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021 On a new geometric homology theory and an

application in categorical Gromov-Witten theory

Hao Yu

December 28, 2021

Abstract

In this note we present a new homology theory, we call it geometrichomology theory (or GHT for brevity). We prove that the homol-ogy groups of GHT are isomorphic to the singular homology groups,which solves an implicit question raised by Voronov et al. We aremainly base on the celebrated stratification and triangulation theoriesof Lie groupoids and their orbit spaces, as well as the extension to Liegroupoids with corners by us. Another ingredient in the proof is areformulation of the definition of singular homology which seems new.GHT has several nice properties compared with singular homology,making it more suitable than singular homology in some situations,especially in chain-level theories. We illustrate one of its applicationsin categorical Gromov-Witten theory, initiatied by Costello. We willdevelop further of this theory in our sequel paper.

Contents

1 Introduction 2

2 Notation 5

3 Orbifolds, orbifolds with corners and Lie groupoids 6

4 Stratification theory on proper Lie groupoids (with corners) 10

5 Reformulations on singular homology 15

6 Main theorem 19

1

7 Applications 22

8 Appendix 26

1 Introduction

In this paper, we introduce a new homology theory, we call it geometrichomology theory, or GHT for brevity. This theory is based on the notionof geometric chains, first introduced by Voronov et al. ([VHZ]). GHT hasseveral nice properties including (but not limited to): first, the homologygroups in GHT are isomorphic to singular homology groups; Secondly, chainsin GHT behave quite well with some natural operations including Cartesianproducts and pull backs along fiber bundles. This is in sharp contrast withthe singular chains, where they do not have canonical definitions when actedby these operations. The third, the fundamental chains of manifolds (ororbifolds) with corners can be canonically defined in GHT, while there iscertainly no such for singular chains. Due to these nice properties, we believeit will be useful in some cases if one use geometric chains in place of singularchains.

Specifically, we borrow the definition of geometric chains from [VHZ]first.

Definition 1.0.1. (geometric chains). Given a topological space X, a ge-ometric chain is a formal linear combination over Q of continuous mapsf : P → X, where P is a compact connected oriented (smooth) orbifold withcorners, modulo the equivalence relation induced by isomorphisms betweenthe source orbifolds P . Here, an orientation on an orbifold with corners is atrivialization of the determinant of its tangent bundle. The dimension of Pis called the dimension of the geometric chain f .

We denote the set of geometric chains of dimension k by GCk(X,Q).

Remark 1.0.2. As we are working with orbifolds, we use coefficient Q insteadof Z. The reader can also check paragraph in [VHZ] to understand why usingQ is necessary.

We will prove the following theorem.

Theorem 1.0.3. (geometric homology is isomorphic to singular homol-ogy). The set of geometric chains of X forms a complex GC∗(X,Q) over Qgraded by the dimensions of chains, with the boundary of a chain f givenby (∂P, f |∂P ), where ∂P is the sum of codimension one faces of P with

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the induced orientation. The homology groups of GC∗(X,Q) are denoted byGH∗(X,Q). Then we have

GH∗(X,Q) ∼= H∗(X,Q) (1)

In particular, if P is restricted to be a manifold with corners, we get asub-complex of geometric chain complex. We denote it by GCm

∗ (X,Q).

Corollary 1.0.4. The homology groups of GCm∗ (X,Q) are isomorphic to

singular homology groups, i.e.

GHm∗ (X,Q) ∼= H∗(X,Q) (2)

This corollary is perhaps more interesting for some readers. After com-pletion of the ideas and methodologies used in the paper, we are aware thatthis corollary has previously appeared in [CD] with the proof also beingbased on triangulations theory. However, no details are given there exceptpointing out some references. We remark that the triangulation theory ofdifferentiable manifolds (or with boundary) has already been a classical topicin topology in its early era, and the development is a long history, the gen-eralization to smooth closed orbifolds is completely nontrivial. This papertakes a further step towards smooth orbifolds with corners that is also non-trivial.The theorem 1.0.3 is somewhat surprising. In [VHZ], Voronov et al.made a remark on this property. We quote their words here ”Our notion ofchains leads to a version of oriented bordism theory via passing to homology.If we impose extra equivalence relations, such as some kind of a suspensionisomorphism, as in [Ja], or work with piecewise smooth geometric chainsand treat them as currents, in the spirit of [FOOO], we may obtain a com-plex whose homology is isomorphic to the ordinary real homology of X”.Although they didn’t state it as a formal conjecture, we nevertheless takethe liberty to regard this as an implicit conjecture. Thus from this view,our theorem 1.0.3 answers their question affirmably, however no additionalconstrains needed.

Similarly, one can define the geometric cohomology groups, as well as cupproducts, cap products, and so on. Essentially, properties hold in singularhomology (cohomology) theory should still hold in GHT. This observationcan be deduced from the very method of the proof of the main theorem 1.0.3.We note that it is a general principle, one still needs to prove each basicproperty one by one. As the first of a potential series papers in developingGHT, we decide not to introduce geometric cohomology theory here, leavingit to our subsequent work.

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For applications, we mention that (which is also a motivation of devel-oping GHT) in [Cos04, Cos05] Costello constructed an algebraic counter-part of the theory of Mirror symmetry, in particular, the construction ofGromov-Witten potential in pure algebra manner. In order to deal withpullbacks of chains, he has to consider S-equivariant chains and S1 homo-topy coinvariants, which is due to the singular chains not being able tocanonically defined under pullbacks as mentioned above. Also, he has touse S1 equivariant chains to define fundamental chains for orbifolds withboundary. More specifically, he needs to construct a Batalin-Vilkovsky al-gebraic structure on the singular chain complexes on the compactificationof moduli spaces of Rienmann surface with marked points, the well-knownDeligne-Mumford spaces, and moreover needs to use fundamental chains assolutions of quantum master equations that encodes the fundamental classesof Deligne-Mumford spaces and thus encodes Gromov-Witten potentials. Byusing geometric chains, we give an extension of his results to open-closedtopological conformal theory and prove the similar results (see [HY]). Thisis also related to the typical motivation of geometric chains theory as ”weneed the unit circle S1 to have a canonical fundamental cycle and the modulispaces which we consider to have canonical fundamental chains, irrespectiveof the choice of a triangulation. On the other hand, we believe it is generallybetter to work at the more basic level of chains rather than that of homol-ogy”, quoted from [VHZ]. We do not need to use equivariant homotopytheory in this case since we are using geometric chains instead of singularchains, everything there looks very natural. Other potential applicationswill be studied as a future work.

We briefly sketch the ideas of the proof as follows. Regarding orbifoldsas proper Lie groupoids s, t : G ⇒ M , with discrete isotropy groups isthe modern views of orbifolds ([Mo]). The method of proof relies heavilyon the celebrated stratifications and triangulations theory of Lie groupoids.Any proper Lie groupoid has a canonical stratification given by the con-nected components of Morita equivalence decomposition. Furthermore, thestratification is a Whitney stratification. The same are also true when pass-ing to the orbit space. The result of Thom and Mather in [Mat70] showsthat any space that has a Whitney-stratification admits an abstract pre-stratification structure. And the result of [V] shows that if a space has anabstract pre-stratification structure, then it has a triangulation compatiblewith the abstract pre-stratification. In particular, the orbit space has anabstract pre-stratification structure and thus has a triangulation. The inno-vative idea of ours is to introduce the notion of Lie groupoid with corners andprove analogous results as for Lie groupoids. One of them is the following

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extension of Bierstone’s theorem ([B]) to the case of manifolds with corners:the orbit space of an Euclidean space with corners acted by a compact Liegroup admits a canonical stratification that is Whitney stratification, and itcoincides with the stratification given by orbit type. This stratification alsorespects the corner structure. With these extensions, the above machinerythen all work, and we can thus prove the theorem 1.0.3.

