Open-Closed String Topology via Fat Graphswe introduce the notion of open-closed fat...

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arXiv:math/0606512v1 [math.AT] 20 Jun 2006 Open-Closed String Topology via Fat Graphs Antonio Ram´ ırez September 10, 2018 Abstract Given a smooth closed manifold M with a family {L i } of closed subman- ifolds, we consider the free loop space LM and the spaces PM (L i ,L j ) of open strings (paths γ : [0, 1] M with γ (0) L i (1) L j ). We construct string topology operations resulting in an open-closed TQFT on the family (h (LM ), {h (PM (L i ,L j ))} i,j ∈B ) which extends the known string topol- ogy TQFT on h (LM ). Here, h is a multiplicative generalized homology theory supporting orientations for M and the L i . To construct the operations, we introduce the notion of open-closed fat graph, generalizing fat graphs to the open-closed setting. 1 Introduction The area of string topology began with a construction by M. Chas and D. Sulli- van [CS] of previously undiscovered algebraic structure on the homology of the free loop space of a closed oriented manifold M. This is is the space LM of all continuous maps from the circle to M. Among other results, Chas and Sullivan found that the homology of LM, with its grading suitably shifted, is a graded com- mutative algebra, much like the homology of an oriented manifold does by virtue of Poincar´ e duality. This operation, the string loop product, can be understood by considering the space Map(P,M ) of maps from a pair-of-pants surface P to M, where P is re- garded as a cobordism from a disjoint union of two circles to a single circle. P is a model for the basic interaction in string theory, in which two strings merge to form a single string. There is a diagram LM × LM i ←− Map(P,M ) j LM, (1) * The author was supported by Stanford University, the Pacific Institute of Mathematical Sciences at the University of British Columbia, and the Mathematical Sciences Research Institute throughout the preparation of this article. 1

Transcript of Open-Closed String Topology via Fat Graphswe introduce the notion of open-closed fat...

Page 1: Open-Closed String Topology via Fat Graphswe introduce the notion of open-closed fat graph,generalizing fat graphs to the open-closed setting. 1 Introduction The area of string topology

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Open-Closed String Topology via Fat Graphs

Antonio Ramırez∗

September 10, 2018

Abstract

Given a smooth closed manifoldM with a family{Li} of closed subman-ifolds, we consider the free loop spaceLM and the spacesPM (Li, Lj) ofopen strings (pathsγ : [0, 1]→M with γ(0) ∈ Li, γ(1) ∈ Lj). We constructstring topology operations resulting in anopen-closedTQFT on the family(h∗(LM), {h∗(PM (Li, Lj))}i,j∈B) which extends the known string topol-ogy TQFT on h∗(LM). Here,h∗ is a multiplicative generalized homologytheory supporting orientations forM and theLi. To construct the operations,we introduce the notion ofopen-closed fat graph,generalizing fat graphs tothe open-closed setting.

1 Introduction

The area of string topology began with a construction by M. Chas and D. Sulli-van [CS] of previously undiscovered algebraic structure onthe homology of thefree loop space of a closed oriented manifoldM. This is is the spaceLM of allcontinuous maps from the circle toM. Among other results, Chas and Sullivanfound that the homology ofLM, with its grading suitably shifted, is a graded com-mutative algebra, much like the homology of an oriented manifold does by virtueof Poincare duality.

This operation, thestring loop product,can be understood by considering thespaceMap(P,M) of maps from a pair-of-pants surfaceP to M, whereP is re-garded as a cobordism from a disjoint union of two circles to asingle circle.P is amodel for the basic interaction in string theory, in which two strings merge to forma single string.

There is a diagram

LM × LMi←−Map(P,M)

j−→ LM, (1)

∗The author was supported by Stanford University, the PacificInstitute of Mathematical Sciencesat the University of British Columbia, and the MathematicalSciences Research Institute throughoutthe preparation of this article.

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wherei andj are restriction maps to the incoming and outgoing boundary of P.The product is then constructed as the composition

H∗(LM) ⊗H∗(LM)i!

−→ H∗(Map(P,M))j∗−→ H∗(LM),

wherei! is an umkehr map.Since the spaces involved are infinite-dimensional, the existence ofi! is not

immediate. However, it can be constructed by observing thatP is homotopy equiv-alent to a “figure-eight” spaceΓ = S1 ∨ S1, so that we may replace (1) by

LM × LMi←− Map(Γ,M)

j−→ LM, (2)

where the maps are finite codimension embeddings in an appropriate sense. Usingessentially diagram (2), Chas and Sullivan constructed theoperation using transver-sality of smooth chains inLM. There have been other constructions since then, no-tably including the homotopy-theoretic approach by R. Cohen and J. Jones [CJ02]in which i! is defined using a Pontrjagin-Thom collapse.

R. Cohen and V. Godin [CG04] showed later that the pair-of-pantsP may bereplaced by any oriented connected cobordismΣ between one-manifolds, providedthat it has nonempty outgoing boundary. The result is a family of operations

µΣ : H∗(LM)⊗p → H∗(LM)⊗q

compatible with gluing of cobordisms. This is a form oftopological quantum fieldtheory(TQFT). The construction exploits the fact that any surface with boundarymay be represented as afat graph,which is a finite graphΓ endowed with extradata that determines a surface with boundary havingΓ as a deformation retract.

The appearance of fat graphs reflects a result, due in its various forms to Harer[Har86], Penner [Pen87], and Strebel [Str84], that moduli spaces of punctured Rie-mann surfaces are homotopy equivalent to spaces of metric fat graphs.

1.1 Open-closed string topology

Open-closed string topologygeneralizes string topology by allowing the strings tobe “open,” that is, paths in a manifoldM which need not be loops. The endpoints ofopen strings are constrained to lie in certain distinguished submanifoldsLb ⊆ M .The basic spaces of open strings considered are then of the form PM (L1, L2) ,standing for the space of pathsγ : [0, 1]→ M such thatγ(0) ∈ L1 andγ(1) ∈ L2.These submanifolds (or typically objects with more structure) are calledD-branesin string theory, and we will shorten it to “branes.”

The idea of open string topology was introduced by Sullivan in [Sul04], usingtransversality of smooth chains. The prototype construction, in homotopy-theoreticterms, is as follows. Consider the diagram

PM (L1, L2)× PM (L2, L3)

ev1×ev0��

PM (L1, L2, L3)j //ioo

ev1/2

��

PM (L1, L3)

L2×L2 L2.∆oo

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Here, we definePM (L1, L2, L3) = {γ ∈ PM (L1, L3) : γ(1/2) ∈ L2}. The mapitakes a path inPM (L1, L2, L3) and splits it at the middle into two paths,j is theinclusion, and the vertical maps are evaluation maps. We maydefine acompositionoperation

H∗(PM (L1, L2))⊗H∗(PM (L2, L3))→ H∗(PM (L1, L3))

asj∗ ◦ i!. WhenM andL2 are oriented, the umkehr mapi! exists by the Pontrjagin-Thom collapse, becausei is a finite codimension inclusion in a suitable sense. SeeSection 6 for a general statement.

Our aim is to extend the string topologyTQFT of [CG04] to the open-closed set-ting. Here the one-manifolds considered may have endpoints, which carry boundaryconditions. Thus each endpoint is labeled by an element of a setB indexing a family{Lb}b∈B of branes. Cobordisms between closed one-manifolds are replaced accord-ingly by the natural cobordisms between one-manifolds withlabeled boundary. Wecall the resulting structuresopen-closedTQFTs, orB-TQFTs to make explicit thedependence onB; see Definition 6. The main result is as follows.

Theorem A Let M be a closed smooth manifold, and let{Lb}b∈B be a family ofsmooth closed submanifolds. Suppose thath∗ is a multiplicative generalized ho-mology theory whose coefficient ringh∗(∗) is a graded field, and suppose thatMand theLb are oriented with respect toh∗. Then, the family

(h∗LM, {h∗PM (La, Lb)}a,b∈B)

supports a positive-boundaryB-TQFT structure over the coefficient ring. This ex-tends the known string topologyTQFT onh∗(LM). ✷

The qualifier “positive boundary” means thatB-TQFT operations only exist forthose cobordisms having nonempty outgoing boundary on eachconnected com-ponent. This restriction is also present in the closed case.

The main tools for constructing the operations will beopen-closed fat graphs(Definition 12). Open-closed fat graphs are a generalization of fat graphs. Theyhave a well-defined notion of “string boundary,” which is a graph isomorphic toa one-manifold withB-labeled boundary. They also have a “fattening” operation,which yields an associated “open-closed surface.” These two properties generalizethe corresponding properties of fat graphs.

The content of this paper is essentially part of the author’sPhD thesis [Ram05].We have recently learned of work by E. Harrelson ([Harb], [Hara]) that overlapswith the present paper. We hope that this will be the subject of future collaboration.

We are very thankful to Professor Ralph L. Cohen for his greatpatience andgenerous guidance throughout this project.

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(a) (b)

Figure 1: WithB = {1, 2, 3, . . .}: (a) An open-closed surface. (b) An open-closedcobordism fromI1,2 ⊔ I1,1 ⊔ I5,3 to I1,2 ⊔ S1 ⊔ I1,3 ⊔ I5,1.

2 Open-closed surfaces and cobordisms

Assume given an arbitrary setB of “formal branes,” which we will eventually useto index the actual branes.

Definition 1 A B-labeled one-manifoldC is an oriented one-manifold with bound-ary together with a functionβ : ∂C → B (theB-labeling). An isomorphismbe-tween twoB-labeled one-manifolds is a diffeomorphism that preservesthe orienta-tion and theB-labeling. LetC∗ stand forC with the opposite orientation. Givena, b ∈ B, let Ia,b be a copy of the unit interval, oriented in the direction from0 to 1,with β(0) = a, β(1) = b; note thatI∗a,b is isomorphic toIb,a.

An open-closed surface(with brane labels drawn fromB) is a smooth orientedsurface with boundaryS, together with distinguished embedded one-dimensionalsubmanifolds with boundary∂sS, ∂fS ⊆ ∂S (thestring boundaryandfree bound-ary, respectively) and a locally constant function∂fS → B (the brane labeling)such that:

1. ∂S = ∂sS ∪ ∂fS, and,

2. ∂(∂sS) = ∂(∂fS) = ∂sS ∩ ∂fS.

