertiesof bordered Floer theory allow us to compute ^fWi :

The invariant

dCFD

H of a handlebody

His rigid, in the sensethat it has no nontrivial graded automorphisms. This allows

one to compute the homotopy equivalences between the re-

sults of making different choices in the computation of cCFY. There is a pairing theorem for holomorphic triangles.

Given these, one can compute ^fW as follows: Using results

from the previous section, we can compute cCFY2 [respectively,cCFY3] using a Heegaard decomposition making the decom-position Y2Y1#kS2 S1 [respectively, Y3Y4#S2 S1]manifest.With respect to this decomposition, themap ^fW1 (respec-

tively, ^fW3 ) is easy to read off.

To compute ^fW2 one works with Heegaard decompositions

of Y2 and Y3 with respect to which the cobordism W2 takes aparticularly simple form, replacing one of the handlebodies Hof a Heegaard decomposition of Y2 with a differently framed

handlebodyH0. It is easy to compute the triangle map dCFDHdCFDH0. By the pairing theorem for triangles, extending thismap by the identity map on the rest of the decomposition gives

the map ^fW2 . Finally, the rigidity result allows one to write down

the isomorphisms between cCFY2 [and cCFY3] computed in

the two different ways. The map ^fW is then the composition of

the three maps ^fWi and the equivalences connecting the two

different models of cCFY2 and of cCFY3.The details will appear in forthcoming work.

Polygon Maps and the Ozsvth

Szab Spectral Sequence. Khovanovintroduced a categorification of the Jones polynomial (60). Thiscategorification associates to an oriented link L a bigraded abe-lian group Khi;jL, the Khovanov homology of L, whose graded

Euler characteristic is q q1 times the Jones polynomial JL.

There is also a reduced version fKhL, whose graded Euler char-acteristic is simplyJL. The skein relation for JL is replaced bya skein exact sequence. Given a linkL and a crossing c ofL, let L0and L1 be the two resolutions of c. Then there is a long exactsequence relating the (reduced) Khovanov homology groups of

L, L1, and L0.Szab and the second author observed that the Heegaard

Floer group dHFDL of the double cover of S3 branched overL satisfies a similar skein exact triangle to (reduced) Khovanovhomology and takes the same value on an n-component unlink(with some collapse of gradings). Using these observations, theyproduced a spectral sequence from Khovanov homology (with F2coefficients) to dHFDL (26). [Because of a difference in con-

ventions, one must take the Khovanov homology of the mirrorrL ofL.] Baldwin recently showed (61) that the entire spectral

sequence fKhrL dHFDL is a knot invariant.Bordered Floer homology can be used to compute this spectral

sequence (62). Write L as the plat closure of some braid B, anddecompose B as a product of braid generators s1i1 s

kik

. Thebranched double cover of a braid generator si is the mappingcylinder of a Dehn twist, and the branched double covers ofthe plats closing B is a handlebody H. So cCFDK is quasi-isomorphic to

A B C

Fig. 4. The operation . (A) Graphical representation of 1 for dCFD. (B) Gra-

phical representation ofm

n1 (n

3

) fordCFA

. (C) Graphical representationof the sum from Eq. 5. In all cases, dashed lines represent module elementsand solid lines represent algebra elements.





8/8

dCFAH dCFDAi1 dCFDAik

dCFDH: [7]

The bordered invariant of a Dehn twist along FZ canbe written as a mapping cone of a map between the identityc'obordism I 0;1 FZ and the manifold Y0 obtained by0-surgery on 0;1 FZ along :

dCFDA Cone dCFDAI0 dCFDAI; [8]

dCFDA1 Cone dCFDAI dCFDAI0: [9]

Applying this observation to the tensor product in Formula 7endows cCFDK with a filtration by f0;1gn. The resultingspectral sequence has E1 page the Khovanov chain complex

and E page dHFDK.

The next step is to compute the groups dCFDAI0 for the

curves corresponding to the braid generators and the mapsfrom Formulas 8 and 9, which again requires a small amountof geometry (62).

Finally, we identify this spectral sequence with the earlierspectral sequence from ref. 26. The key ingredient is anotherpairing theorem identifying the algebra of tensor products ofmapping cones with counts of holomorphic polygons.

ACKNOWLEDGMENTS. We thank Mathematical Sciences Research Institutefor hosting us in the spring of 2010, during which part of this researchwas conducted. R.L. was supported by National Science Foundation (NSF)Grant DMS-0905796 and a Sloan Research Fellowship. P.S.O. was supportedby NSF Grant DMS-0505811. D.P.T. was supportedby NSF Grant DMS-1008049.

1. Ozsvth PS, Szab Z (2004) Holomorphic disks and topological invariants for closedthree-manifolds. Ann Math 159(2):10271158 arXiv:math.SG/0101206.