The organization of the remaining of this paper is as follows. In section3, we review relevant concepts and properties on orbifolds, orbifolds withcorners and Lie groupoids and introduce new notions of Lie groupoids withcorners. We prove an equivalence between orbifolds with corners and properLie groupoids with corners by studying a local model of Lie groupoids withcorners. Section 4 discusses the celebrated stratification theory on proper Liegroupoids and presents generalization to proper Lie groupoids with corners.Interestly, we reformulate the definition of singular homology in section 5which seems new. This formulation has its independent interest and is alsoa key ingredient in this paper. We then prove our main results in section 6.An application in categorical Gromov-Witten theory is discussed in section7.

2 Notation

Let Rn denote the n-dimensional Euclidean space. Rn+ := {(x1, x2, . . . , xn)|xi ≥

0, i = 1, . . . , n} denotes the subset of Rn consisting of points whose coordi-nates are all nonnegative reals. And let Rn

k := {(x1, x2, . . . , xn)|xi ≥ 0, i =1, . . . , k} denote the subset of Rn with the first k coordinates non-negative.And let Rn

+k denote a subset of Rn+ consisting of points whose coordinates

have exact n− k 0.We need a well-behaviored category of smooth manifolds (and orbifolds)

with corners. According to Dominic Joyce [Jo], there are several definitionsof smooth manifolds with corners and smooth morphisms between them inthe existing literature. We heretoforth adopt the definition of category ofsmooth manifolds with corners in that paper, and the natural extension toorbifolds with corners, in this work. We highly refer the readers to that paperfor the full account of details. For any smooth manifold with corners M , wedenote by Bordk(M) a subset of M consisting of points each of which hasa neighbourhood diffemorphic to an open set in Rn

k . sending that point to0. It is called the k-corner of M . Each connected component of Bordk(M)is a smooth manifold (with the induced topology) of which the closure is amanifold with corners (the latter is the notion of k-corners defined in [Jo]).

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M together with all Bordk(M) is called the corner structure of M . Whena Lie group G acts continuously on a topological space M , its orbit space isdenoted by |M/G|. A smooth action of a Lie group G on a manifold withcorners M is a smooth map

f : G×M →M (3)

such that f |G×Bordk(M) ⊆ BordkM . Moreover, if M = Rnk and G acts on

each Bordl(M) lineally, we say Rnk is a linear corner representation of G.

We frequently, in the remaining part of the paper, expand symmetricallyalong the first k axes of Rn

k to make it into Rn in an obvious way. We callthis operation doubling.

3 Orbifolds, orbifolds with corners and Lie groupoids

We review the preliminaries on orbifolds, orbifolds with corners and Liegroupoids. The standard reference for orbifolds and Lie groupoids are [ALR,Mo, MP] as well as the relevant work [CM, MM], see also the pioneering work[S]. We start discussing orbifolds by presenting its original definition, thenwe show that they can be equivalently reformulated as proper Lie groupoidswith discrete isotropy groups.

Definition 3.0.1. (Orbifolds). Let X be a topological space, and fix n ≥ 0.

• An n-dimensional orbifold chart on X is given by a connected opensubset U ⊆ Rn, a finite group G acting smoothly on U , and a mapφ : U → X so that φ is G-invariant and induces a homeomorphism ofU/G onto an open subset U ⊆ X.

• An embedding λ : (U ,G, ϕ) → (V ,H, ψ) between two such charts is asmooth embedding λ : U → V with ψλ = ϕ. An orbifold atlas on Xis a family U = (U ,G, ϕ) of such charts, which cover X and are locallycompatible: given any two charts (U ,G, ϕ) for U = ϕ(U ) ⊂ X and(V ,H, ψ) for V ⊆ X, and a point x ∈ U ∩ V , there exists an openneighbourhood W ⊆ U ∩ V of x and a chart (W ,K, µ) for W suchthat there are embeddings (W ,K, µ) → (U ,G, ϕ) and (W ,K, µ) →(V ,H, ψ).

• An atlas U is said to refine another atlas V if for every chart in U thereexists an embedding into some chart of V. Two orbifold atlases aresaid to be equivalent if they have a common refinement.

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An orbifold X of dimension n or an n-orbifold is a paracompact Hausdorffspace X equipped with an equivalence class U of n-dimensional orbifoldatlases.

As geometric chains are built on parts of orbifolds with corners, we needto similarly extend the definition of 3.0.1, as what we did from manifolds tomanifolds with corners.

Definition 3.0.2. (orbifolds with corners). Let X be a topological space,and fix n ≥ 0.

• A generalized n-dimensional orbifold chart on X is given by a con-nected open subset U ⊆ Rn

+, a finite group G acting smoothly on U ,

and a map ϕ : U → X so that ϕ is G-invariant and if for any 0 ≤ k ≤ nUk := U ∩ Rn

+k 6= ∅ then φ restricted to any connected component of

Uk is G-invariant, and φ induces a homeomorphism of U/G onto anopen subset U ⊆ X.

• An embedding λ : (U ,G, ϕ) → (V ,H, ψ) between two such charts isa collection of smooth embeddings λk : Uk → Vk. with ψλ = ϕ. Anorbifold atlas on X is a family U = (U ,G, ϕ) of such charts, whichcover X and are locally compatible: given any two charts (U ,G, ϕ) forU = ϕ(U ) ⊆ X and (V ,H, φ) for V ⊆ X and a point x ∈ U ∩V , thereexists an open neighbourhood W ⊆ U ∩ V of x and a chart (W ,K, µ)for W such that there are embeddings (W ,K, µ) → (U ,G, ϕ) and(W ,K, µ) → (V ,H, ψ).

• An atlas U is said to refine another atlas V if for every chart in U thereexists an embedding into some chart of V. Two generalized orbifoldatlases are said to be equivalent if they have a common refinement.

An orbifold with corners X of dimension n or an n-orbifold with corners isa paracompact Hausdorff space X equipped with an equivalence class U ofn-dimensional generalized orbifold atlases.

Remark 3.0.3. It is possible to give a notion of orbifolds with corners forwhich the first condition is modified to allow G not necessarilly keep eachconnected component of k-corner invariant but only keep the whole k-cornerinvariant. However, actually these two definitions are equivalent. We willleave the proof to the readers.

From the definition we see that the notion of orbifolds are tied with thenotion of spaces with finite group action, and more generally smooth spaces

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with Lie groups actions. A generalization of the latter is the notion of Liegroupoid.

Definition 3.0.4. (Lie groupoid). A Lie groupoid, consists of the followingdata: two smooth manifolds G and M , called the space of sources and spaceof objects, two smooth maps s, t : G → M, called the source and targetmaps, such that s or t is a surjective submersion, a partial defined smoothmultiplication m : G(2) → G, defined on the space of composable arrowsG(2) = {(g, h) ∈ G|s(g) = t(h)} (by being the surjective submersion of s ort, this set is a smooth manifold), a unit section u :M → G, and an inversioni : G → G, satisfying group-like axioms. We denote the above Lie groupoidby G ⇒M .