The restriction ofβ to ∂(∂sS) makes∂sS aB-labeled one-manifold. ✷

Example 1 Figure 1(a) shows an open-closed surface with genus two, fivestringboundary components (only one of which is closed), and sevenfree boundary com-ponents, drawn dotted, with brane labeling inB = {1, 2, . . .}. ✷

Definition 2 GivenB-labeled one-manifoldsC−, C+, an open-closed cobordismfromC− toC+ is aB-surfaceS together with a decomposition∂sS = ∂−S ⊔ ∂+Sand orientation-preserving isomorphismsι− : C− → ∂−S, ι+ : C∗

+ → ∂+S ofB-labeled one-manifolds. We call∂−S, ∂+S the incomingandoutgoingboundary,respectively. ✷

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Example 2 Figure 1(b) shows an open-closed cobordismS, where the incomingboundary is at the bottom and the outgoing boundary at the top. Notice that a singletopological boundary component may contain both incoming and outgoing stringboundary subintervals. ✷

3 Open-closed TQFT

Recall that an ordinaryTQFT consists of a vector spaceV together with a homo-morphismµΣ : V ⊗p → V ⊗q for every oriented cobordismΣ from

p S1 to

q S1.

The mapsµΣ are required to be diffeomorphism-invariant and to satisfynaturalcompatibility conditions with respect to disjoint union and composition of cobor-disms. This can be described succinctly using the language of PROPs: aTQFT isan algebra overCob, which is aPROPof oriented cobordisms between circles (see,for example, Voronov [Vor]). We will introduce an analogousnotion to describeTQFTs involving open and closed strings.

Definition 3 DefineMB to be the free abelian monoid on the symbolsS1 andIa,bfor a, b ∈ B. Define aB-PROP to be a symmetric strict monoidal category havingMB as its monoid of objects. ✷

This can be elaborated as follows. The symmetric monoidal axioms [ML98]imply that each object

x = nS1 +∑

a,b∈B

ma,bIa,b ∈MB

of aB-PROPC carries an actionΣx → AutC (x), where the group

Σx = Σn ×∏

a,b∈B

Σma,b

is a permutation group associated tox. It follows that each setC (x, y) has anaction byΣx on the right and a commuting action byΣy on the left. Moreover, foranyx, y, z ∈ Ob(C ), the composition map◦ : C (x, y) × C (y, z) → C (x, z) isequivariant with respect to theΣx andΣz actions, and it satisfiesp◦ (σq) = (pσ)◦qfor everyp ∈ C (y, z), q ∈ C (x, y), σ ∈ Σy.

Definition 4 An algebraover aB-PROPC is defined to be a monoidal functor fromC to the symmetric monoidal category ofR-modules for some ringR. ✷

In more detail, an algebra overC is specified by giving:

1. aB-family V = (V, {Va,b}a,b ∈ B) of R-modules, and,

2. for everyx, y ∈MB, a mapC (x, y)→ Hom(V (x),V (y)), where we define

V (nS1 +∑

a,b∈B

ma,bIa,b) := V ⊗n ⊗⊗

a,b∈B

V⊗ma,b

a,b .

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These maps are required to satisfy the necessary functoriality and equivariance con-ditions.

Definition 5 Given anyx = nS1 +∑

a,b∈Bma,bIa,b ∈MB, let |x| be theB-labeled

one-manifold(S1)⊔n ⊔⊔

a,b∈B I⊔ma,b

a,b (in particular,|0| = ∅).

Define theB-PROPCobB (thecobordismB-PROP) by letting a morphism fromx to y be an equivalence class of triples(S, L−, L+), where:

1. S is aB-cobordism with∂−S ∼= |x| and∂+S ∼= |y|∗, together with a choiceof parametrization for each string boundary component by eitherI orS1; thisparametrization is orientation-preserving on the incoming components andorientation-reversing on the outgoing components.

2. L− is a functionπ0(∂−S)→ Nwhich restricts to a bijection with{1, . . . , ma,b}on incoming components of typeIa,b and to a bijection with{1, . . . , n} on in-coming components of typeS1. Similarly forL+ : π0∂

+S → N.

3. Two such triples are considered equivalent if they are related by an isomor-phism ofB-cobordisms which preserves the boundary parametrizations andthe orderingsL± of boundary components.

The composition of(S, L−, L+) ∈ CobB(x, y) and(S ′, L′−, L

′+) ∈ CobB(y, z)

is given by(S ∪|y| S ′, L−, L′+), whereS ∪|y| S ′ is theB-cobordism that results from

identifying each outgoing componentc of ∂+S to the unique incoming componentc′ ⊂ ∂−S ′ of the same type such thatL+(c) = L−(c

′). The identification is withrespect to the boundary parametrizations which are part of the data. The monoidalstructure is given by letting(S, L−, L+)⊗ (S ′, L′

−, L′+) := (S ⊔S ′, L−, L+), where

each labelingL± is given by ordering the boundary components ofS ′ after thoseof S in eachB-labeling type. The groupΣy × Σop

x then acts on(S, L−, L+) ∈CobB(x, y) by lettingΣx permute the labelingL− and lettingΣy permuteL+. ✷

Definition 6 An open-closed topological quantum field theory with branesB (whichwe will abbreviateB-TQFT) is an algebra over theB-PROPCobB. ✷

This definition restricts to the usual definition of aTQFT whenB = ∅.

The string topology operations do not yield a wholeB-TQFT, since there are nooperations associated toB-cobordisms to the empty one-manifold. The appropriatevariant is as follows.

Definition 7 Let Cob+B be the subcategory ofCobB consisting ofB-cobordisms in

which every connected component has nonempty outgoing boundary. This categoryinherits aB-PROPstructure fromCobB. A positive boundaryB-TQFT is an algebraoverCob+

B . ✷

Remark 1 We regard this definition chiefly as an ad-hoc device, and we donotclaim that it is the most adequate definition of an open-closed field theory. Forother treatments, we refer the reader to A. D. Lauda and H. Pfeiffer [LP] and to K.Costello [Cos]. ✷

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4 Graphs

Fat graphs have been used extensively to study moduli spacesof punctured andbordered Riemann surfaces. A fat graph consists of a finite graph together witha cyclic ordering of the edges (more precisely, half-edges)incident at every givenvertex. This extra structure specifies a canonical “fattening” of the graph having theform of an oriented surface with boundary with the graph as a deformation retract,in which the punctures correspond to certain cycles of oriented edges.

We will define the notion ofopen-closed fat graph,which is a fat graph withextra structure, which will induce an open-closed surface structure on the fattening.To fix notation, we recall some basic definitions.

Definition 8 A graphΓ = (V (Γ), H(Γ), s, r) consists of

• a setV (Γ) of vertices,

• a setH(Γ) of half-edges,

• a “source” maps : H(Γ) → V (Γ) taking a half-edge to the vertex that itattaches to, and,

• a “reversal” involutionr of H(Γ), having no fixed points, understood to takea half-edge to the opposite half-edge of their common edge.

The edgesE(Γ) are the orbits ofr. We introduce the notationH(v) := s−1(v)for the set of half-edges incident with a given vertex. Thedegree(or valence) of avertexv ∈ V (Γ) is defined to be the number#H(v).

Definet as the compositions ◦ r, which is the “target” map taking a half-edgeto its destination vertex (the source of its reversal).

Say that a graph isdiscreteif it has no edges; any set (in particular,B) can thenbe regarded as a discrete graph, and we will do so without mention.

Given two graphsΓ1,Γ2, define a morphismϕ : Γ1 → Γ2 of graphs to be a pair(ϕV , ϕH) of functionsϕV : V (Γ1)→ V (Γ2), ϕH : H(Γ1)→ H(Γ2)⊔V (Γ2), suchthat

1. ϕV (s(e)) = s(ϕH(e)) for all e ∈ H(Γ1), and,

2. ϕH(r(e)) = r(ϕH(e)) for all e ∈ H(Γ1),

where we extend the structure mapss andr to V (Γ2) as the identity. When unam-biguous, we will refer to both mapsϕV , ϕH simply byϕ. DefineGraph to be thecategory of finite graphs, with these morphisms.

Define asubgraphof a graphΓ to be a graphΓ′ with H(Γ′) ⊆ H(Γ), V (Γ′) ⊆V (Γ) and for which the edge reversal and source maps are given by restriction fromthose ofΓ; we writeΓ′ ⊆ Γ. ✷

All our graphs (possibly excludingB) will be finite.

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Remark 2 A half-edgee ∈ H(Γ) can be equally regarded as an oriented edge,oriented (for definiteness) in the direction that points away from its source vertexs(e). We will use this point of view when convenient. ✷

Given Γ′ ⊆ Γ, we would like a complement graphΓ \ Γ′. This is obtainednaively by removing fromΓ all vertices and edges belonging toΓ′. After this,though, everye ∈ H(Γ) \H(Γ′) attached to a vertexs(e) ∈ V (Γ′) ends up with nosource vertex. We repair the result by formally introducinga new vertex for eachsuche. Precisely:

Definition 9 GivenΓ′ ⊆ Γ, define thecomplementgraphΓ \ Γ′ as follows:

• Vertices: V (Γ \ Γ′) := (V (Γ) \ V (Γ′)) ⊔ δ(Γ \ Γ)′, whereδ(Γ \ Γ′) :=s−1Γ (V (Γ′)) \H(Γ′) is the set of half-edges attached to a vertex ofΓ′ but not

lying in Γ′.

• Half-edges:H(Γ \Γ′) := H(Γ) \H(Γ′), with edge-reversal involution givenby restriction from that ofΓ.

• Incidence of half-edges:DefinesΓ\Γ′(e) :=

{

e, if e ∈ s−1Γ (V (Γ′))

sΓ(e), otherwise. ✷

Remark 3 There is a natural mapΓ \ Γ′ → Γ which is injective on edges but not

necessarily on vertices. It is easy to see that there is a pushout squareΓ Γ\Γ′oo

Γ′

OO

δ(Γ\Γ′)oo

OO

in Graph. Topologically,Γ \ Γ′ is the complement inΓ of an open neighborhoodof Γ′. ✷

Any other graph-related notions we use (such astree, geometric realizationandconnected components) will be assumed well-known.