2. Ozsvth PS, Szab Z (2004) Holomorphic disks and three-manifold invariants:Properties and applications. Ann Math 159(2):11591245 arXiv:math.SG/0105202.

3. Ozsvth PS, Szab Z (2006) Holomorphic triangles and invariants for smoothfour-manifolds. Adv Math 202:326400 arXiv:math.SG/0110169.

4. Ozsvth PS, Szab Z (2004) Holomorphic disks and genus bounds. Geom Topol8:311334 arXiv:math.GT/0311496.

5. Ozsvth P, Szab Z (2005) Heegaard Floer homology and contact structures. DukeMath J 129:3961.

6. Ghiggini P (2008) Knot Floer homology detects genus-one fibred knots. Amer J Math

130:1151

1169.7. Ni Y (2007) Knot Floer homology detects fibred knots. Invent Math 170:577608.8. Juhsz A (2006) Holomorphic discs and sutured manifolds. Algebr Geom Topol

6:14291457 arXiv:math/0601443.9. Juhsz A (2008) Floer homology and surface decompositions. Geom Topol12:299350

arXiv:math/0609779.10. Ozsvth P, Szab Z (2004) Holomorphic triangle invariants and the topology of

symplectic four-manifolds. Duke Math J 121:134.11. Ni Y (2009) Heegaard Floer homology and fibred 3-manifolds. Amer J Math

131:10471063.12. Ai Y, Peters TD (2010) The twisted Floer homology of torus bundles. Algebr Geom

Topol 10:679695.13. Ai Y, Ni Y (2009) Two applications of twisted Floer homology. Int Math Res Notices

(19):37263746.14. Ozsvth PS, Szab Z (2003) Knot Floer homology and the four-ball genus. Geom Topol

7:615639.15. Ozsvth P, Szab Z (2005) Knots with unknotting number one and Heegaard Floer

homology. Topology 44:705745.

16. Owens B (2008) Unknotting information from Heegaard Floer homology. Adv Math217:23532376.17. Lisca P, Stipsicz AI (2009) On the existence of tight contact structures on Seifert fibered

3-manifolds. Duke Math J 148:175209.18. Ozsvth P, Szab Z (2005) On knot Floer homology and lens space surgeries. Topology

44:12811300.19. Greene JE (2010) The lens space realization problem. arXiv:1010.6257.20. Kronheimer P, Mrowka T, Ozsvth PS, Szab Z (2007) Monopoles and lens space

surgeries. Ann Math 165(2):457546 arXiv:math.GT/0310164.21. Kronheimer P, Mrowka T (2007) Monopoles and three-manifolds. New Mathematical

Monographs (Cambridge Univ Press, Cambridge, UK), Vol 10.22. Hutchings M (2002) An index inequality for embedded pseudoholomorphic curves in

symplectizations. J Eur Math Soc 4:313361.23. Usher M (2006) Vortices and a TQFT for Lefschetz fibrations on 4-manifolds. Algebr

Geom Topol 6:16771743 (electronic).24. Perutz T (2007) Lagrangian matching invariants for fibred four-manifolds I. Geom

Topol 11:759828.25. Perutz T (2008) Lagrangian matching invariants for fibred four-manifolds. II. Geom

Topol 12:1461

1542.26. Ozsvth PS, Szab Z (2005) On the Heegaard Floer homology of branched double-covers. Adv Math 194:133 arXiv:math.GT/0309170.

27. Grigsby JE, Wehrli SM (2010) On the colored Jones polynomial, sutured Floer homol-ogy, and knot Floer homology. Adv Math 223:21142165 arXiv:0907.4375.

28. Hedden M (2009) Khovanov homology of the 2-cable detects the unknot. Math ResLett 16(6):991994.

29. Dunfield NM, Gukov S, Rasmussen J (2006) The superpolynomial for knot homologies.Experiment Math 15(2):129159.

30. Lipshitz R, Ozsvth PS, Thurston DP (2008) Bordered Heegaard Floer homology:Invariance and pairing. arXiv:0810.0687.

31. Sarkar S, Wang J (2010) An algorithm for computing some Heegaard Floer homolo-gies. Ann Math 171(2):12131236 arXiv:math/0607777.

32. Lipshitz R, Manolescu C, Wang J (2008) Combinatorial cobordism maps in hatHeegaard Floer theory. Duke Math J 145:207247 arXiv:math/0611927.

33. Ozsvth P, Stipsicz A, SzabZ (2009) CombinatorialHeegaard Floerhomology and niceHeegaard diagrams. arXiv:0912.0830.

34. Manolescu C, Ozsvth P, Sarkar S (2009) A combinatorial description of knot Floerhomology. Ann Math 169(2):633660 arXiv:math.GT/0607691.