We assume that the readers have some familiarities with basic notionsand properties of Lie groupoids. For comprehensive introduction we referthe book by Mackenziesee ([Mac]).

As conventional, for an open set U ⊆M we denote by GU the restrictedLie groupoid of G to U . The isotropy group of G at x, in particular, isdenoted by Gx. And Ox denotes the orbit of x ∈ M under the canonicalaction of G on M .

In [Mo] it established a very important and nice property that an orbifoldis equivalently described as a proper Lie groupoid with discrete isotropygroups (and also equivalently formulated as a proper Lie groupoid with etaleisotropy groups if considering Morita equivalence). To work in the categoryof orbifolds with corners, we introduce a new notion of Lie groupoids withcorners.

Definition 3.0.5. (Lie groupoid with corners). A Lie groupoid with cornersconsists of the following data: two smooth manifolds with corners G and M ,and all the data as in the definition of Lie groupoids, such that the source andtarget maps respect the corner structures of G and M , i.e. for each k ≥ 0,the inverse image of each connected component of Bordk(M) is a unionof several connected components of Bordl(G) for a possible list of ls, andthe image of each connected component of Bordl(G) lies in one connectedcomponent of Bordk(M).

One can deduce then that m,u, i all respect the corners structures of Gand M .

We often write G⋉M for a Lie groupoid (resp. with corners) associatedto a Lie group action on a manifold (resp. with corners) M .

Naturally, one expect there is also an equivalence between orbifolds withcorners and proper Lie groupoids with corners and discrete isotropy groups.

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For proving this, we need the following local model of proper Lie groupoidswith corners around a point.

Proposition 3.0.6. (local model of proper Lie groupoids with corners abouta point). Let G ⇒M be a proper Lie groupoid with corners over M . And letx ∈ M be a point. There is a neighbourhood U of x in M , diffemorphic toO ×W , where O ⊆ Ox is an open set, W ⊆ Nx is an Gx invariant open setin the normal space Nx to Ox at x, such that under this diffemorphism GU

is isomorphic to the product of the pair groupoid O ×O ⇒ O with Gx ⋉W .

Proof. W.l.o.g assume s is a surjective submersio. For any x ∈ M , chooseany g ∈ Gx, we can then find neighbourhoods U around g and V around x,such that on U s is given by projection onto the second factor:

s|U : Rlm ×Rn

k → Rnk (4)

for some l,m and n, k. GV is an non-empty open set of U since it contains g.Now we use ”doubling” argument to extend manifolds with corners GV andV to manifolds without corners. The idea is simply to symmetrically expandRn

k along first k coordinate directions to make it into Rn, so that GV willexpand to be an open set G := Sym(GV ) in R

l+n, and V becomes an open setSym(V ) in Rn. The s, t maps on G are defined to be the natural extensionof s, t on GV that become invariant under this symmetry. G ⇒ Sym(V ) isa proper Lie groupoid. By a local model for proper Lie groupoids (cf. [CM]prop. 2.30) around x we know that there is a neighbourhood N around xin Sym(V ) such that when restricting to N , there is a Gx invariant open setW in Nx (the normal space in Sym(V ) to the orbit Ox) and an open set Oin Ox ∩Sym(V ), such that GN is isomorphic to the product of linear action

groupoids Gx⋉ W with pair groupoid O×O ⇒ O. So when restricting backto the original Lie groupoid with corners GV and V , we see that there is aneighbourhood N ⊆ V around x, an (GV )x invariant open set W in Nx (thenormal space in V to the orbit Ox, which is Rn

k) and an open set O in Ox∩Vso that (GV )N is isomorphic to the product of a linear action groupoid withcorners (GV )x ⋉W with product groupoid O × O ⇒ O. This is the localmodel of Lie groupoid with corners. Also (GV )N is Morita equivalence to(GV )x ⋉ W . Moreover, (GV )N is also Morita equivalence to (GV )x ⋉ Nx.For this we use also that Nx

∼= Rnk admits arbitrarily small (GV )x-invariant

open neighbourhoods of the origin which are equivariantly diffemorphic toNx. The latter can be deduced from the corresponding statement for Rn byusing doubling argument straightforwardly.

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Theorem 3.0.7. There is an equivalence between orbifolds with cornersdefined in the sense of 3.0.2 and proper Lie groupoids with corners withdiscrete isotropy groups.

Proof. =⇒ If X is given by the original definition 3.0.2, locally, X is givenby the action groupoid φ ⋉ U , i.e. locally X is homeomorphic to the orbitspace |U/φ|. We can glue such action groupoids together to form a (proper)Lie groupoid with corners whose isotropy groups are all finite groups.

⇐= If X is homeomorphic to the orbit space of a proper Lie groupoidwith corners with discrete isotropy groups, then by the proposition 3.0.6, weknow that locally G is Morita equivalent to a linear action groupoid Gx⋉Ux,for an invariant open set Ux ⊆ Nx. As Gx is discrete and compact, it is afinite group. Thus locally X is homeomorphic to |Ux/Gx|. These Gx ⋉ Ux

give the generalized orbifold charts of X in the sense of 3.0.2. We thusconclude the proof of the theorem 3.0.7.

One can also prove an analogue of Linearization theorem for Lie groupoidswith corners, along with the lines of using Zung’s theorem ([Z]) (for orbit asingle point) and Weinstein’s trick ([W]). As we do not need it in this paper,and the proposition 3.0.6 above is enough for our main result, theorem 1.0.3,we will prove it in our sequel paper.

A somewhat surprising property of orbifolds is that it can always bedescribed as a global quotient.

Proposition 3.0.8. Every classical n-orbifold X is diffemorphic to a quo-tient orbifold for a smooth, effective, and almost free O(n)-action on a framebundle (which is a smooth manifold) Fr(X).

We refer the readers to [ALR] for details on this result.

4 Stratification theory on proper Lie groupoids(with corners)

In this section, we discuss the stratification and triangulation theories de-veloped particularly for orbifolds, and more generally for orbispaces. It wasgeneralized to Lie groupoids setting, leading to a unified and more compacttheory.

For stratified spaces we understand they are Hausdorff second-countabletopological spaces X endowed with a locally finite partition S = {Xi|i ∈ I}such that

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(1) Each Xi endowed with the subspace topology, is a locally closed, con-nected subspace of X and a smooth manifold.

(2) (locally finiteness) every point x ∈ X has a neighbourhood that inter-sects finite members of S.

(3) (frontier condition) the closure ofXi is the union ofXi with the membersof S of strict lower dimension.

If X has a differentiable structure, for example a differential manifold (ormore generally a differentiable space developed in [NS]), one usually furtherrequires it being a Whitney stratification (cf. [CM]).

Definition 4.0.1. (Whitney stratification). Let M be a smooth manifold(without boundary). Let N be a subset of M . Let S be a stratification ofN . Then S is said to be a Whitney stratification if the following conditionshold

(A) For any strata R,S ∈ S, given any sequence {xi} of points in R suchthat xi → y ∈ S and TXxi

converges to some r-plane (r=dim(R))τ ⊂ TMy, we have TYy ⊂ τ .

(B) Let {xi} be a sequence of points in R, converging to y and {yi} asequence of points in Y , also converging to y. Suppose TXxi

convergesto some r-plane τ ⊂ Rn and that xi 6= yi for all i and the secants (xiyi)converge (in projective space Pn−1) to some line l ⊂ Rn . Then l ⊂ τ .