4.1 Fat graphs

We will use the usual definition of fat graph, except for the restriction on vertices tobe at least trivalent.

Definition 10 A fat graph is a finite graphΓ equipped with a cyclic ordering oneach of the setsH(v), v ∈ V (Γ). We will encode this as a permutationσ of H(Γ)with disjoint cycle decomposition given by theH(v), and we will use the notationσfor the compositionσ ◦ r : H(Γ)→ H(Γ), wherer is edge reversal. Theboundarycyclesof Γ are the cycles ofσ. ✷

It will be useful to represent the boundary ofΓ as an associated graph∂Γ having anatural morphism∂Γ→ Γ.

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Definition 11 Given a fat graphΓ, define∂Γ as follows. Let the vertices of∂Γ beV (∂Γ) := H(Γ). Let the half-edges be given byH(∂Γ) := H(Γ) × {0, 1}, withedge-reversal involutionr(e, i) := (e, 1 − i). Define the attachment of half-edgesby the source maps(e, 0) := e, s(e, 1) := σ(e). Define a morphismι : ∂Γ → Γ byletting ι(e) := s(e) on vertices andι(e, 0) := e, ι(e, 1) := r(e) on half-edges. ✷

It is immediate that the boundary cycles ofΓ are in bijection with the connectedcomponents of∂Γ, each of which is a cyclic graph. Moreover,∂Γ has a naturalorientation induced byσ.

4.2 Open-closed fat graphs

Now we extend the notion of fat graph to include free boundarywith labels inB.

Definition 12 An open-closed fat graph (with brane labels drawn fromB) is a fatgraphΓ together with:

1. a distinguishedfree boundarysubgraph∂fΓ ⊆ ∂Γ such that the restriction ofι : ∂Γ→ Γ is an embeddingιf : ∂fΓ → Γ, and,

2. a labelingβ : ∂fΓ→ B assigning an element ofB to each connected compo-nent of∂fΓ.

Given an open-closed fat graph, we define itsstring boundary∂sΓ as ∂Γ \ ∂fΓ(Definition 9), and we letιs : ∂sΓ→ Γ be the restriction ofι. ✷

Each of∂fΓ and ∂sΓ is necessarily a disjoint union of linear and cyclic graphs.In the case of∂fΓ, the definition includes the possibility of components thatareisolated vertices. Moreover, the two graphs∂fΓ and∂sΓ intersect by definition onlyon the setδ∂sΓ of vertices which are endpoints of linear components of∂sΓ, andthe restriction ofβ then makes|∂sΓ| aB-labeled one-manifold.

Example 3 Consider the fat graph

Γ = ,

with the usual convention that the cyclic ordering at each vertex is counterclock-wise. Its boundary graph is

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∂Γ = .

We can specify an open-closed structure onΓ by choosing a subgraph∂fΓ of ∂Γwith aB-labeling of its components. This can be represented as follows:

1

2

3

,

where the dotted lines and the hollow vertex are∂fΓ and the numbers indicate thelabeling inB = {1, 2, . . .}. This givesΓ the structure of a genus one open-closedfat graph, having string boundary of typeI3,3 ⊔ I1,1 ⊔ S1 ⊔ S1. ✷

Example 4 Another example of an open-closed fat graph, using the same conven-tions, is

2

1

3

.

With a boundary partitioning, this represents the “composition” operation from theintroduction. ✷

As is well-known, given two graphsΓ1,Γ2 with Γ1 a fat graph, a morphismϕ : Γ1 → Γ2 induces a fat graph structure onΓ2 if it is simple. A simple morphismis one that can be written as a composition of “edge collapses,” that is, morphismswhose effect on theB-graph, up to isomorphism, is to collapse a single non-loopedge down to a vertex, leaving the rest of the graph intact. More precisely:

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Definition 13 Call a morphism inGraphϕ : Γ1 → Γ2 simpleif the inverse imagesubgraph of any vertex ofΓ2 is a tree andϕH is injective onϕ−1

H (H(Γ2)). A mor-phismϕ : Γ1 → Γ2 of fat graphs is a simple morphism of graphs which takes thefat graph structure ofΓ1 to that ofΓ2.

Given open-closed fat graphsΓ1, Γ2, define amorphismfrom Γ1 to Γ2 to be amorphismϕ : Γ1 → Γ2 of the underlying fat graphs such that the induced morphism∂Γ1 → ∂Γ2 restricts to a labeling-preserving simple morphism∂fΓ1 → ∂fΓ2. LetFatB be the resulting category of open-closed fat graphs. ✷

Definition 14 A partitioningof an open-closed fat graph is a decomposition of∂sΓas a disjoint union of graphs∂−Γ (incoming)and∂+Γ (outgoing). We denote byι±the restriction to∂±Γ of the morphismι : ∂Γ→ Γ. ✷

5 B-labeled graphs and their mapping spaces

Now we will let the setB index actual branes{Lb}b∈B in a manifoldM . Then, theB-labels carried by an open-closed fat graphΓ (and by its boundary graph) maybe used to specify constraints on maps from|Γ| toM . This can be generalized asfollows.

Definition 15 A B-labeled graph (orB-graph for short) is a diagram

Γθ←− B(Γ)

β−→ B

of graphs in whichθ is an embedding. TheB-graphs form a categoryGraphB, inwhich a morphismϕ : Γ1 → Γ2 of B-graphs is defined to be a pairϕ : Γ1 → Γ2,ϕB : B(Γ1)→ B(Γ2) of morphisms inGraphmaking the diagram

B(Γ1)β1 //

θ1

��

ϕB&&MMMM B

B(Γ2)β2

;;wwww

θ2

��Γ1

ϕ &&MMMM

MM

Γ2

commute. ✷

This is another way of saying that aB-graph is a graphΓ together with a distin-guished familyB(Γ) of disjoint connected subgraphs, each labeled by an elementof B. We put it in this way to clarify the morphisms. We have two examples ofB-graphs.

Example 5 Given an open-closed fat graphΓ, and lettingB(Γ) = ∂fΓ, Γ is natu-

rally aB-graph via the diagramΓιf← ∂fΓ

β−→ B. ✷

Example 6 The string boundary graph∂sΓ of an open-closed fat graph becomes aB-graph by lettingB(∂sΓ) be the discrete subgraphδ∂sΓ consisting of the endpointsof the linear components of∂sΓ. There is a morphismιs : ∂sΓ→ Γ of B-graphs.✷

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Given aB-graphΓ and a family of closed submanifoldsLb of a given smoothclosed manifoldM , we may consider maps from|Γ| to M which respect theB-labeling.

Definition 16 (mapping spaces ofB-graphs) We will callM = (M, {Lb}b∈B) a

B-brane systemin M . Given aB-graphΓθ←− B(Γ)

β−→ B, define[[Γ]]M to be the

space of continuous mapsf : |Γ| → M such that for every connected componentC of B(Γ), the compositionf ◦ |θ| : |C| → M has image inLβ(C) ⊆M . ✷

For a fixedM, this is a contravariant functor fromGraphB to topological spaces.We will denote its action on a morphismϕ by [[ϕ]]M. We will write [[— ]] instead of[[— ]]M when it is clear by context.

6 Generalized Pontrjagin-Thom collapse

The construction of the open-closed string topology operations will make use of a

homology-level umkehr homomorphism for the restriction map [[Γ]]M[[ι−]]−−→ [[∂−Γ]]M,

whereΓ is a partitioned open-closed fat graph. As is the case in the closed case,this homomorphism will arise from a Pontrjagin-Thom collapse. Here, we gatherwithout proof some properties of a generalized form of the Pontrjagin-Thom col-lapse. In the following section, we specialize to the case ofthe maps that we areinterested in.

Basic fact Assume given a homotopy pullback square

Xf //

p

��

Y

q

��P g

// Q

such thatP andQ are smooth closed manifolds, andg is a smooth map. There is astable backwards mapf ! : Σ∞Y+ → XTQ−TP, where by abuse of notationXTQ−TP

stands for the Thom spectrum of the virtual bundlep∗g∗TQ− p∗TP onX. ✷

If h∗ is a generalized homology theory and the manifoldsP andQ areh∗-oriented,we may apply the Thom isomorphism theoremh∗(Xξ)

∼=−→ h∗−dim ξ(X) to obtain a

homomorphismh∗(Y )→ h∗−d(X), also denotedf !, whered = dimQ− dimP .

6.1 Sketch of the construction

This construction and its properties do not seem to be published in this general-ity; however, an upcoming article by R. Cohen and J. Klein [CK] will include fullderivations. We include only a sketch of the construction.

First assume that the square is a pullback square withg an embedding andq alocally trivial fiber bundle. In that case, we may take a tubular neighborhoodUg

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for g and pull it back to an open setUf ⊆ Y . Then,Uf will be a tubular neighbor-hood forf , in the sense that the pair(Uf , Uf \ f(X)) is homeomorphic to the pair(p∗νf , p

∗νf \X), whereνf ց P is the normal bundle ofQ andX includes inp∗νfas the zero section. The desired mapf ! then comes about as usual, by collapsingthe complement ofUf to a point and mapping the rest homeomorphically.

If g is not an embedding, then we choose an embeddingi of P into a high-dimensional Euclidean spaceRN and replaceg andq respectively by(g, i) : P →Q× R

N andq × id : Y × RN → Q× R

N . Different choices of (sufficiently large)N yield target Thom spaces that differ by a suspension, and theresult is a stablemap into the Thom spectrum.

Finally, if q is not locally trivial, it may be replaced by a Serre fibrationusingthe standard mapping path space construction (see, e.g., [May99]). This fibrationhas sufficient structure to allow a tubular neighborhood forg to be lifted to one forf using parallel transport along paths.

6.2 Naturality properties

The generalized Pontrjagin-Thom collapse satisfies the following two naturalityproperties.

Proposition 1 (Functoriality) Consider a diagram

Xf1 //

p

��

Yf2 //

q

��

Z

r

��P g1

// Q g2// R,

where thegi are smooth maps betweenh∗-oriented closed manifolds and bothsquares are homotopy pullbacks.

Then, the umkehr homomorphisms satisfy(f2 ◦ f1)! = f !

