research statement - city university of new...

Research Statement Kate Poirier

This document contains a detailed outline of my work. A shorter version is available on mywebsite: math.berkeley.edu/~poirier/Research . Experts may wish to skip the first section.

My research addresses the question, “what is the algebraic topology of a manifold?” I study thealgebraic, geometric, and combinatorial topology of manifolds—specifically, their string topology—and the moduli space of Riemann surfaces.

1. Background and Introduction

One goal of algebraic topology is the classification of topological spaces using algebraic invariants.Certain classes of spaces admit invariants with richer algebraic structure than other classes do,so these classes lend themselves more easily to classification. For example, the homology of atopological space is an abelian group. However, if the space is a manifold then its homology hasan extra product giving it the structure of a ring. Sometimes, new algebraic language must bedeveloped to capture the structure that we see.

String topology studies algebraic structures arising from intersections of loops in a manifold. Thesestructures are often phrased in terms of an action of some algebraic object on a vector space. Itis interesting to ask what the appropriate algebraic object is for capturing the structure in aparticular situation. The structures studied in string topology are governed by the moduli spaceof Riemann surfaces.

A Riemann surface is a manifold locally modeled on the complex plane. A Riemann surface withboundary is a manifold with boundary locally modeled on the upper-half plane. See Figure 1.

Figure 1. A Riemann surface and a Riemann surface with boundary

The moduli space M(g, n) parametrizes Riemann surfaces with genus g and n boundary compo-nents. It is a classical object of interest in topology, geometry, and physics. While it has beenstudied from these different perspectives since Riemann’s time, many of its properties remainmysterious. For example, the homology of moduli space is well understood only when g is small.

While we do not have a complete understanding of moduli space, it provides several examplesof algebraic structures. These structures arise via actions of moduli space on sets or spaces—oractions of linear models of moduli space on vector spaces or chain complexes. The language of

Research Statement (page 2 of 12) Katherine Poirier

quantum field theory or of PROPs is convenient in this setting. We consider surfaces have genusg, k boundary components designated as incoming, and the remaining ` boundary componentsdesignated as outgoing.

We denote the moduli space of these surfaces by M(g, k, `). Two surfaces may be composed byidentifying outputs of one with inputs of the other as in Figure 2.

Figure 2. Composing a (2, 3, 2) surface with a (1, 2, 1) surface to obtain a (4, 3, 1) surface

Roughly, an action of moduli space on a set S is an assignment of an operation S×k → S×` toeach Riemann surface in M(g, k, `) so that composition of surfaces is assigned to composition ofoperations.

Based on previous work of Chas and Sullivan [CS99, CS04], Cohen and Godin [CG04], and Chataur[Cha05], Godin [God07] and Kupers [Kup11] each describe an action of the homology of modulispace on the homology of the free loop space of a manifold M .

The loop space LM of a manifold M is the space of continuous maps from the circle to M .

The action of the homology of moduli space on the homology of the free loop space is a general-ization of a structure discovered by Goldman [Gol86] and Turaev [Tur91] by intersecting curveson surfaces.

Fix an oriented surface Σ and let V be the Q-vector space generated by free homotopy classes ofloops on Σ. For free homotopy classes α and β, choose representative loops a and b that intersectone another only in transverse double points p. For such an intersection point, let a ·p b be theloop obtained by cutting a and b at p and let α ·p β be the free homotopy class of a ·p b. See Figure3.

Goldman defined a 2-to-1 operation [ , ] : V ⊗ V → V on generators by

[α, β] =∑p∈a∩b

α ·p β.

He showed that this operation is well defined on free homotopy classes and gives V the structureof a Lie algebra [Gol86].


p

Figure 3. Cutting and reconnecting two loops to obtain one loop

Analogously, Turaev defined a 1-to-2 operation ∆ : V → V ⊗V by cutting and reconnecting loopsat self-intersection points as in Figure 4. He showed that V , together with [ , ] and ∆, is a Liebialgebra [Tur91].

Figure 4. Cutting and reconnecting one loop to obtain two loops

These 2-to-1 and 1-to-2 operations may be generalized to k-to-` operations V ⊗k → V ⊗`. Theseare defined when k loops intersect in a prescribed way and can be cut and reconnected to form `loops.

Chas and Sullivan’s first string topology operation, now known as the Chas-Sullivan product,generalizes Goldman’s bracket for manifolds M of any dimension. Let M be a closed, orientedmanifold of dimension d. Roughly, the Chas-Sullivan product

H∗(LM)⊗H∗(LM)→ H∗−d(LM)

is defined by combining intersection theory in M and composition of loops in M sharing a base-point.