When M is a manifold with corners, the condition (A) should be replacedby:

(A) (for manifold with corners). We require that when S is on a corner ofR, then Tx(M) should be understood as the R-dimensional vector spacetangent to x on the doubling of the local model Rn

k around x which isRn.

A closely related notion of abstract pre-stratification is a very essentialconcept in the stratification and triangulation theory(cf. [Mat70, V]).

Any Lie groupoid has a canonical stratification induced by the Moritatype equivalence classes, denoted by x ∼M y, i.e. x ∼M y ⇐⇒ (Gx, Nx) ∼=(Gy, Ny) whereNx, Ny are normal (normal spaces to the orbits Ox and Oy re-spectively) representations of Gx,Gy, respectively. Since Morita type equiv-alence relation is invariant on any orbit, it also induces a stratification onthe orbit space. We refer the readers to [CM] for details on Morita typeequivalence and canonical stratification.

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There are other equivalence relations, for example, isotropy isomorphismequivalence that is defined by: x ∼= y ⇔ Gx

∼= Gy. However when passingto connected components of each equivalence class they all give the samestratification.

If the Lie groupoid happens to be an action groupoid, then the stratifica-tion also agrees with the stratification induced by some equivalence relationsspecific to group actions. For example, orbit type equivalence that is definedby x ∼ y ⇔ Gx ∼ Gy (i.e. Gx and Gy are conjugate in G), or isotropyisomorphism equivalence. We refer the interested readers to [CM] for moredetails.

The key result in the theory on the stratification of proper Lie groupoidis that the canonical stratification on M and X are both Whitney stratifi-cations (cf. [CM] prop. 5.7), which we will generalize to Lie groupoids withcorners.

Theorem 4.0.2. Let G ⇒ M be a proper Lie groupoid over M , then thecanonical stratifications of M and X are Whitney stratifications.

Any Morita equivalence between two proper Lie groupoids induces anisomorphism of differetiable stratified space between their orbit spaces.

The most innovative ingredient of this paper is to generalize the resultabove to Lie groupoids with corners. We need to talk first on what theMorita equivalence relation means in this setting.

Definition 4.0.3. (Morita equivalence for Lie groupoids with corners). LetG be a Lie groupoid with corners over M and H a Lie groupoid with cornersover N . A morita equivalence between G and H is given by a principle G−Hbi-bundle, i.e. a manifold with corners P endowed with

1. Surjective submersion α : P → N,β : P → M that compatible withcorner structures of P,M and N .

2. A left action of H on P along the fibers of α, compatible with thecorner structure of H and P , which makes β : P →M into a principleH-bundle.

3. A right action of G on Q along the fibers of β, compatible with thecorner structure of G and Q, which makes α : Q→ N into a principleG-bundle.

4. the left and right action commute.

G and H are Morita equivalence if such a bi-bundle exists.

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The following statement is the ”corner” case extension of a standardresult on the stratification of proper action groupoids (cf. [CM] thm. 4.30).

Proposition 4.0.4. Let G ⇒M be a proper Lie groupoid with corners overM . The canonical decompositions of M and X := |M/G| by the Moritatype equivalence classes on each k-corner of M and X are stratifications.Moreover, this stratification on M is a Whitney stratification.

Given a Morita equivalence between two Lie groupoids with corners Gand H, the induced homeomorphism at the level of orbit space preserves thecanonical stratification.

Proof. For proper Lie groupoids over manifolds, this property is shown tobe true in [CM]. To generalize it to ”corner” case, however, the ”doubling”argument does not work at all, since in general each k-corner will formindependent stratum, regardless of their orbit types. We can prove it bycontinuity argument: If T is a strata on k-corner of M or X with orbit typeH, and if T ∩ ∂X 6= ∅, then any point in this intersection has orbit type H,due to the continuity of Gx on M .

The manifold condition on M (resp. X) are obvious satisfied, since eachstrata on each k-corner of M (resp. X) is a smooth manifold. For locallyfiniteness condition, let x ∈ M . By the local model of Lie groupoids withcorners 3.0.6, locally there are neighbourhoods U of x and W of Nx suchthat the conclusion in 3.0.6 holds. Then the strata that intersects withU corresponds to strata on Gx ⋉ W ⊆ Nx

∼= Rnk . Using the ”doubling”

technique, we expand along n − k axes in Rnk to turn the latter symmet-

rically into Rn, with the action of Gx × Zn−k2 of which the second factor

acts by reflection along corresponding axis. Let g ∈ Gx, τ 6= 1 ∈ Z2, from(g, 1) × (1, τ) = (1, τ) × (g, 1), we deduce that Gx × 1 keeps all corners in-variant, then Gx ⋉Rn

k is equivalent to (Gx ×Zn−k2 )⋉Rn with the constrain

that Gx × 1 keeps Rnk invariant. The strata expands accordingly. By the

standard fact (cf. [DK]) the canonical decomposition on (Gx × Zn−k2 )⋉Rn

is a stratification, thus locally finite. Back to Gx ⋉W , locally the strata onit are the intersections of these finite strata with all corners of W , whichis also finite. Now we verify the condition of frontier. Let T1 be a stratain a connected component of k-corner Xk, and let Ul := T1 ∩ ∂lXk, andT2 := T1 ∩Xk/T1, U2l := T2 ∩∂lXk. Since Ul has orbit type H, strata in U2l

has orbit type the same as the corresponding strata in T2, and the union ofclosures of all strata in Xk covers its boundary, U1 ∪ U2l will be the lowerdimensional stratum consisting of strata in ∂lXk lying in the closure of T1.Since on Xk it satisfies the condition of frontier ([CM]), we conclude thatthe condition of frontier holds. The continuity of the canonical projection

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π : M → X ensures that the condition of frontier also holds on the orbitspace.

To prove that it is a Whitney stratification on M , we observe that if(R,S) are two strata on the same k-corner of M , then [CM] shows thatWhitney condition (A) is true. If (R,S) lie on a k-corner of M and itsboundary l-corner respectively, then for any xn ∈ R converging to x ∈ S,and the tangent spaces Txn(R) converging to τ ⊂ Tx(M) (locally around x,Tx(M) is given by doubling the local model Rn

k , so Tx(M) is R-dimensional),Tx(S) must lie in τ , due to the very structure of manifold with corners.

The statement of Morita invariance on orbit space follows from the factthat Morita equivalence relation is invariant on any orbit O on any k-cornerof M .

We will prove that the above stratification is also a Whitney stratificationon X. To achieve that, we need a ”corner” case extension of a result ofBierstone on the stratification of orbit space of a proper action groupoid.

Proposition 4.0.5. Let G ⋉M be a proper action groupoid with corners, then the orbit space X := |M/G| has a Whitney stratification, and it co-incides with the canonical stratification induced by Morita type equivalenceclasses.