1 ◦ f!2. ✷

Proposition 2 (Compatibility with induced maps on homology) Consider a com-mutative diagram

P1>>p1

}}}}

}}}}

g1 //

s

��

Q1>>

q1}}

}}}}

}

��

X1

u

��

f1

// Y1

v

��

P2>>

p2}}

}}}}

}}

g2 // Q2>>

q2}}

}}}}

}

X2 f2

// Y2

such that eachpi is a homotopy pullback ofqi via gi, thePi andQi areh∗-orientedmanifolds, and the virtual bundless∗(g∗2TQ2 − TP2) andg∗1TQ1 − TP1 are stably

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equivalent. Then the diagram

h∗(X1)

u∗

��

h∗(X2)

v∗

��

f !1oo

h∗(Y1) h∗(Y2)f !2

oo

commutes. ✷

7 Umkehr maps induced by morphisms ofB-graphs

We need a good family of morphismsϕ : Γ1 → Γ2 of B-graphs for which theinduced map[[ϕ]] : [[Γ2]]M → [[Γ1]]M admits a homology umkehr map.

First, observe that pushout squares inGraphB become pullback squares of map-ping spaces after applying[[— ]]M. The proof is standard and we omit it.

Next, we identify a class of morphismsϕ : Γ1 → Γ2 in GraphB with the propertythat the induced map[[ϕ]]M fibers naturally over a smooth map of manifolds. Forthis, we introduce the following construction.

Definition 17 Given aB-graphΓ define itsvertexB-graphV(Γ) as theB-graphthat results from removing all the edges ofΓ, keeping only the vertices and their

labels. Formally, ifΓ is given by a diagramΓθ←− B(Γ)

β−→ B, defineV(Γ) by the

diagram

V (Γ)θ|V (B(Γ))←−−−−− V (B(Γ))

β|V (B(Γ))−−−−−→ B.

This defines a functorGraphB → GraphB having a natural inclusionV(Γ) → Γ.

Proposition 3 Let ϕ : Γ1 → Γ2 be a morphism inGraphB such that

1. ϕH is a bijectionH(Γ1)→ H(Γ2), and,

2. ϕ carries unlabeled edges to unlabeled edges; that is,ϕ takes edges not in theimage ofB(Γ1) to edges not in the image ofB(Γ2).

Then, the diagramΓ1

ϕ //Γ2

V(Γ1)

OO

V(ϕ)//V(Γ2)

OOis a pushout square inGraphB. ✷

Proof The hypothesis says that, up to isomorphism compatible withϕ, Γ2 is ob-tained fromΓ1 by identifying together vertices with a common preimage andthenadjoining (possibly labeled) isolated vertices corresponding toV (Γ2) \ ϕ(V (Γ1)).But this is equivalent to this square being a pushout. �

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Note that[[V(Γ)]]M ∼=∏

v∈V (Γ) Lv, whereLv is the submanifold ofM corre-sponding to the unique label carried by the vertexv, or M if v is unlabeled. Inparticular,[[V(Γ)]]M is a smooth manifold. In addition, ifϕ : Γ1 → Γ2 is a mor-phism inGraphB, thenϕ induces a smooth map[[V(Γ2)]]M → [[V(Γ1)]]M; in fact,this map is a cartesian product of coordinate projections ofthe formL × L′

։ Land diagonal inclusions of the formsL → Lp ×M q andM → Mp.

In view of this and Proposition 3, we can make the following definition.

Definition 18 Let ϕ : Γ1 → Γ2 be a morphism ofB-graphs satisfying the hypoth-esis of Proposition 3. Suppose given aB-brane systemM = (M, {Lb ⊆M}b∈B)which ish∗-oriented; that is, such thatM and each of theLb is oriented with respectto a multiplicative homology theoryh∗.

Define[[ϕ]]!M : h∗([[Γ1]]M)→ h∗([[Γ2]]M)

as the umkehr homomorphism associated by the generalized Pontrjagin-Thom col-lapse to the pullback square

[[Γ1]]M oo [[ϕ]]M[[Γ2]]M

[[V(Γ1)]]M��

oo [[V(Γ2)]]M.��

This uses that the normal bundle ofL inM ish∗-oriented if bothTM andTL are.✷

7.1 Enlarging the class of morphisms: the categoryGraph!B

Thus a morphism ofB-graphs which is a bijection on edges and preserves unlabelededges induces an umkehr map in a natural way. However, we willneed umkehrhomomorphisms for a larger class ofB-graph morphisms:

Definition 19 Let Graph!B be the subcategory ofGraphB consisting of morphismsϕ : Γ1 → Γ2 of B-graphs such that:

1. ϕH induces an injectionH(Γ1)→ H(Γ2) of half-edges, and,

2. ϕ carries unlabeled edges to unlabeled edges in the sense of Proposition 3.✷

We will construct the desired homomorphisms by showing thatmorphisms inGraph!Bcan be naturally factored up to homotopy into morphisms satisfying the hypothesisof Proposition 3.

Let ϕ : Γ1 → Γ2 be a morphism inGraph!B. Let

Ξϕ =⊔

e∈E(Γ2)\ϕ(E(Γ1))

e,

where eache is aB-graph consisting of two vertices joined by a single edge, whichis labeled by the same label carried bye in Γ2, or unlabeled ife is unlabeled. Let

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ξϕ be theB-graph obtained fromΞϕ by collapsing eache to a vertex carrying thesame label, if any. Then,ϕ decomposes as

Γ1ϕ′

−→ Γ1 ⊔ ξϕϕ′′

←− Γ1 ⊔ Ξϕϕ′′′

−−→ Γ2,

whereϕ′ is the natural inclusion,ϕ′′ extends the defining quotient mapΞϕ ։ ξϕ bythe identity onΓ1, andϕ′′′ is the morphism which extendsϕ by takinge to e.

The morphismsϕ′ andϕ′′′ are readily seen to satisfy the hypothesis of Proposi-tion 3. Moreover,ϕ′′ induces a homotopy equivalence[[Γ ⊔ ξϕ]]M → [[Γ ⊔ Ξϕ]]M,and[[ϕ]]M is homotopic to[[ϕ′]]M ◦ [[ϕ

′′]]−1M ◦ [[ϕ

′′′]]M.

Example 7 We illustrate this for simplicity whenB(Γ1) = B(Γ2) = ∅. Consider

the morphismϕ−→ , which clearly lies inGraph!B. In this case, the

factorization takes the form

� � ϕ′

// ϕ′′

oooo ϕ′′′

// // .

We may now make the following definition.

Definition 20 Given a morphismϕ ∈ Graph!B and anh∗-orientedB-brane systemM we define theumkehr homomorphism

[[ϕ]]!M : h∗([[Γ1]]M)→ h∗([[Γ2]]M)

as the composition[[ϕ′′′]]!M ◦ ([[ϕ′′]]M)∗ ◦ [[ϕ

′]]!M, where the homomorphisms[[ϕ′]]!Mand[[ϕ′′′]]!M are given by Definition 18. ✷

Remark 4 Sinceξϕ is discrete, the map[[ϕ′]]M is a projection[[Γ1]]M×N → [[Γ1]]Mwith N a closedh∗-oriented manifold. Its corresponding umkehr map is simplycrossing with the fundamental class ofN . In these terms, the map[[ϕ′′]]M is theinclusion[[Γ1]]M×N → [[Γ1]]M×PN , wherePN stands for the space of arbitrarycontinuous paths inN , withN included as the constant paths. ✷

Finally, we observe that this homomorphism behaves well under the appropriatenotion of simple morphism forB-graphs, as well as under pushouts ofB-graphembeddings.

Definition 21 Say that a morphismϕ = (ϕ, ϕB) : Γ1 → Γ2 of B-graphs issimpleif each ofϕ : Γ1 → Γ2 andϕB : B(Γ1)→ B(Γ2) is a simple morphism of graphs.✷

A morphism of open-closed fat graphs is in particular a simple morphism of theunderlyingB-graphs, and it induces a simple morphism of the associated stringboundaryB-graphs.

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Remark 5 A simple morphism ofB-graphs is of course a composition of edge col-lapses. However, a non-loop edge may be collapsed only if it is “wholly monochro-matic,” that is, if the subgraph consisting of the edge and its two endpoints is eitherdisjoint fromB(Γ) or contained inB(Γ). ✷

Proposition 4 LetΓ1

α //γ1 ��

Γ2

γ2��Γ1

β// Γ2

be a commutative diagram inGraphB. Suppose that

α andβ lie in Graph!B, and that either:

1. the diagram is a pushout andα is an embedding, or,

2. theγi are simple.

Then, the diagram

h∗([[Γ1]]M)[[α]]! // h∗([[Γ2]]M)

h∗([[Γ1]]M)[[β]]!

//

[[γ1]]∗

OO

h∗([[Γ2]]M)

[[γ2]]∗

OO(3)

commutes. ✷

Proof Case 1:the diagram is a pushout andα is an embedding. In this case, theγiinduce a commutative diagram

Γ1

��

α′

// Γ1 ⊔ ξ

��

Γ1 ⊔ Ξ

��

α′′

oo α′′′

// Γ2

��

Γ1

β′

// Γ1 ⊔ ξ Γ1 ⊔ Ξβ′′

oo β′′′

// Γ2,

whereξ = ξα ∼= ξβ, Ξ = Ξα ∼= Ξβ, and in which the leftmost and rightmostsquares are pushouts inGraphB. It follows that these squares become pullbacksupon applying[[— ]]M. Moreover, each of these pullbacks fibers, in the sense ofProposition 1, over the corresponding pullback square of manifolds obtained byapplying[[V(—)]]M. The result follows because the hypothesis on normal bundlesof Proposition 1 is easily verified to hold for the latter squares.

Case 2:γ1 andγ2 are simple. By induction we can assume thatγ2 collapses asingle edge. If the collapsed edge is in the image ofα, the result reduces to case 1.Assume then thatγ2 collapses a single edgee which is not in the image ofα. Wehave up to isomorphism thatΓ2 = Γ2/e andΓ1 = Γ1. Let us change notation forclarity, writing Ξ andξ for Ξβ, ξβ respectively. We clearly haveΞα ∼= Ξ ⊔ e andξα ∼= ξ ⊔ ∗e, where∗e stands for a one-vertex graph carrying the same label ase, ifany. Also writeγ for γ2; γ1 becomes the identity in this case.