Consider a singular i-simplex and singular j simplex of LM . These determine an i-dimensional andj-dimensional family of loops in M respectively. Assume that the i-dimensional and j-dimensionalfamilies of basepoints intersect one another transversally in M as in Figure 5.

The intersection locus parametrizes an i + j − d-dimensional family of composable loops. Com-posing the loops along the intersection locus gives an i+ j − d-dimensional family of loops as inFigure 6.

This defines a partial product on singular chains

C∗(LM)⊗ C∗(LM)→ C∗−d(LM)

which determines the fully defined product on homology

H∗(LM)⊗H∗(LM)→ H∗−d(LM).

The homology of the free loop space also comes equipped with a BatalinVilkovisky (BV) operatorH∗(LM)→ H∗+1(LM). Chas and Sullivan showed that H∗(M), together with the Chas-Sullivanproduct and BV operator, is a BV algebra.


Figure 5. Two 2-dimensional families of loops in a 3-dimensional manifold andtheir families of basepoints, intersecting transversally

Figure 6. Composable loops along the intersection locus, and the family of com-posed loops

Cohen and Godin [CG04] generalized a homotopy-theoretic realization of the Chas-Sullivan prod-uct developed by Cohen and Jones [CJ02] to define k-to-` operations. Cohen and Godin’s op-erations use a special class of fatgraphs they call Sullivan chord diagrams, which keep track ofintersections of loops.

A Sullivan chord diagram is a particular type of fatgraph which determines an orientable surfacewith incoming and outgoing boundary.

A boundary component of a surface Σ is a copy of a circle, so if a surface has k incoming boundarycomponents and ` outgoing, a map of Σ into M determines k incoming loops and ` outgoing loopsin M . Figure 7 shows pairs of pants giving rise to 2-to-1 and 1-to-2 operations and a generalsurface giving rise to a k-to-` operation.

Let Maps(Σ,M) be the space of maps of Σ into M and let in and out denote the maps given byrestricting to incoming and outgoing boundary respectively:

LM×kin←−Maps(Σ,M) out−→ LM×`.


incomingoutgoing

2 incoming 1 outgoing 2 outgoing1 incoming

Figure 7. Surfaces giving 2-to-1, 1-to-2, and k-to-` operations

Cohen and Godin apply a Pontryagin-Thom construction to obtain a wrong-way map on homology

H∗(LM)⊗k in!−→ H∗+χd(Maps(Σ,M))

where χ is the Euler characteristic of Σ.

They compose in! with the map induced on homology by out to obtain a k-to-` string topologyoperation

H∗(LM)⊗k → H∗(LM)⊗`.In the case that the Σ is a pair of pants with two incoming and one outgoing boundary component,Cohen and Godin’s construction recovers the Chas-Sullivan product

H∗(LM)⊗H∗(LM)→ H∗−d(LM).

The assignment of a string topology operation respects composition of surfaces.

Further, the operation depends only on the topological type of the surface determined by theSullivan chord diagram and the space of Sullivan chord diagrams is connected. Letting S be thespace of Sullivan chord diagrams, this gives an action of H0(S) on H∗(LM). The action givesH∗(LM) the structure of a Frobenius algebra without counit and provides an example of a positiveboundary topological quantum field theory.

Chataur later extended this to an action of H∗(S) on H∗(LM) [Cha05].

The space of Sullivan chord diagrams S is a subspace of the moduli space of Riemann surfacesM.Godin extended the action to one of H∗(M) on H∗(LM) [God07]. Kupers has recently simplifiedthe description of this action. [Kup11].

Much of my recent and current work addresses two related questions.

First, it is a general principle that algebraic structure on homology is inherited from a richerstructure on chain complexes computing the homology.

Question 1. What is a concrete geometric construction realizing this richer structure on a chaincomplex computing the homology of the free loop space?

In what follows, we use the complex of singular chains on the free loop space of the manifold M .

Second, the moduli space is non-compact in general. We expect that the appropriate space forcapturing the richer algebraic structure is a compactification of moduli space.


Question 2. What is the compactification of moduli space that is appropriate for string topology?

2. Recent Work

2.1. Compactifying string topology [PR11], with Nathaniel Rounds. Using ideas in mythesis [Poi10], we define a family of compact cell complexes of string diagrams and describehow cells give operations on the singular chains of the free loop space LM of a closed, orientedd-dimensional Riemannian manifold M .