Proof. It was shown in the proposition 4.0.4 that the partition by Moritatype equivalence classes on X is a stratification. Now we prove it is aWhitney stratification. As this is a local property, by the local model 3.0.6,the orbit space X is locally homeomorphism to the orbit space of a linearcorner representation of a compact Lie group Gx. Consider, now, Gx actinglinear on Rn

k compatible with the corner structure. According to [B], theorbit space of any linear action of a compact Lie group H on Rn can berealized as a semi-algebraic subset (for semi-algebraic set see [L]) of an affinevariety that is induced by the map:

Φ : Rn → Rh (5)

where Φ = (φ1, . . . , φh) is a set of generator of H invariant polynomials onRn, and can be assumed to be homogeneous. Rn/H is homeomorphic tothe image Φ(Rn) that is a semi-algebraic subset of the affine variety V (I)where I is the idea of algebraic relations among φ1, φ2, . . . , φh. Using the”doubling” technique, we expand along n− k axes in Rn

k to turn the lattersymmetrically into Rn, with the action of Gx × Zn−k

2 of which the secondfactor acting by reflection along corresponding axis. Let g ∈ Gx, τ 6= 1 ∈ Z2,

14

from (g, 1) × (1, τ) = (1, τ) × (g, 1), we deduce Gx × 1 keeps all corners in-variant, then Gx ⋉Rn

k is equivalent to (Gx ×Zn−k2 )⋉Rn with the constrain

that Gx × 1 keeps Rnk invariant. Then, as in [B] we can use Gx × Zn−k

2

to find a mapping Φ = (φ1, φ2, . . . , φl) : Rn → Rl that induces a home-omorphism from Rn/(Gx × Zn−k

2 ) to a semi-algebraic subset of an affinevariety in Rl. One can freely add (x1, x2, . . . , xn) to Φ, so we may assume(φ1, φ2, . . . , φn) = (x1, x2, . . . , xn). Restricting to Rn

k , we see that Φ inducesa homeomorphism from Rn

k/Gx to a semi-algebraic set in Rlk, and this home-

omorphism is compatible with the corner structure, i.e. restricting to eachBordt(R

nk )/Gx is a homeomorphism to a semi-algebraic set in Bordt(R

lk).

Since by the result of [B] the image on each t-corner of Rlk has a Whit-

ney stratification that coincides with the canonical stratification for eacht-corner of Rn

k , and with the same reason as the proof on proposition 4.0.4,for (R,S) lying on different corners of Rl

k the Whitney condition (A) is alsosatisfied (with ambient manifold Rl). We conclude the proposition.

We can now give a ”corner” case result corresponding to theorem 4.0.2.

Proposition 4.0.6. Let G ⇒M be a proper Lie groupoid with corners, thenM and the orbit space X := |M/G| has a Whitney stratification, and it coin-cides with the canonical stratification induced by the Morita type equivalenceclasses.

Any Morita equivalence between two proper Lie groupoids with cornersinduces a homeomorphism between their orbit spaces that preserves the canon-ical stratifications.

Proof. Let n = dimM . It was already shown in the proposition 4.0.4 thatthe canonical decompositions on M and X are stratifications and it is aWhitney stratification on M . It is a local problem to verify it is a Whitneystratification on X. By the local model 3.0.6 of G around a point x ∈M , weknow that it is enough to check this property for a proper action groupoidwith corners Gx ⋉ U where U ⊂ Nx

∼= Rnk for some k. The proposition

4.0.5 already shows that this is true. The Morita invariance on X is alsoestablished in proposition 4.0.4.

5 Reformulations on singular homology

This section presents several new results on singular chains and homologygroups which are new and useful. We will use them in the proof of the maintheorem. some new lemmas on singular chains and homology that will beused in the proof of the main theorem, which are important in itself.

15

First, we give a reformulate of the definition of singular chains and ho-mology groups, based on arbitray simplex, that is a bit more convenient insome situations.

Definition 5.0.1. (Reformulation of singular homology) A singular sim-plex in a topological space X is a continuous map f : ∆ → X from an-dimensional simplex ∆n in some Euclidean space Rm(m ≥ n), that is,a convex combination of a collection of n + 1 affine independent vectors(v0, v1, . . . , vn) in R

m, which means (v1 − v0, v2 − v0, . . . , vn − v0) are linearindependent, with a specific orientation, to X. Note that ∆n doesn’t haveto be a standard simplex. To emphasize this difference, we use notationδn for standard simplex. ∆n is the domain of f , which is also denoted byDom(f). The boundary of a singular simplex is the restriction of f to thefaces of ∆n in the standard way. A singular chain over Q is a finite ratio-nal linear combination of singular simplexes. And the singular homologygroups are defined parallel to standard definition of singular homology. Wedenote this new reformulation of singular chains and homology groups byCnewm,∗ (X,Q) an Hnew

m,∗ (X,Q) respectively. Note that for a specific m thesechains and homology groups are zero for dimension greater than m, as thereis no singular chain in these dimensions.

There is a canonical homomorphism from Hnewm,∗ (X,Q) to Hnew

m+1,∗(X,Q),

induced by the standard embedding of Rm into Rm+1. These homomor-phisms form a direct system. Denote Hnew

∗(X,Q) := lim→Hnewm,∗ (X,Q).

Lemma 5.0.2. Let k be a positive integer. The natural homomorphismf : Hnew

k,∗ (X,Q) → Hnewk+1,∗(X,Q) induced by the standard embedding from

Rk into Rk+1 is an isomorphism in dimension ≤ k.

Proof. Injectiveness: If c ∈ Hnewk,∗ (X,Q), d ∈ Hnew

k+1,∗(X,Q), and assume∂d = c by regarding c as its image Hnew

k+1,∗. One can slightly perturb the

simplexes in d when necessary and then project them into Rk, so that theresulting images P (d) are also simplexes in Rk. And all the simplexes inP (d) are homeomorphic to their corresponding ”inverse image” (under P )simplexes, denote these homeomorphisms by h. Then one can define a sin-gular chain complex d′ : P (d) → X by composition the hs with singularsimplexes in d. f is clearly that ∂d′ = c.

Surjectiveness: The general idea is: for a closed singular chain c ∈Hnew

k+1,∗(X,Q), one can take two steps: 1. perturb it slightly to get a new

chain c2 that is homologous to c, and 2. project c2 into Rk to get a chainc′ ∈ Hnew

k,∗ (X,Q) that is homologous to c2, identified c′ with its image inHnew

k+1,∗(X,Q). Thus the inverse image of [c] is c′ ∈ Hnewk,∗ (X,Q).

16

For step 1., we slight perturb Dom(c) to arrive at Dom2(c) such thatevery simplex in Dom2(c) is homeomorphic to corresponding simplex inDom(c). The perturbed singular chain with domain Dom2(c) is defined tobe the composition of these homeomorphisms with c. For the second step,if the perturbation is sufficiently small, one can guarantee that the projec-tion is also a closed simplex chain in Hnew

k,∗ (X,Z), which we denote it asDom(c′). The simplexes in Dom(c′) are homeomorphic to its corresponding”inverse image” simplexes in Dom(c2). The singular chain c′ is constructedas what we have done several times before by compositions of these home-orphisms with c2. The singular chain d′ ∈ Cnew

k+1,∗(X,Q) with ∂d′ = c2 ∪ c′is constructed as follows: for any simple σ in c2 in Rk+1, assume σ′ is itsprojection simplex in Rk, we form the ”band” set B(σ′) between σ and σ′

{(x, y)|x ∈ σ′, y ∈ [0, lc(x)]} where lc(x) is the last coordinate of the point inσ corresponding (inverse image) to x. Set a continuous map b : B(σ′) → Xwith b(x, y) = c′(x). We then divide B(σ′) into several simplexes in standardway (like how we divide rectangular into simplexes), and make sure thesesimplexes are compatible on the boundaries of these B(σ′)s. So when wesum over these bs restricted on divided simplexes, we get a singular chain d′

. It is clear that ∂d′ = c2 ∪ c′.

The following lemma shows that our new formulation of singular homol-ogy is isomorphic to the original one. In the rest part of this paper we willuse this new definition of singular homology.