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We have the diagram

Γ1α′

//

β′

��<<

<<<

<<<<

<<<

<<<

Γ1 ⊔ ξ ⊔ ∗e Γ1 ⊔ Ξ ⊔ eα′′

oo α′′′

// Γ2

γ

��Γ1 ⊔ ξ Γ1 ⊔ Ξ

β′′

ooβ′′′

// Γ2/e

and we are to show that[[α′′′]]! ◦ [[α′′]]∗ ◦ [[α′]]! = [[γ]]∗ ◦ [[β

′′′]]! ◦ [[β ′′]]∗ ◦ [[β′]]!. For this,

complete the diagram as follows:

Γ1α′

//

β′

��88

88

88

88

88

88

88

88

8Γ1 ⊔ ξ ⊔ ∗e Γ1 ⊔ Ξ ⊔ e

α′′

oo α′′′

//

γ′

��

Γ2

γ

��

Γ1 ⊔ Ξ ⊔ ∗e

ϕ

hhPPPPPPPPPPPP

ψ

&&LLLLLLLLLL

Γ1 ⊔ ξ

δ

OO

Γ1 ⊔ Ξβ′′

ooβ′′′

//

δ′

OO

Γ2/e

Here,γ′ collapsese to ∗e, ϕ extendsβ ′′ by the identity on∗e, ψ extendsβ ′′′ bymapping∗e to the vertex to whiche is collapsed, andδ and δ′ are the obviousinclusions. Note thatδ, δ′ andψ satisfy the hypothesis of Definition 18.

By Remark 5 above,e is wholly monochromatic; this ensures that[[δ]] and[[δ′]]have the same normal bundle data (in the sense of Proposition2) when [[— ]] isapplied (the stable normal bundle is a pullback of−TL for both, whereL is eitherM or the brane submanifold corresponding to the label carriedby e). This is alsotrue for the pair[[α′′′]], [[ψ]].

The equality of the two homomorphisms then follows by applying the naturalityproperties:

[[γ]]∗ ◦ [[β′′′]]! ◦ [[β ′′]]∗ ◦ [[β

′]]! = ([[γ]]∗ ◦ [[ψ]]!) ◦ [[δ′]]! ◦ [[β ′′]]∗ ◦ [[β

′]]!

= [[α′′′]]! ◦ [[γ′]]∗ ◦ ([[δ′]]! ◦ [[β ′′]]∗) ◦ [[β

′]]!

= [[α′′′]]! ◦ [[γ′]]∗ ◦ [[ϕ]]∗ ◦ ([[δ]]! ◦ [[β ′]]!)

= [[α′′′]]! ◦ ([[γ′]]∗ ◦ [[ϕ]]∗) ◦ [[α′]]!

= [[α′′′]]! ◦ [[α′′]]∗ ◦ [[α′]]!,

as desired. �

8 Definition of the operations

With the constructions of the previous section, the definition of the string topologyoperation associated to an open-closed fat graph is straightforward.

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Definition 22 Say that a partitioning of an open-closed fat graphΓ is admissibleifthe morphismι− : ∂−Γ → Γ lies in Graph!B. If this is the case, we say thatΓ iswell-partitioned.

Define a categoryFat⋆B as follows. An object ofFat⋆B is a well-partitioned open-closed fat graph. The morphisms fromΓ1 to Γ2 are the morphismsϕ : Γ1 → Γ2 ofopen-closed fat graphs which respect the partitioning, in the sense that the inducedmorphism∂sΓ1 → ∂sΓ2 takes∂−Γ1 to ∂−Γ2 and∂+Γ1 to ∂+Γ2. ✷

Definition 23 Let M be anh∗-orientedB-brane system, and letΓ be a well-partitioned open-closed fat graph. Define the homomorphismΓ∗ as

Γ∗ := [[ι+]]∗ ◦ [[ι−]]! : h∗([[∂

−Γ]]M)→ h∗([[∂+Γ]]M).

This homomorphism is theopen-closed string topology operationcorrespondingto Γ. ✷

The operationsΓ∗ are invariant under morphisms of open-closed fat graphs:

Proposition 5 If Γ1,Γ2 ∈ Fat⋆B and there is a morphismϕ : Γ1 → Γ2, then(Γ2)∗ =[[∂+ϕ]]−1

∗ ◦ (Γ1)∗ ◦ [[∂−ϕ]]∗. ✷

Proof In the commutative diagram

∂+Γ1

∂+ϕ //

ι1+��

∂+Γ2

ι2+��

Γ1ϕ // Γ2

∂−Γ1

ι1−

OO

∂−ϕ // ∂−Γ2,

ι2−

OO

of B-graphs, the morphismsι1−, ι2− lie in Graph!B and the horizontal morphisms are

simple. Then, with an application of Proposition 4, we have that

(Γ2)∗ = [[ι2+]]∗ ◦ [[ι2−]]

!

= ([[∂+ϕ]]−1∗ ◦ [[ι

1+]]∗ ◦ [[ϕ]]∗) ◦ ([[ϕ]]

−1∗ ◦ [[ι

1−]]

! ◦ [[∂−ϕ]]∗)= [[∂+ϕ]]−1

∗ ◦ [[ι1+]]∗ ◦ [[ι

1−]]

! ◦ [[∂−ϕ]]∗= [[∂+ϕ]]−1

∗ ◦ (Γ1)∗ ◦ [[∂−ϕ]]∗,

as desired �

9 Gluing

Now we describe the combinatorial counterpart to gluing of cobordisms.Suppose givenΓ1,Γ2 ∈ Fat⋆B together with isomorphisms∂+Γ1

γ1←− ∆

γ2−→

∂−Γ2, where∆ is aB-graph. Assume thatγ2 ◦ γ−11 is orientation-reversing. We

may construct aB-graphΓ1#Γ2 by identifying the outgoing boundary ofΓ1 with

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the incoming boundary ofΓ2 according to their common identification with∆.More precisely, we can defineΓ1#Γ2 by the pushout diagram

Γ1#Γ2 Γ2oo

Γ1

OO

∆,α1

oo

α2

OO (4)

in GraphB, whereα1 = ι1− ◦ γ1 andα2 = ι2+ ◦ γ2.While the pushoutΓ1#Γ2 exists, it does not necessarily inherit an open-closed

fat graph structure having the correct isomorphism type. For that, we need an extracondition on the partitioning:

Definition 24 Say that a partitioning of an open-closed fat graphΓ ∈ Fat⋆B is veryadmissibleif the inclusionι− : ∂−Γ → Γ is an embedding ofB-graphs (in thatcase,Γ is very well-partitioned). ✷

Remark 6 This condition is somewhat analogous to the chord diagram constraintof [CG04]. However, we don’t require the complement of the incoming boundaryto be a forest; we may do this because the factorization described in Section 7.1makes it unnecessary to collapse this complement. ✷

Lemma 6 Suppose givenΓ1,Γ2 ∈ Fat⋆B, with Γ2 very well-partitioned. TheB-graphΓ1#Γ2, as defined by pushout diagram (4), has an induced partitioned open-closed fat graph structure with∂−Γ1

∼= ∂−(Γ1#Γ2), ∂+Γ2∼= ∂+(Γ1#Γ2) asB-

graphs. ✷

Proof To describe the fat graph structure permutationσ of Γ1#Γ2, we may firstdescribe its associated permutationσ (Definition 10), and then verify that, if wedefineσ asσ ◦ r, the result is a fat graph.

Describingσ is equivalent to constructing a graph∂Γ and a morphismι : ∂Γ→Γ in Graph, taking edges to edges, such that:

1. ∂Γ is a disjoint union of cyclic graphs, each with a chosen orientation, and,

2. every oriented edgee ∈ H(Γ) is the orientation-preserving image of exactlyone edge in∂Γ.

In the process, we will construct aB-labeled free boundary subgraph∂f(Γ1#Γ2) ⊆∂(Γ1#Γ2) and the partitioning∂s(Γ1#Γ2) = ∂−(Γ1#Γ2) ⊔ ∂

+(Γ1#Γ2).Since pushouts inGraphB can be constructed setwise, we have thatB(Γ1#Γ2) =

B(Γ1) ∪B(∆) B(Γ2). The graphB(Γ1#Γ2) is a disjoint union of cyclic and linear(possibly degenerate) graphs because each of theB(Γi) is by definition one suchgraph, and the discrete graphB(∆) identifies pairs of endpoints of linear compo-nents. Thus we may identify the discrete graphδB(Γ1#Γ2) as consisting of the

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endpoints of the linear components ofB(Γ1#Γ2), with multiplicity two for the de-generate ones. Define∂(Γ1#Γ2) by the pushout

∂(Γ1#Γ2) ∂−Γ1 ⊔ ∂+Γ2

oo

B(Γ1#Γ2)

OO

δB(Γ1#Γ2).oo

OO

(this is a case of the pushout in Remark 3). It is easy to see that the resulting graphis a disjoint union of cyclic graphs. It inherits a preferredorientation from theorientations on the∂Γi, containsB(Γ1#Γ2) as a subgraph, and satisfies∂(Γ1#Γ2)\B(Γ1#Γ2) ∼= ∂−Γ1 ⊔ ∂

+Γ2.It remains to verify that the permutationσ := σ ◦ r implicit in this construction

has exactly one disjoint cycleH(v) per vertex ofΓ1#Γ2. Here, we use the factthatΓ2 is very well partitioned. This implies that the mapV (Γ1) → V (Γ1#Γ2) isinjective, and hence that the vertices of the pushoutΓ1#Γ2 are of only two types:

1. verticesu corresponding to thosev ∈ V (Γ2) not in the image ofV (∆), and,

2. verticesu corresponding tow ∈ V (Γ1), resulting as the identification ofwwith each of the verticesα2(α

−11 (w)) ⊆ V (Γ2).

In the first case, we haveH(u) ∼= H(v), and the cyclic ordering is given by thecyclic ordering inH(v).