Definition 1. A simple string diagram of type (g, k, `) is a marked metric fatgraph constructedfrom k input circles and χ intervals with endpoints attached along the circles. The fatgraphstructure is such that the resulting surface has genus g and k + ` boundary cycles, k of whichcorrespond to the k input circles. The remaining ` boundary cycles are called output circles. SeeFigure 8.

Figure 8. The surface associated to a simple string diagram of type (1, 3, 3).Input and output marked points are indicated by vertices the graph together witha line joining each to the boundary of the surface.

We denote the space of simple string diagrams of type (g, k, `) by SD(g, k, `) and show that it isa cell complex. We then describe a construction that produces a map

ST : C∗(SD(g, k, `))⊗ C∗(LM)⊗k −→ C∗−χd(LM)⊗`

for cellular chains of SD(g, k, `), singular chains of LM , and coefficients in any commutative ring.

Theorem 1. ST : C∗(SD(g, k, `))⊗ C∗(LM)⊗k −→ C∗−χd(LM)⊗` is a chain map.

In particular, our construction produces, for each cellular chain of SD(g, k, `), a k-to-` operationon the singular chains of the loop space.

Previous string topology constructions make a transversality assumption to use intersection the-ory. Rather than making such an assumption, our construction uses a “diffuse intersection”theory; we use short geodesic segments joining nearby points on loops rather than requiring theloops to intersect. This is an important ingredient in allowing us to define operations at the chainlevel, rather than just on homology.

Theorem 2. With field coefficients, the operations induced on homology by ST recover knownoperations as defined by Chas-Sullivan and Cohen-Godin.

We consider an equivalence relation ∼ on SD(g, k, `) called slide equivalence. While the chain-level operations described above are not well defined on SD(g, k, `)/∼, we make a series of choicesto construct a map inducing well-defined operations on homology with field coefficients:

H∗(SD(g, k, `)/∼)⊗H∗(LM)⊗k −→ H∗−χd(LM)⊗`.


This work details the first step of a non-equivariant story which parallels the equivariant oneappearing in my thesis. While only part of the story, it is still a considerable improvement. Forexample, the operations here are defined for the complex of singular chains on LM as opposed toa more complicated chain complex. Additionally, the new chain-level construction for SD worksfor any coefficient ring, as opposed to field coefficients.

2.2. String diagrams and moduli space [Poi]. Simple string diagrams provide a connectionbetween string topology operations and moduli spaces of decorated Riemann surfaces. In thiswork, I add extra combinatorial data and additional parameters in (0, 1] called spacing parametersto the simple string diagrams as described above and define spaces LD(g, k, `) of these newdiagrams. I also extend the description of slide equivalence to LD(g, k, `) and show that thequotient gives a compactification of moduli space.

Theorem 3. LetM(g, k, `) be the moduli space of Riemann surfaces of genus g with k+` boundarycomponents. There is a continuous map

M(g, k, `) ↪→ LD(g, k, `)/∼

whose image is open and dense. In fact, the image is a union of open cells.

Let LD(g, k, `)/ ∼ be the compactification M(g, k, `). The proof uses harmonic functions onRiemann surfaces. Later, we will use M(g, k, `) as a guide for building a solution to the mas-ter equation. This compactification of moduli space satisfies a number of properties that areconvenient in this regard.

Proposition 1. (1) M(g, k, `) is a pseudomanifold with boundary.(2) Points at the boundary of M(g, k, `) satisfy at least one of two conditions:

(a) an output circle contains no edges from input circles, or(b) there is a spacing parameter equal to 1.

(3) The space M(g, k, `) contains SD(g, k, `)/∼ as a deformation retract.

3. In Progress

3.1. Compactified combinatorial string topology [DCPR], with Gabriel C. Drummond-Cole and Nathaniel Rounds. The construction provided in [PR11] described above suggeststhat there is more structure than what is captured by the spaces SD(g, k, `). For example,

• the spaces SD(g, k, `) are too small to capture relations among operations arising fromslide-equivalent cells of SD(g, k, `) (the operations “differ by a homotopy”),• composition of simple string diagrams (or of chains on the spaces of string diagrams) is not

defined, so it is not clear what algebraic structure the construction defines on C∗(LM).In current work we address these two issues [DCPR]. Building on ideas appearing in my thesisand in [PR11], we define a much larger space of fatgraphs which we refer to as string diagrams.The definition of a string diagram is somewhat technical, but it is similar to that of a simplestring diagram. The main difference is that in a string diagram we attach endpoints of metrictrees (satisfying a metric condition) instead of just intervals. We also add spacing parameters in(0, 1] as we did for diagrams in LD(g, k, `) above.