Lemma 5.0.3. Homology groups Hnew∗(X,Q) are canonically isomorphic

to singular homology groups H∗(X,Q).

The proof uses the following nice result.

Lemma 5.0.4. Given an intger m ≥ 0, for any i-dimensional simplex inRm(i ≤ m), there is a continuous bijection from standard i-simplex δi to ∆i

such that all these maps together are compatible with boundary mappings.

Proof. We prove this by induction on dimensions and give an explicit con-struction. If i = 0,this case is trivial. For i = 1, we can first arbitrarilly set

an orientation for any ∆1. Assume ∆1 =−−→AB, one can set the values of points

in 1-standard simplex: (1, 0) maps to A, (0, 1) maps to B, and requires themap being linear on ∆1 which determines the values of all other points.Assume that we have set the continuous bijection for all i-dimensional sim-plex. Given a i+1-dimensional simplex ∆i+1, the way to set the continuousbijection from standard i+1-simplex to ∆i+1 is via the linear interpolation

17

of the values (which are the vertexes of ∆i+1)) of vertex of δi+1, i.e., for apoint P ∈ δi+1, get its barycentric coordinate xP = (x1, x2, · · · , xn), andmap it to the linear combination of the values of its vertexes with weightxP . It is straightforward to verify that what we construct is a continuousbijection.

Proof of Lemma 5.0.3. For a given integer m > 0, let us construct a map g :Cnewm,∗ (X,Q) → C∗(X,Q). For a singular simplex f : ∆i → X, denote g(f) :=

f ◦ h : δi → X where h is the continuous bijection from δi to ∆i definedabove. Then extend g linearly on Cnew

m,∗ (X,Q). It is straightforward to verifyg is a chain map, so it induces a homomorphism between Hnew

m,∗ (X,Q) andH∗(X,Q). We still denote this homomorphism by g. We should prove thesurjectiveness and injectiveness of g for dimensions < m, so establishing thatit is an isomorphism in these dimensions.

Surjectiveness: it is obvious, since any standard simplex is a simplex.Injectiveness: Fix an integer m > 0. Assume c =∈ Cm,k(X,Q) is a

boundary, c = ∂d, and there is a c′ ∈ Cnewm,k (X,Q) such that g(c′) = c. We

should prove that there is a d ∈ Cnewm,k+1(X,Q) with ∂d′ = c′ and g(d′) = d.

Let P1, P2, · · · , Pk be the vertexes of standard k + 1-simplexes in d that isnot in the vertexes of c, counted with multiplicity. We can choose distinctproperty placed vertexes Q1, Q2, · · · , Qt in R

m which is one to one correp-sponding to P1, P2, · · · , Pt respectively, and form a collection of simplexes,denoted by Dom′

k+1, by using them together with k − 1-dimension simplexchain c′ in mimic the configuration of the simplexes of d. One can thenconstruct d′ by compositions of singular simplexes in d′ and the continuousbijections constructed above. More precisely, if ∆k+1 is one k+1-dimensionalsimplex in Dom′

k+1, the singular simplex f ′ : ∆k+1 → X is defined to bef ′ := (f ◦ h−1 where g is the corresponding singular simplex in c and h isthe continuous bijection from δk+1 to ∆k+1. It is straightforward to verifythat d′ satisfies ∂d′ = c′ and g(d′) = d.

By lemma(5.0.2), it is clear now that Hnew∗(X,Q) is isomorphic to

H∗(X,Q).

Next we will describe a notion of singular simplicial complex.

Definition 5.0.5. A singular chain is called singular simplicial complex ifthere is a continuous map: g : SC → X, where SC is a simplicial complex insome euclidean space Rm, such that f =

∑∆∈SC g|∆, where ∆ is a primary

simplex, i.e., a simplex that is not a face of another simplex in SC.

18

Lemma 5.0.6. If a singular simplicial complex f is a boundary, i.e., thereis a singular chain g such that ∂g = f , then g is also a singular simplicialcomplex.

Proof. Recall that a simplicial complex is a collection of simplexes suchthat any face of a member of it is in it and any two members well intersect,i.e., they intersect along both of their common faces. Let g =

∑ni=1 fi

where fi : ∆i → X are singular simplex. The union of boundaries of fiare divided into two parts. First part consists of singular simplexes in fand another part otherwise. The singular simplexes in the second part mustcancel each other, which means the domain simplexes of g intersect wellalong the domain of these singular simplexes. For the first part, since fis already a singular simplicial complex, all the domain simplexes of thecorresponding singular simplexes in g intersect well. Note that it may hasmultiple singular simplexes in g corresponding to a singular simplex in f ,but they cancel out thus still well intersect. The domain of g must consistsof primary simplexes, as what of f does.

6 Main theorem

Now we prove our main theorem 1.0.3. Before that we take this opportunityto briefly mention a former effort of us to tackle this problem. We triedto use the global quotient theorem 3.0.8 and a result of Yang ([Y]) on thetriangulability of the orbit space of transformation groups. However, unfor-tunately we finally found that his result is incorrect due to using a resultof S. S. Cairns ([CSS]) on the triangulability of so called regular locallypolyhedral spaces that is known to be false now. And the global quotientconstruction on orbifolds with corners is also not very pleasant.

Proof of Theorem 1.0.3. Let G be a proper Lie groupoid with corners overM , being the orbifold structure of an orbifold with corners P in our ge-ometric chains. Proposition 4.0.4 shows that the canonical stratification(induced by the Morita equivalence 4.0.3) on the orbit space P ∼=M/G is astratification. And the proposition 4.0.6 shows that it is further a Whitneystratification. A classical result of Thom and Marden (cf. [Mat70]) showsthat any Hausdorff paracompact topological space embedded in a manifoldwithout boundary and with a Whitney stratification admits an abstractpre-stratification structure. The notion of abstract pre-stratification is anaxiomatization of the notion of control data in [Mat70]. See appendix fora precise definition. Although our orbit space P does not necessary be

19

embedded in a manifold, however, locally it did, as from the proof on theproposition 4.0.5. Or one can use the definition of Whitney stratification ona general differentiable space (cf. [CM]). So it indeed admits an abstractpre-stratification structure. A result of Verona (cf. [V]) then shows that anyHausdorff paracompact topological space with an abstract pre-stratificationstructure (in particular it has a stratification) admits a triangulation subor-dinate to the pre-stratification (the stratification underlying it). By a trian-gulation of a topological space X we mean a homeomorphism tri : X → |L|from X onto a underlying space of a simplicial complex L, and we say thatthis triangulation is subordinate to a stratification S of X if for any strataR ∈ S, tri|R is the underlying space of a simplicial subcomplex of L. Notethat the canonical stratification is compatible with the corner structure ofP .