In the second case, for eachv ∈ α2(α−11 (w)),H(v) has a preferred linear order-

ing given by opening its cyclic ordering at the uniquee ∈ H(v) lying in ∂−(Γ2); it isunique since otherwise∂−Γ2 → Γ2 would not be injective. Then,H(u) is obtainedfromH(w) by inserting the half-edgesH(v) in this linear order in spaces betweenhalf-edges inH(w) determined by the image of∆ in Γ1. This yields a cyclic order-ing onH(u) given by “cyclically splicing” the linear orders on theH(v) into thecyclic ordering onH(w).

These cyclic orders are directly seen to agree with the ones induced by∂(Γ1#Γ2)above. �

Remark 7 Verifying thatσ gives a cyclic ordering to eachH(u), u ∈ V (Γ1#Γ2) isa necessary step. IfΓ2 is notveryadmissible, then, while we may still construct therequisite∂(Γ1#Γ2) → Γ1#Γ2, it may induce a permutationσ for which a singleH(u) contains multiple cycles ofσ. ✷

9.1 Compatibility with gluing

We will show that the open-closed string operations are compatible with the gluingconstruction, in the sense that(Γ1#Γ2)∗ = (Γ2)∗ ◦ (Γ1)∗, appropriately interpreted,whenever theΓi are very well-partitioned open-closed fat graphs that fit ina gluingsetting.

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Proposition 7 Let gluing data(Γ1,Γ2, ∂+Γ1

γ1←− ∆

γ2−→ ∂−Γ2) be given withΓ2

very well-partitioned, and let be formed accordingly. Then,

(Γ1#Γ2)∗ = (Γ2)∗ ◦ [[γ−11 ◦ γ2]]∗ ◦ (Γ1)∗. ✷

Proof We have the diagram

∂−Γ1

ι1− //

ι− $$IIIIIIIIIΓ1� _

δ1��

∆α1oo

� _

α2

��Γ1#Γ2 Γ2

δ2oo

∂+Γ2.

ι+

ddJJJJJJJJJι2+

OO

where the upper-right square is the defining pushout. Proposition 4 (case 1) impliesthat this square has the property

[[α2]]! ◦ [[α1]]∗ = [[δ2]]∗ ◦ [[δ1]]

! (5)

in homology. Then we have:

(Γ1#Γ2)∗ = [[ι+]]∗ ◦ [[ι−]]!

= [[ι2+]]∗ ◦ [[δ2]]∗ ◦ [[δ1]]! ◦ [[ι1−]]

!

= [[ι2+]]∗ ◦ [[α2]]! ◦ [[α1]]∗ ◦ [[ι

1−]]

!

= [[ι2+]]∗ ◦ [[ι2− ◦ γ2]]

! ◦ [[ι1+ ◦ γ1]]∗ ◦ [[ι1−]]

!

= [[ι2+]]∗ ◦ [[ι2−]]

! ◦ [[γ2]]! ◦ [[γ1]]∗ ◦ [[ι

1+]]∗ ◦ [[ι

1−]]

!

= (Γ2)∗ ◦ [[γ−11 ◦ γ2]]∗ ◦ (Γ1)∗. �

10 The string topologyB-TQFT

Let h∗ be a multiplicative homology theory whose coefficient ringh∗ is a gradedfield, that is, a graded ring in which all nonzero homogeneous elements are invert-ible. Given anh∗-orientedB-brane systemM = (M, {Lb}b∈B), we have aB-family

VM = (h∗LM, {h∗PM (La, Lb)}a,b∈B)

over the coefficient ring ofh∗. HereLM is the free loop space ofM . The con-straints onh∗ have the effect that there is a product maph∗(X) ⊗h∗ h∗(Y ) →h∗(X × Y ) (becauseh∗ is multiplicative) which is moreover an isomorphism ofgradedh∗-modules (because the graded field condition onh∗ makes the Kunnethspectral sequence collapse).

In this section, we will describe the positive-boundaryB-TQFT structure onVM

arising from the open-closed string topology operations from Definition 23.

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Definition 25 Recall that, given a fat graphΓ, there is an associated oriented sur-faceS(Γ) having|Γ| as a deformation retract, and having an identification∂S ∼=|∂Γ|. If Γ is additionally an open-closed fat graph,S becomes naturally an open-closed surface by decreeing the image of|∂fΓ| in ∂S to be the free boundary,1 withthe inducedB-labeling. A partitioning ofΓ then makesS(Γ) into an open-closedcobordism. ✷

Proposition 8 • If two partitioned open-closed fat graphs are related by a mor-phism inFat⋆B, then their corresponding open-closed cobordisms are isomor-phic

• Gluing of partitioned open-closed fat graphs translates, up to isomorphism,into gluing of the corresponding open-closed cobordisms. ✷

Proof sketch The first statement is clear from the corresponding result for fat graphs.For the second statement, we may observe that the defining pushout diagram (4) forgluing realizes to a homotopy pushout square equivalent to the counterpart dia-gram that arises when gluing cobordisms; this determines the Euler characteristicof S(Γ1#Γ2). By a comparison of boundary components, also the genus andB-labeling structure are seen to correspond. �

We will make use of the following lemma, whose proof we defer to the appendix.

Lemma 9 The connected components ofFat⋆B are in one-to-one correspondencewith isomorphism types of open-closed cobordismsS for which ∂+S intersectsevery connected component ofS. ✷

Using the notation of Definition 5, we first show how a morphismS : x− → x+in Cob+

B (with x± ∈ MB) produces a homomorphismµS : VM(x−) → VM(x+).We may assume thatS is connected, and induce the remaining operations by tensorproduct.

By Lemma 9, we can represent the morphismS by a well-partitioned open-closed fat graphΓ ∈ Fat⋆B having|∂−Γ| ∼= |x−| and|∂+Γ| ∼= |x+|∗ asB-labeledone-manifolds, together with:

1. a choice of orientation-preserving parametrization of each component of∂Γ,and,

2. a choiceL− of linear orderings of the incoming boundary components ineachB-labeling type, and a similar choice ofL+ for the outgoing boundarycomponents.

A boundary parametrization may be specified by a starting point on each closedstring boundary component; it is then determined by starting at that point and

1Strictly speaking we take a small closed neighborhood of|∂fΓ| in ∂S in order to deal withdegenerate components of∂fΓ.

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parameterizing each edge in a piecewise-linear fashion at constant speed. Stringboundary intervals have a natural parametrization in the same way.

The parametrizations and linear orderings (together with our restrictions onh∗)give isomorphismsh∗(∂±Γ) ∼= VM(x±), and therefore the operationΓ∗ defines ahomomorphismVM(x−)→ VM(x+), which we callµS.

Lemma 10 The homomorphismµS is well-defined; that is, it is independent ofthe choice of representing well-partitioned open-closed fat graph and the choice ofboundary parameterization. ✷

Proof We will temporarily useµΓ for the operation defined with a particular choiceof boundary-parametrized open-closed fat graphΓ. Define a categorymFat⋆B inwhich an object is a well-partitioned open-closed fat graphtogether with a choiceof starting point in each string boundary cycle, with morphisms being those mor-phisms of partitioned open-closed fat graphs which preserve the choices of start-ing points. We can argue as in [CG04] to show that the obvious forgetful functormFat⋆B → Fat⋆B is a torus fibration on each component, and thus the connectedcomponents ofmFat⋆B are also in one-to-one correspondence with isomorphismtypes of open-closed cobordisms. Thus, it is enough to show that ifϕ : Γ1 → Γ2 isa morphism inmFat⋆B with Γ1 andΓ2, thenµΓ1 = µΓ2 .

Consider the diagrams

|∂−Γ1|

|∂−ϕ|

��

|∂+Γ1|

|∂+ϕ|

��

|x−|

γ−1∼=

;;wwwwwwwww

γ−2

∼=

##GGGG

GGGG

G|x+|

γ+1∼=

ccHHHHHHHHH

γ+2

∼=

{{vvvv

vvvv

v

|∂−Γ2| |∂+Γ2|.

The γ±i are isomorphisms ofB-labeled one-manifolds induced by the choices ofstarting points on the boundary cycles. The two triangles commute up to homotopyrelative to the boundary. It follows that, when we apply the functor [[— ]]M, theinduced homomorphisms on homology satisfy

[[γ±2 ]]∗ = [[γ±1 ]]∗ ◦ [[∂±ϕ]]∗.

By definition, we have

µΓ2 = [[γ+2 ]]∗ ◦ (Γ2)∗ ◦ [[γ−2 ]]

−1∗ ,

and therefore

µΓ2 = [[γ+1 ]]∗ ◦ [[∂+ϕ]]∗ ◦ (Γ2)∗ ◦ [[∂

−ϕ]]−1∗ ◦ [[γ

−1 ]]

−1∗ .

By Proposition 5, we have that(Γ1)∗ = [[∂+ϕ]]∗ ◦ (Γ2)∗ ◦ [[∂−ϕ]]−1

∗ , and thus

µΓ2 = [[γ+1 ]]∗ ◦ (Γ1)∗ ◦ [[γ−1 ]]

−1∗ .

But this is equal toµΓ1 by definition. �

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Lemma 11 The assignmentS 7→ µS is functorial. ✷

Proof This is directly implied by Proposition 7. �

As a corollary, we have our theorem.

Theorem A If h∗ is a multiplicative generalized homology theory for which thecoefficient ring is a graded field, then, lettingVM be theB-family

(h∗LM, {h∗PM (La, Lb)}a,b∈B)

overh∗(∗), there is a positive-boundaryB-TQFT structure onVM which extends theknown positive-boundary string topologyTQFT structure onh∗(LM). ✷

A Connected components ofFat⋆BHere we turn to the proof of Lemma 9. We will show that in each connected compo-nent ofFat⋆B there is, after making a few controlled choices, a graph in a particular“normal” form, and that this form is uniquely determined, upto these choices, bythe isomorphism type of its associated open-closed cobordism. This will show thatthe connected components ofFat⋆B are as desired. Our proof of this relies heavilyon the corresponding construction for chord diagrams in theclosed string case, aspresented by R. Cohen and V. Godin [CG04].

Remark 8 We include this proof for completeness, but it will be superseded bya result stating that the categoryFat⋆B realizes to a space homotopy equivalent toan appropriate moduli space of open-closed Riemann surfaces; this will recoverLemma 9 upon applyingπ0. ✷

A.1 Isomorphism invariants of open-closed surfaces

There are a few invariants which together determine the isomorphism type of aconnected open-closed surface. They are as follows:

• The genusg of S.