We denote the space of string diagrams of type (g, k, `) by SD(g, k, `) and define a composition ofstring diagrams

SD(g, k, `)× SD(g′, k′, `′) −→ SD(g′′, k′′, `′′).


Keeping in mind that, for the master equation package, we are seeking a space that parametrizeschain-level string topology operations and has prescribed boundary, we have the following conve-nient properties of SD(g, k, `).

Proposition 2. (1) The space SD(g, k, `) is pseudomanifold with boundary.(2) An string diagram that can be written as a composition of extended string diagrams lies

in the boundary of SD(g, k, `).(3) Composition of string diagrams induces a properad structure on the collection of complexes

of cellular chains of SD(g, k, `).

Denote this properad by C∗(SD).

There are more technical points to consider in describing a string topology construction for SDthan there were for SD, but we address them to define chain maps

ST : C∗(SD(g, k, `))⊗ C∗(LM)⊗k −→ C∗−χd(LM)⊗`

which respect composition. Again, we use cellular chains on SD(g, k, `) and singular chains on LMand apply a construction which uses diffuse intersection and generalizes the geodesic constructiondescribed above.

Theorem 4. The complex of singular chains of the loop space C∗(LM) is an algebra over theproperad C∗(SD).

In particular, we witness more algebraic structure on C∗(LM) than was evident from the opera-tions arising from SD.

3.2. String topology and the quantum master equation [DCP], with Gabriel C. Drummond-Cole. The spaces SD(g, k, `) are still not quite the right spaces for the chain-level structure weexpect to see. This time, they are too big; certain cells contain parameters which do not playa role in the string topology construction. We call such parameters “redundant.” We believe weunderstand the general situation. Here is an interesting example.

The cellular chains on SD(0, 2, 2) parametrize genus zero, two input, two output operations. Thespace SD(0, 2, 2) is a torus bundle over a 3-dimensional space which is a product of a disk withtwo holes and a closed interval as shown in Figure 9. The homotopy type of this space is thesame as that of the moduli space M(0, 2, 2). In certain cells of SD(0, 2, 2) there are redundantparameters. If we forget these redundant parameters the space we obtain has a very nice descrip-tion: it is homeomorphic to the compactified moduli space of Riemann surfaces as described byLD(g, k, `)/∼ in Section 2.2 above.

Figure 9. The space SD(0, 2, 2), the quotient SD(0, 2, 2)/∼ given by forgettingredundant parameters, and the quotient of SD(0, 2, 2)/∼ given by collapsing eachsmall output component of the boundary to a point.


An analogous example appeared as part of the equivariant story in my thesis but this is again asignificant improvement. There, the construction of operations depended on some non-canonicalchoices while here there are no choices; it is more natural to forget the redundant parameters.

We expect a complete combinatorial characterization of all redundancies for general (g, k, `).

Composition is well defined in the quotient by redundancies SD/ ∼ and induces a properadstructure on the cellular chains, denoted C∗(SD/ ∼).

Theorem 5. The complex of singular chains of the loop space C∗(LM) is an algebra over theproperad C∗(SD/ ∼).

The quotient by redundancies reveals more of the algebraic structure on C∗(LM) given by thestring topology construction.

Conjecture 1. (1) The quotient SD(g, k, `)/∼ of SD(g, k, `) by forgetting redundant param-eters is a pseudomanifold with boundary.

(2) Cells at the boundary give rise to operations coming from compositions and to operationsproducing “small outputs.”

For solving the master equation, “composition boundary” is good, while “small output boundary”is somehow bad. The small output part of the boundary can be mysterious in the non-equivariantstory. There is a sense in which the operations producing them are essentially trivial, but anothersense in which these operations describe much of the algebraic topology of M itself. There havebeen ways around this issue, as in Basu’s transversal string topology [Bas11] or working relativeto small loops as in the usual equivariant story [Poi10].

Our goal is to understand and deal with the small outputs produced by our diffuse intersectionconstruction. The collection of string topology operations given by fundamental chains of quotientspaces produced from SD(g, k, `)/∼ by collapsing boundary cells producing small outputs can bedescribed in terms of a master equation package. In particular, the master equation packagedescribes the algebraic structure on C∗(LM) obtained from these operations.