In the above we proved that an orbifold with corners admits a triangu-lation compatible with the corner structure. For any geometric ”simplex”f : P → X, we choose such a triangulation trif . Define the following homo-morphism from homogenuous geometric chains of dimension m to singularchains Cnew

m+1,m(X,Z) in the sense of section 5:

gtrif : GCm(X,Q) → Cnewm+1,m(X,Q) (6)

(f : P → X) → Στf(tri−1f )|τ (7)

where P is a connected compact orbifold with corners of dimension m, τ isany simplex in the triangulation trif . Since P is compact, the right hand of(6) is finite. g commutes with differential d on both side. Thus it inducesa homomorphism from geometric homology groups of X GHm(X,Q) to

Hnewm+1,m(X,Q). We denote it by g

trifh .

gtrifh is surjective, since any connected simplex is a connected compact

orbifold with corners. Now let c ∈ Cnewm+1,m(X,Q) be a singular boundary

chain induced by f and trif whose domain Dom(c) = trif (P ), so thereis a d ∈ Cnew

m+1,m+1(X,Z) with ∂d = c. As by definition c is a simplicialsingular chain, lemma 5.0.6 shows d is also a simplicial singular chain. Onecan first remove all the faces in Dom(d) but not in Dom(c) to ”connect”corresponding simplexes, then one smooth out some corners (i.e.faces) inDom(c) according to its inverse image tri−1

f (c), a kind of ”detriangulation”.Then one get an orbifold with corners R whose face is homeomorphic to P ,and also an induced geometric chain fd : R→ X with its face fd|P equivalent

to f . We therefore show that gtrifh is injective. Using lemma 5.0.3 we deduce

that GHm(X,Q) ∼= Hm(X,Q).

20

Furthermore we can show that gtrifh is independent of the chosen trian-

gulations trif s. Let tri1f : P → L1 and tri2f : P → L2 be two triangulations

corresponding to a geometric ”simplex” f : P → X. As simplicial com-plexes, the simplicial homology groups of L1 and L2 are isomorphic that arealso isomorphic to the singular homology groups H∗(P,Q). There exists acommon refinement L of L1 and L2. Then the homology class of top dimen-sion simplexes in L is the same as the ones in L1 and L2. This means thatthe homology classes of top dimension simplexes in L1 and L2 represent the

same class in HdimP (P,Q). Thus gtri1

f

h (f) and gtri2

f

h (f) represent the sameelement in H∗(X,Q). This completes the proof.

Proof of Corollary 1.0.4. With the same steps as in the proof of the theorem

1.0.3, we arrive at a homomorphism gtrifh from GCm

∗ (X,Q) to C∗(X,Q).The subjectiveness and infectiveness of this homomorphism follow from theobservation that any simplex is a manifold with corners (a simplicial complexis thus a geometric complex with bases restricting to manifolds with corners).The canonical definiteness of gh also follows directly from the same reasonderived in the proof above.

As stated in the introduction, the geometric chains behavior nicely withsome natural operations. We list two of them.

1. (Pullback). Let X,Y be topological spaces and f : Y → X be afiber bundle with fiber F a smooth connected oriented manifold pos-sibly with corners (e.g. Sn), it induces a pullback f∗ : GC∗(X,Q) →GC∗(Y,Q) as usual.

Definition 6.0.1. For (x : P → X) ∈ GC∗(X,Q), f∗(x) : Ry−→ Y is

an element of GC∗(Y,Q) such that the following diagram commutes.

Z

��✴

✴

✴

✴

✴

✴

✴

✴

✴

✴

✴

✴

✴

✴

''PP

P

P

P

P

P

P

P

P

P

P

P

P

P

��

R //

y��

P

x��

Yf

// X

and it satisfies the universal property: if Z is another topological spacesatisfying the same property, then there is a unique morphism Z → Rmaking the above diagram commutes.

21

By universality, f∗(x) is unique up to a unique isomorphism. Theexistence can be deduced as follows: f∗(x) exists as a topological space.It is enough to check locally, in which case it is Y = X × F with fa projection onto the first factor. In this case f∗(x) is isomorphic to

P × F(f,id)−−−→ Y where P × F is a smooth connected oriented orbifold

with corners.

2. (Cartesian product). Let X,Y be two topological spaces, then thereis a functor prod : GC∗(X) × GC∗(Y ) → GC∗(X × Y,Q), defined asfollows.

Definition 6.0.2. if P and Q are two smooth connected orbifoldswith corners, then prod((fX , fY )) := (fX × fY )(R) where R is thecartesian product of orbifolds with corners P and Q with the inducedorientation, which is also an orbifold with corners.

As stated in the section of introduction, we expect our GHT has appli-cations in some situations where if using singular chains there will result insome difficulties. In the following section, we demonstrate this by using it inthe author’s work on Batalin-Vilkovsky (hereafter BV for short) algebraicstructures in open-closed topological conformal field theory.

7 Applications

As was briefly mentioned in the introduction, we find that the geometricchains is quite suited for the construction of BV structure in the compact-ified moduli space of Riemann surfaces with boundary and marked points,extending the work of Costello ([Cos04, Cos05]) for the algebraic defini-tions of Gromov-Witten potentials. Let us give a bit more details on thisconstruction. Costello originally tries to construct a BV structure on the sin-gular chains on compactified moduli spaces of Riemann surface with markedpoints. Unfortunately, the BV ∆ operator is defined via pull backs of chainsalong smooth fiberations, where it doesn’t exist for singular chains. He cir-cumvented this problem by using homotopy coinvariants with respect to S1

actions on singular chains. Equivariant homotopy constructions make ∆well defined then, it certainly requires much more efforts though.

An alternative way to define this BV structure found by us is to usegeometric chains instead of singular chains. It is quite simpler than toolsin S1 equivariant homotopy theory and is essentially elementary. A betterthing is that the main theorem (1.0.3) ensures that the further detailed

22

quantitative analysis in [Cos05] can be essentially extended directly to open-closed cases. Let us explain this a bit more. The following statements areon the BV structure on the space of geometric chains mentioned above.

Definition 7.0.1. Let V be a graded linear space over field k. A dg-BValgebraic structure on V is a quadruple (V, •, d, δ), satisfying the followingthree conditions:

1. (V, •, d) is a differential, graded, (graded)commutative, (graded)associativealgebra over k. The differential d is of degree 1 and d(1) = 0.

2. ∆ is a second order differential operator with respect to •, i.e. thedegree of ∆ is 1, ∆2 = 0, ∆(1) = 0, and for any given a, b, c ∈ V ,

∆(a • b • c) = ∆(a • b) • c+ (−1)|a|a •∆(b • c) + (−1)(|a|+1)|b|b •∆(a • c)−(∆a) • b • c− (−1)|a|a • (∆b) • c− (−1)|a|+|b|a • b • (∆c)

where || is the degree of an element.

3. graded commutator [d,∆] = d∆+∆d = 0.

If V is concentrated in degree 0 (with d trivial), then the dg-BV algebraicstructure is a BV algebraic structure.

Remark 7.0.2. Condition 2 is equivalent to the fact that the deviation of thederivative ∆ from being derivation, which is defined by

{, } := ∆(ab)−∆(a)b− (−1)|a|a∆(b)

is a (graded)Lie bracket and {, } is a (graded)derivation for each variables,i.e.