• The subset∂S1S of π0(∂sS) consisting of closed string boundary components.

• The subset∂IS of π0(∂sS) consisting of string boundary intervals.

• The functionξ : ∂IS → B assigning to a string boundary interval the label ofits final endpoint, where “final” is with respect to the orientation.

• A permutationψ ∈ Sym(∂IS) taking a string boundary intervalc to the onefollowing c in the same boundary component, in the direction induced by theorientation.

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• The subset∂wS of π0(∂fS) consisting of closed components (“windows”).

• TheB-labelingβ : ∂wS → B induced fromβ : ∂fS → B

If S is an open-closed cobordism, we can further identify:

• The partitions∂S1S = ∂−S1S ⊔ ∂

+S1S, ∂IS = ∂−I S ⊔ ∂

+I S.

Denote byX(S) the tuple

X(S) := (g, ∂S1S, ∂IS, ξ, ψ, ∂wS, β).

For an open-closed cobordism, denote byY (S) the tuple

Y (S) := (g, ∂S1S, ∂IS, ξ, ψ, ∂wS, β, ∂±S1S, ∂

±I S).

Definition 26 In the absence of an open-closed surface or cobordismS, we con-sider tuples

(g, ∂S1, ∂I , ξ, ψ, ∂w, β)

whereg ≥ 0 is an integer,∂S1 , ∂I and∂w are arbitrary finite sets,ξ : ∂I → B andβ : ∂w → B are arbitrary functions, andψ is a permutation of∂I . We call theseopen-closed data tuples. A partitionedopen-closed data tuple is one of the form(X, ∂±

S1 , ∂±I ), whereX = (g, ∂S1, ∂I , . . .) is an open-closed data tuple and the∂±

S1 ,∂±I are partitionings of∂S1 , ∂I .

Two such tuples (partitioned or not) areisomorphicif they have the sameg andthere are bijections of the corresponding sets which preserve theB-labelings, thepermutationψ, and the partitions if present. ✷

Proposition 12 1. Two open-closed surfaces (resp. cobordisms) are isomorphicif and only if their data tuples (resp. partitioned data tuples) are isomorphic.

2. Given an arbitrary (resp. partitioned) open-closed datatuple, there is an open-closed surfaceS withX(S) isomorphic to it (resp. an open-closed cobordismwith Y (S) isomorphic to it). ✷

Proof sketch This is mostly clear, so we only provide a sketch of the constructionfor the second statement, which should make the first statement obvious.

Choose an ordinary oriented surfaceS of genusg having its boundary compo-nents in bijection with∂S1 ⊔ ∂I/ 〈ψ〉 ⊔ ∂w. Label each entire component corre-sponding tow ∈ ∂w by β(w), making it part of the free boundary. Given a cyclec = (x1, . . . , xk) of ξ, let Ac ⊆ ∂S1 be the corresponding boundary component.Choose an embedding{x1, . . . , xk} × [0, 1] → Ac which is orientation-preservingsuch that the cyclic order of thexi induced from the orientation ofAc coincideswith the cyclic order given byξ. Declare the image of this embedding to belong tothe string boundary. Decree the components ofAc \ ({x1, . . . , xk} × [0, 1]) to be inthe free boundary, and label them according to the rule that the component comingafter{x} × [0, 1] in the cyclic order carries the labelξ(x). The boundary compo-nents corresponding to∂S1 are decreed to be part of the string boundary. We omitthe easy verification that this yields an open-closed surface havingX(S) ∼= X. �

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Now, given an open-closed fat graphΓ, X(S(Γ)) is entirely determined byΓ,and in fact we can writeX(Γ) for an open-closed data tuple obtained directly fromΓ, as follows:

X(Γ) := (g(Γ), ∂S1Γ, ∂IΓ, ξ, ψ, ∂wΓ, βΓ|∂wΓ),

where

• ∂IΓ, ∂S1Γ ⊆ π0(∂sΓ) are the sets of free boundary intervals and cycles, re-spectively.

• ∂wΓ ⊆ π0(∂fΓ) is the set of closed free boundary components.

• g(Γ) := 1− #π0(∂Γ)+χ(Γ)2

.

• For c ∈ ∂IΓ, ξ(c) is theB-label carried by the final endpoint ofc

• ψ takes a string boundary intervalc to the one appearing immediately after itin the component of∂Γ containingc.

If Γ has a partitioning, we can further defineY (Γ) = (X(Γ), ∂±S1Γ, ∂

±I I). The

following is clear.

Proposition 13 Y (S(Γ)) ∼= Y (Γ). ✷

A.2 Preliminary reductions

Let Γ ∈ Fat⋆B and letb ⊆ ∂fΓ be a linear component of the free boundary.

Definition 27 Denote byb+ ⊆ ∂sΓ (resp.b−) the string boundary component thatappears immediately afterb (resp., beforeb) in ∂fΓ according to the orientation. Wecan classifyb as belonging to one of four types:

• say thatb ∈ B−(Γ) if b− ⊆ ∂−Γ andb+ ⊆ ∂−Γ,

• say thatb ∈ B+(Γ) if b− ⊆ ∂+Γ andb+ ⊆ ∂+Γ,

• say thatb ∈ B−+(Γ) if b− ⊆ ∂−Γ andb+ ⊆ ∂+Γ,

• say thatb ∈ B+−(Γ) if b− ⊆ ∂+Γ andb+ ⊆ ∂−Γ. ✷

We now identify a convenient class of graphs containing a representative of eachconnected component.

Definition 28 Call Γ specialif the following conditions hold:

1. Each linear component of∂fΓ consists of a single vertex.

2. Each cyclic component of∂fΓ has exactly one edge.

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3. The image loop inΓ of each cyclic free boundary componentb is attachedto a trivalent vertex; this vertex is therefore incident with the two half-edgesforming the loop and with a third edge; the loop is then aballoonattached toΓ by this edge.

4. Eachb ∈ B−+(Γ) ∪ B+−(Γ) ∪B+(Γ) is attached to a univalent vertex ofΓ

5. Eachb ∈ B−(Γ) is attached to a bivalent vertex ofΓ.

6. There are no other bivalent or univalent vertices. ✷

A special open-closed fat graphΓ then has distinguishedB-labeled bivalent ver-tices, leaves withB-labeled endpoint, andB-labeled balloons.

Proposition 14 Any Γ ∈ Fat⋆B is connected to a special graph by a sequence ofmorphisms inFat⋆B. ✷

Proof The first two and last conditions can be attained by collapsing the image inΓ of a maximal forest in∂fΓ; after that the three other conditions can be attained byexpanding suitable vertices into trees. �

Remark 9 We cannot have ab ∈ B−(Γ) attached to aunivalentvertex—this vertexwould in turn be the endpoint of an edge having both its orientations in∂−Γ, andthe partitioning would not be admissible. ✷

A.3 Normal forms

In view of previous section, it is enough to show that twospecialgraphsΓ,Γ′ ∈Fat⋆B with Y (Γ) = Y (Γ′) are in the same connected component.

Our strategy for finding suitable normal forms will be to use the algorithm byR. Cohen and V. Godin in [CG04], henceforth referred to as “algorithmV .” Theidea is, roughly, to runΓ through this algorithm, having it work on the underlyingfat graphU(Γ).However, it will not be quite this simple, since we have to be carefulfor two reasons:

• The algorithm in [CG04] assumes that the starting graph is a chord diagram.Since we use the laxer admissibility condition from Definition 22, we willwork to achieve this form; see Lemma 17 below.

• Open-closed fat graphs may have incoming and outgoing string boundaryintervalsin addition to cycles.

• Whenever the algorithm expands an edge ofU(Γ), there is at least one way toexpand the corresponding edge ofΓ while keepingΓ special; however, whenthe algorithm collapses an edge ofU(Γ), the corresponding edge ofΓ maynot be collapsable, since it may result in joining two components of the imageof ∂fΓ.

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To describe the different cases that arise when finding the normal forms, weintroduce some terms.

Definition 29 Say thatΓ ∈ Fat⋆B is clean if it is special and, additionally,∂fΓ =B−+(Γ) ∪ B+−(Γ).

Given a specialΓ ∈ Fat⋆B, letw(Γ) be the open-closed fat graph obtained fromΓ by

1. removingB+(Γ) ∪ B−(Γ) from ∂fΓ,

2. removing all the cyclic components of∂fΓ, as well astheir image loops inΓ,

3. removing fromΓ any leaves or bivalent vertices created by the previous twosteps.

Notice thatw(Γ) inherits a boundary partitioning fromΓ, since the only re-moved free boundary intervals lie between string boundary components on the sameside of the partitioning ofΓ.

We define aweak string boundary componentof Γ to be a string boundarycomponent ofw(Γ), and we let∂±wΓ := ∂±w(Γ). ✷

The result we aim for is as follows.

Lemma 15 Let Γ ∈ F+B be connected.

• Case 1. Suppose thatΓ has a weak incoming boundary cycle. Then,Γ isconnected inFat⋆B to an open-closed fat graph of the following form:

c0···

···

· · ·

.

• Case 2.Suppose thatΓ has no weak incoming boundary cycles, but it hasmore than one topological boundary component. Then,Γ is connected to agraph of the form:

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c0···

··· .

• Case 3.Suppose thatΓ has no weak incoming boundary cycles, and thatΓhas exactly one topological boundary component. Then,Γ is connected to agraph of the form:

c0

···

.

The symbols used in the pictures are as follows:

• A serrated portion of an edge ( ) stands for a (possibly empty)sequence of bivalent vertices carrying elements ofB−(Γ); the triangles pointtowards the topological boundary component containing them.

• The symbol represents a (possibly empty) sequence · ··

in which where each element stands for a structure of the form

d

c

ak

a2a1

· ··

.

Here, c stands for an element ofB+−(Γ), d for one ofB−+(Γ), and eachai stands for an element ofB+(Γ) (the meaning of the serrated edge is asbefore).

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• The symbol stands for a (possibly empty) sequence of balloons ofthe form

wk

w2

w1

···

(that is, loops in the image of∂fΓ labeled by elementswi ∈ B.)