35. Manolescu C, Ozsvth PS, Szab Z, Thurston DP (2007) On combinatorial link Floerhomology. Geom Topol 11:23392412 arXiv:math.GT/0610559.

36. Manolescu C, Ozsvth P, Thurston D (2009) Grid diagrams and Heegaard Floerinvariants. arXiv:0910.0078.

37. Manolescu C, Ozsvth P (2010) Heegaard Floer homology and integer surgeries onlinks. arXiv:1011.1317.

38. Ozsvth P, Stipsicz A, Szab Z (2009) Floer homology and singular knots. J Topol

2:380

404.39. Ozsvth P, Szab Z (2009) A cube of resolutions for knot Floer homology. J Topol2:865910.

40. Kutluhan , Lee Y-J, Taubes CH HF=HM I : Heegaard Floer homology and SeibergWitten Floer homology. arXiv:1007.1979.

41. Kutluhan , Lee Y-J, Taubes CH HF=HM II: Reeb orbits and holomorphic curves for theech/Heegaard-Floer correspondence. arXiv:1008.1595.

42. Kutluhan , Lee Y-J, Taubes CH HF=HM III: Holomorphic curves and the differentialforthe ech/Heegaard Floer correspondence. arXiv:1010.3456.

43. Colin V, Ghiggini P, Honda K (2011) Equivalence of Heegaard Floer homology andembedded contact homology via open book decompositions. Proc Natl Acad SciUSA 108:81008105.

44. Mark TE (2008) Knotted surfaces in 4-manifolds. arXiv:0801.4367.45. Lipshitz R, Ozsvth PS, Thurston DP (2010) Bimodules in bordered Heegaard Floer

homology. arXiv:1003.0598.46. Lipshitz R, Ozsvth PS, Thurston DP (2010) Heegaard Floer homology as morphism

spaces. arXiv:1005.1248.47. Auroux D (2010) Fukaya categories of symmetric products and bordered Heegaard-

Floer homology. arXiv:1001.4323.

48. Penner RC (1987) The decorated Teichmller space of punctured surfaces. CommunMath Phys 113:299339.

49. Zarev R (2009) Bordered Floer homology for sutured manifolds. arXiv:0908.1106.50. Heegaard P (1898) Forstudier til en topologisk teori for de algebraiske Fladers

Sammenhng. PhD thesis, Kbenhavn, Danish. French translation: Sur lAnalysis situs.Soc Math France Bull 44 (1916) pp 161242.

51. Scharlemann M (2002) Heegaard splittings of compact 3-manifolds. Handbook ofGeometric Topology (North-Holland, Amsterdam), arXiv:math/0007144, pp 921953.

52. Lipshitz R (2006) A cylindrical reformulationof HeegaardFloer homology. Geom Topol10:9551097 arXiv:math.SG/0502404.

53. Lipshitz R (2006) A Heegaard-Floer invariant of bordered 3-manifolds. PhD thesis(Stanford University, Palo Alto, CA).

54. Lipshitz R, Ozsvth PS, Thurston DP (2009) Slicing planar grid diagrams: A gentle in-troduction to bordered Heegaard Floer homology. Proceedings of Gkova Geometry-Topology Conference 2008 [Gkova Geometry/Topology Conference (GGT), Gkova],eds A Akbulut, T nder, and RJ Stern (International Press, Somerville, MA), pp 91119arXiv:0810.0695.

55. Lipshitz R, Ozsvth PS, Thurston DP (2010) A faithful linear-categorical action of the

mapping class group of a surface with boundary. arXiv:1012.1032.56. Nstsescu C, Van Oystaeyen F (2004) Methods of graded rings. Lecture Notes in

Mathematics (Springer, Berlin), Vol 1836.57. Bene AJ (2010) A chord diagrammatic presentation of the mapping class group of a

once bordered surface. Geom Dedicata 144:171190 arXiv:0802.2747.58. Bene AJ (2009) Mapping class factorization via fatgraph Nielsen reduction.

arXiv:0904.4067.59. Lipshitz R, Ozsvth PS, Thurston DP (2010) Computing cHF by factoring mapping

classes. arXiv:1010.2550.60. Khovanov M (2000) A categorification of the Jones polynomial. Duke Math J

101:359426.61. Baldwin JA (2008) On the spectral sequence from Khovanov homology to Heegaard

Floer homology. arXiv:0809.3293.62. Lipshitz R, Ozsvth PS, Thurston DP (2010) Bordered Floer homology and the spectral

sequence of a branched double cover I. arXiv:1011.0499.


robert lipshitz,peter s. ozsváth and dylan p. thurston- tour of bordered floer theory

Documents