{a, bc} = b{a, c} + (−1)|b|{a, b}c{ab, c} = a{b, c} + (−1)|a|{a, c}b

The condition 3 is equivalent to d being a (graded)derivation for the Liebracket, , i.e.

d{a, b} = {da, b} + (−1)|a|{a, db}The open-closed analogy of the moduli spaces of Riemann surfaces with

marked points is the moduli spaces of boarded Riemann surfaces with markedpoints lying both on the interior and boundaries. More precisely, considerthe moduli spaces of isomorphism classes of stable bordered Riemann sur-faces of type (g, b) with (n, ~m) marked points and certain extra data, namely,decorations by a real tangent direction, i.e., a ray, in the complex tensor

23

product of the tangent spaces on each side of each interior node. We denoteit by M b,~m

g,n . We will be working on the moduli spaces:

M b,mg,n / = (

∏

~m:∑

mi=m

M b,~mg,n /Zm1

× · · · × Zmb)/b × n

of stable bordered Riemann surfaces as above with unlabelled boundarycomponents and marked points, that is, the quotient of the disjoint unionM b,m

g,n / =∏

~m:∑

mi=mMb,~mg,n of moduli spaces with labelled boundaries and

marked points by an appropriate action of the permutation group =(∏

~m:∑

mi=m Zm1

× · · · × Zmb) × b × n. M

b,~mg,n and M b,m

g,n / are both ori-

ented orbifolds with corners with dimension equals 6g + 6 + 2n + 3b +m.The latter is equipped with a local system Qǫ coming from a certain signrepresentation of on a one-dimensional vector space L over Q, which isthe orientation sheaf of M b,m

g,n /. Qǫ is the orbifoldic construction from the

trivial determinant bundle onM b,mg,n (as it is oriented) acted by . See [VHZ]

for details.The space we will introduce a BV algebraic structure is the space of

geometric chain complexes.

V :=⊕

b,m,n

Cgeom∗ (M b,m

n /;Qǫ)

whereM b,mn / is the moduli space of real compactification of stable bordered

Riemann surfaces with b boundary components, n interior marked points,and m boundary marked points, just like the Riemann surfaces in M b,m

g,n /,but in general having multiple connected components of various genera.

One can canonically define an operator ∆ = ∆c +∆o +∆co on V . It isproved that this is a BV operator.

Theorem 7.0.3. The operator ∆ = ∆c+∆o+∆co is a graded second-orderdifferential on the dg graded commutative algebra V and thereby defines thestructure of a dg BV algebra on V .

We will not present a detail explanation of the construction of ∆, in-stead, we refer the interested readers to [VHZ] for a very clear account. Wejust want to emphasize that the BV operator is more naturally and easilyconstructed in the space of geometric chain complexes than the space ofsingular chain complexes. This is also a reason we prefer using geometricchains in this case.

This nice result is essentially not completely new, as it is the mathe-matical rigorous presentation of the ideas developed originally in physics by

24

physicist Sen and Zwiebach. For details, see ([SZ]) for closed case and [Zwi]for open-closed case.

To develop a pure algebraic treatment of Gromov-Witten invariants, oneneeds first replace the fundamental classes of the compactified moduli spacesof Riemann surfaces with boundary and marked points by a certain solutionof Quantum master equation associated with the BV algebra constructedabove. The fundamental result is the following existence and uniqueness ofsolutions of the associated quantum master equation.

Let V b,mg,n be the image of the inclusion of Cgeom

∗ (M b,mg,n /;Q

ǫ) → V .

Theorem 7.0.4. For each g, n,m, b with 2− 2g−n− b−m/2 < −12 , there

exists an element Sb,mg,n ∈ V b,m

g,n of degree 0, with the following properties.

1. S0,00,3 is a 0-chain in the moduli space of Riemann spheres with 3 un-

parameterised, unordered closed boundaries and with no open or freeboundaries.

2. Form the generating function

S =∑

g,n,b,m≥02g+n+b+m/2−2>0

~p−χλ−2χSb,mg,n ∈ λF (M)[[

√~, λ]]

here p = 1− m+n2 , χ = 2− 2g − n− b− m

2 .

S satisfies the Batalin-Vilkovisky quantum master equation:

deS/~ = 0

(Remember that in dg BV algebra B[[~, λ]], we let d = d+~∆). Equivalently,

dS +1

2{S, S} = 0

Further, such an S is unique up to homotopy through such elements.

This result has the same form as the one for closed case. By our maintheorem, the proof is essentially identical to the proof in [Cos05]. For moredetails see [HY].

As for closed cases, this solution also encodes the fundamental chains ofmoduli spaces of Riemann surfaces with boundary and marked points, asdemonstrated in [VHZ]:

25

Theorem 7.0.5. For the notation of above theorem, if we take Sb,mg,n :=

[M b,mg,n /] ∈ Cgeom

0 (M b,mg,n /;Q

ǫ) be the fundamental (geometric) chains. ThenS satisfies the quantum master equation:

(d+ ~∆)eS/~ = 0

Together with results in [Cos05], these theorems essentially state thatthe fundamental classes of moduli spaces of Riemann surfaces with markedpoints, or the (geometric) fundamental chains of moduli spaces of boardedRiemann surfaces with marked points, are encoded in the unique (up to ho-motopy) solution of the quantum master equation associated with a canon-ically defined BV algebraic structure on the singular (or geometric) chaincomplexes. From this point, one can proceed to define Gromov-Witten po-tential for any (open-closed) topological conformal field theory, a subjectcalled categorical GW theory initiated by Costello, motivated by the ho-mological picture of mirror symmetry of Kontesvich. The closed case wascompleted, the open-closed case is still open, I guess.

Finally, we remark that the theory in this paper only hold for differ-entiable orbifolds with corners. We don’t expect similar results hold intopological category, i.e. topological orbifolds with corners (in whateversense), at least these can not be deduced using methods in this paper, sinceC.Manolescu ([Man]) showed that there is a topological manifold in eachdimension ≥ 5 that is not triangulable. We will leave it as an interestingtopic for future study.

8 Appendix

A closely related notion to the notion of Whitney stratification is abstractpre-stratification. It is a very essential concept in the stratification andtriangulation theory (cf. [Mat70, V]).

Definition .0.1. An abstract pre-stratified set is a triple (V,S,J ) satisfyingthe following axioms.

1. V is as Hausdorff, locally compact topological space with a countablebasis for its topology.

2. S is a family of locally closed subsets of V , such that V is the disjointunion of the members of. The members of S will be called the strataof V .

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3. Each stratum of V is a topological manifold (in the induced topology),provided with a smoothness structure.

4. The family S is locally finite.

5. The family S satisfies the axiom of the frontier: if X,Y and Y ∩X 6= ∅,then Y ⊆ X . If Y ⊆ X and Y 6= X, we write Y < X. This relation isobviously transitive: Z < Y and Y < X imply Z < X.

6. J is a triple {(TX ), (πX), (ρX)}, where for each X ∈ S, TX is an openneighbourhood of X in V , πX is a continuous retraction TX of ontoX, and ρX : X → [0,∞) is a continuous function. We will TX call thetubular neighbourhood of X (with respect to the given structure of aprestratified set on V ), πX the local restriction of TX onto X and ρXthe tubular function of X.

7. X = {v ∈ TX : ρX(v) = 0}. If X and Y are any strata, we letTX,Y = TX ∩ Y , πX,Y = πX |TX,Y , and ρX,Y = ρX |TX,Y ,. Then πX,Y

is a mapping of TX,Y into X and ρX,Y is a mapping ρX,Y into (0,∞).Of course, TX,Y may be empty, in which case these are the emptymappings.

8. For any strataX and Y the mapping (πX,Y , ρX,Y ) : TX,Y → X×(0,∞)is a smooth submersion. This implies dim X < dim Y when TX,Y 6= ∅.

9. For any strata X,Y , and Z, we have πX,Y πY,Z(v) = πX,Z(v)ρX,Y πY,Z(v) = ρX,Z(v) whenever both sides of this equation are satis-fied, i.e., whenever v ∈ TY,Z and πY,Z(v) ∈ TX,Y .

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Hao Yu, School of Mathematics, University of Minnesota, Minneapolis, MN55455, USA, and

AI center of Cyclone, Chaoyang District, Beijing 100000, China.dustless2014@163.com

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