Note that those boundary components that contain a serratededge are weak incom-ing boundary components, and the ones that contain a are topologicalboundary components which contain part of the outgoing boundary.

In the first two cases, we will use terminology partially adapted from [CG04],as follows:

• The component markedc0 in the pictures will be called theouter compo-nent (this is called the big incoming circle in [CG04], but itis an outgoingcomponent in our case2).

• The topological boundary components in the top- and bottom-right quadrantsin cases 1 and 2 will be calledsimple outgoing cycles.

• The topological boundary component in the top-right quadrant will be calledthecomplicated outgoing cycle.

• The weak incoming cycles obtained by going clockwise aroundthe smallcircles on the lower left of case1 will be calledsmall incoming cycles.

The uniqueness of these normal forms is as follows:

• In case1 we may choose an arbitrary weak incoming cyclec0 to be the outercomponent, and we may choose an arbitrary topological boundary compo-nent containing part of the outgoing boundary to be the complicated outgo-ing cycle. Moreover, we can specify arbitrarily the order inwhich the simpleoutgoing cycles and the small incoming cycles occur in the cyclic orderingaround the central vertex.

• In case2 we may choose an arbitrary topological boundary componentc0(necessarily containing part of the outgoing boundary) to be the outer com-ponent, and one to be the complicated outgoing cycle. As in case1, we canspecify the order of the simple outgoing cycles arbitrarily.

• In all cases, we can choose an arbitrary linear ordering (consistent with theunderlying cyclic order) for theB-labels to appear in each or

cluster.

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• In all cases, we can choose an arbitrary order for the labelswi of the balloonsappearing in the .

In each case, the normal form is uniquely determined, after the correspondingchoices have been made, by the combinatorial data carried bythe invariantY (Γ),and therefore uniquely determined by the isomorphism type of S(Γ). ✷

Lemma 16 If Γ ∈ Fat⋆B has nonempty∂−Γ, then it is connected by morphisms toa special graphΓ′ for which every edge ofw(Γ′) hasexactlyone of its orientationsin the incoming string boundary. ✷

Proof Assume without loss of generality thatΓ is connected and special. We aimto get rid of edges ofw(Γ) having two outgoing orientations (sinceΓ is well-partitioned, there are no edges having two incoming orientations). We will do thisinductively, by showing that we can reduce the number of such“bad” edges viamorphisms that introduce only “good” edges.

Suppose there is at least one bad edge. SinceΓ is connected, there is a vertexnot in the image of∂fΓ, and incident to both a bad edge and to at least one edgetaking part in the incoming boundary. The cyclic ordering atthis vertex hence lookslike:

⊖⊕⊕

,

where the symbols⊖,⊕ denote incoming and outgoing boundary, respectively. Wecan modifyΓ in two steps, as follows:

⊖⊕⊕

← ⊖⊕⊕

→ ⊖⊕

The graph in the middle maps to the other two graphs by obviousmorphisms inFat⋆B; the one on the left isΓ and the one on the right has one less bad edge thanΓ.We leave it to the reader to verify that this works even if the bad edge is a loop.�

Lemma 17 If Γ ∈ Fat⋆B has nonempty∂−Γ, then it is connected by morphisms inFat⋆B to a specialΓ′ ∈ Fat⋆B for which∂−w(Γ′) is embedded inw(Γ) in such a waythat the complement graph is a forest. ✷

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Proof We may assume thatΓ is special and satisfies the condition in the conclusionof Lemma 16. So the set of edges of∂−w(Γ) is already embedded in the set ofedges ofw(Γ); it remains to make the vertices embedded too. For every vertex vnot in the image of∂fw(Γ), the angles aroundv must alternate between incomingand outgoing when traversed according to the cyclic ordering atv. Because of this,the following transformation (illustrated in the case thatv has valence 6) may beused to replace the vertexv by a tree, resulting inΓ′ ∈ Fat⋆B mapping toΓ by amorphism:

Γ Γ′

⊖⊕⊖⊕⊖⊕

⊖⊕⊖

⊕ .

It is clear that applying this transformation to every vertex achieves the desiredcondition. �

With these preliminary result in place, we are ready to proveour main lemma.

Proof of Lemma 15 We may assume thatΓ is special. Notice that we may dis-regard the balloons throughout, since any two special open-closed fat graphs thatdiffer only on the location of balloons along the outgoing boundary are connectedby a sequence of morphisms; we leave this as an exercise. Thus, we can let themmove around the graph arbitrarily, and we can collect them atthe end to form asingle at the correct location.

Assume first thatΓ is clean, so thatΓ = w(Γ). By Lemma 17, we may assumethat∂−Γ is embedded inΓ with a forest complement. Following [CG04], we willcall the edges of the complement “ghost edges.”

We can apply much of AlgorithmV toΓ, by using an arbitrary (open or closed)incoming boundary componentc0 of Γ instead of the “big incoming component,”and we may treat the other incoming boundary intervals much of the way as ifthey were incoming boundary cycles. We leave it to the readerto verify thatΓcan be transformed, following the first steps of AlgorithmV , into a form in whichexactly one non-univalent vertexv0 is incident with more than one ghost edge, andin which every incoming boundary component other thanc0 is connected by exactlyone ghost edge tov0.

The next step in AlgorithmV is to push the “small incoming cycles” all the wayto the “right” in the cyclic ordering atv0. This relies on these boundary componentsbeing cycles; thus we are not able to do it with the incoming boundary intervals.However, after the previous step, we may collapse the uniqueghost edge attachedto each of the incoming boundary intervals (other thanc0). After doing this, wecreate a structure of type for each incoming interval. Call the resultinggraph State⋆.

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Every time a structure of type is created, we will treat it thereafteras a single unit, so its constituent edges will not be collapsed. We will call edgesthat arenot in any of these structures “active;” we will call a vertex active if all itsincident edges are active.

Now, suppose thatc0 is an incoming boundarycycle. Then, we can in factcontinue applying AlgorithmV to the end, and we obtain, after relocating structuresof type by means of expansion/collapse pairs if necessary, precisely thenormal form in case 1.

Suppose that we are in case 2, so thatc0 must be a incoming boundary inter-val. After State⋆ there are no incoming boundary cycles, and we have collapsedall the incoming weak boundary intervals but one, namelyc0. To proceed, collapsec0 except for its two endpoints (which carry labels inB−+(Γ) andB+−(Γ) respec-tively); this is possible sincec0 is still embedded inΓ. This creates a structure oftype . After this, we collapse a maximal active subtree ofΓ . The resultis a special open-closed fat graph with all its active half-edges belonging to∂+Γ,having more than one topological boundary cycle and exactlyone active vertex .

Change notation, lettingc0 be any topological boundary cycle, which will actas our “big cycle” from now on. Using Lemma 16, we can replaceΓ by one inwhich each edge has exactly one of its orientations inc0 (to do this, we introduce atemporary boundary partitioning onΓ which makesc0 incoming and the remainingcomponents outgoing, and then apply the lemma; here it is essential thatΓ hasmore than one topological boundary component). Then, as we did in Lemma 16for the incoming boundary, we can further replaceΓ by a graph in whichc0 → Γis an embedding and the complement ofc0 in Γ is a forest. From this point on, weapply AlgorithmV , treatingc as if it were the “big incoming cycle,” resulting in therequired normal form.

In case 3, we have after State⋆ and after collapsing the remaining incominginterval and a maximal active tree thatΓ has one active vertex, a single topologicalcomponent and all its active half-edges in∂+Γ. Since∂Γ is connected, we canensure that all the labeled univalent vertices are contiguous by using a sequenceof expansions/collapse pairs. The normal form for this casethen follows directlyfrom Lemma 18 below on the structure of fat graphs with a single vertex and asingle boundary component. (This case does not arise in [CG04] because they donot consider fat graphs with empty incoming boundary.)

If Γ is not clean, then we apply the preceding procedure tow(Γ). Every timeour algorithm expands a vertex ofw(Γ) into an edge, the corresponding operationmay be carried out onΓ. When the algorithm contracts an edge ofw(Γ), we mustbe careful because this edge might come from a sequence of edges inΓ separatedby bivalent vertices carrying labels inB−(Γ). However, we can still collapse thecorresponding edges inΓ provided that we first move those bivalent vertices furtheralong in the boundary ofΓ; this can be done by a straightforward sequence ofexpansion/collapse pairs. At the end, we are in a state for which w(Γ) is in one ofthe special forms; we can then use expansion/collapse pairsso that all theB−(Γ)labels are contiguous (forming serrated edges), and so thatall the univalent verticescarryingB+(Γ) labels are also contiguous, and form part of a structure of type

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.We omit the proof of the uniqueness statement, which followsfrom a computa-

tion of the open-closed data tuple of each of the normal form and from observingthat they cover distinct isomorphism types. �

Lemma 18 Let Γ be a fat graph having a single boundary component and a singlevertex. Then,Γ it is connected by a sequence of morphisms to a fat graph havingsingle vertexv in which the cyclic ordering of half-edges atv is of the form

(e1, e2, r(e1), r(e2), e3, e4, r(e3), r(e4), . . . , e2k−1, e2k, r(e2k−1), r(e2k))

(where as usualr stands for the edge-reversal involution onΓ). ✷

The proof is an easy induction.

Lemma 9 is now direct corollary of Lemma 15.

References

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[CJ02] Ralph L. Cohen and John D. S. Jones. A homotopy theoretic realizationof string topology.Math. Ann., 324(4):773–798, 2002.

[CK] Ralph L. Cohen and John R. Klein. To appear.

[Cos] Kevin J. Costello. Topological conformal field theories and Calabi-Yaucategories, arXiv:math.QA/0412149.

[CS] Moira Chas and Dennis Sullivan. String topology.Annals of Math.,arXiv:math.GT/9911159. To appear.

[Hara] Eric Harrelson. PhD thesis, University of Minnesotta. In preparation.

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[Sul04] Dennis Sullivan. Open and closed string field theoryinterpreted in classi-cal algebraic topology. InTopology, geometry and quantum field theory,volume 308 ofLondon Math. Soc. Lecture Note Ser., pages 344–357.Cambridge Univ. Press, Cambridge, 2004.

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