Additionally, we plan to follow details of the parallel equivariant story, where this collection offundamental chains introduces a possible model for a free resolution of the properad governinginvolutive Lie bialgebras and describes such a structure on reduced equivariant chains of LM .

3.3. String topology operations for mapping spaces [AP], with David Ayala. In a newcollaboration, we are discussing a version of diffuse intersection that gives rise to much moregeneral string-topology-type operations. In this case, we focus on mapping spaces Maps(K,M)where M is a closed, oriented manifold and K is any topological space. Just as traditional stringtopology allows us to study the manifold M by studying its free loop space LM , these operationsallow us to study M by studying mapping spaces Maps(K,M) for different topological spaces K.

In this setting, we create an analog of extended string diagrams called pinch configurations andan analog of the moduli space that provides operations. A string diagram Γ comes equippedwith canonical maps

⊔k S1 → Γ and⊔` S1 → Γ as boundary cycles. Here, we include maps of

topological spaces K and L into a pinch configuration as part of the data.

Roughly, a pinch configuration consists of the following data:(1) topological spaces K and L, called input and output respectively,(2) a finite set S of K equipped with a partition T ,


(3) a continuous map from L to a space formed by adjoining cones of each subset of T to K.

Note there is a canonical inclusion of P into a pinch configuration. Let Pinch(K,L) denote thespace of pinch configurations with input K and output L.

Presently, we have a sketch of a construction that produces, from particular points inPinch(K,L)×Maps(K,M) a map in Maps(L,M). We are working on completing the construc-tion in such a way that it recovers the operations for loop spaces above.

Theorem 6. The chains on the space of pinch configurations act on the chains on the space ofmaps into M .

The construction recovers known structures for loop spaces. It also recovers, for example, theknown En-algebra structure on H∗(Maps(Sn,M)) for n ≥ 1.

We have begun developing the appropriate algebraic language that captures this action.

4. Future Research

4.1. Open-closed string topology . For the most part, we have focused on “non-open” stringtopology, that is the non-equivariant story for Maps(S1,M). The diffuse intersection constructioncan also be applied to spaces of open strings in M , that is, spaces of paths in M , perhaps beginningand ending in specified submanifolds (branes). Drummond-Cole and I currently have an explicitdescription of open string topology operations when endpoints of paths may lie anywhere in M .I plan to generalize this description to include endpoints labeled by any collection of branes todevelop the full open-closed chain-level theory.

Upon a full description, it will be natural to ask whether the open-closed theory arises from apurely open theory, as outlined in Kevin Costello’s work [Cos07] and in Nathalie Wahl and CraigWesterland’s work [WW11].

4.2. Homotopy invariance of string topology . It is unknown to what extent the algebraicstructure arising from the full open-closed theory, or from some part of it, or from the equivarianttheory, could be invariant under homotopy equivalence. Two recent results suggest that it mightbe possible that some version of string topology could be sensitive to the homeomorphism typeof the manifold M .

• Basu’s transversal string topology uses a construction that is different from, but not whollyunrelated to, the generalized geodesic one above. His construction recovers an algebraicstructure that is known to distinguish two lens spaces that are homotopy equivalent butnot homeomorphic [Bas11].• The Lie bracket of equivariant string topology reduces to the Goldman bracket in the case

that M has dimension two and the homological dimension is zero. Gadgil has recentlyshown that a homotopy equivalence between two compact, oriented surfaces with boundaryis homotopic to an orientation-preserving homeomorphism if and only if it commutes withthe Goldman bracket [Gad11].

With these results in mind, I am very interested in pursuing the question of homotopy invarianceof the algebraic structures as described above.


4.3. Compactified moduli space and the space of string diagrams. The spaces of stringdiagrams SD(g, k, `) above, parametrize chain-level string topology operations. We see from theexample (g, k, `) = (0, 2, 2) that the space of string diagrams has a strong relationship with themoduli space of Riemann surfaces.

Conjecture 2. The space SD(g, k, `) is homeomorphic to a compactification of moduli spaceM(g, k, `) that has the homotopy type of M(g, k, `) itself.

While the combinatorics of the cell complex SD(g, k, `) can be rather complicated, even in small-genus examples, it could still be useful for computing the homology of open moduli space. Usinga different model for moduli space, Carl-Friedrich Bodigheimer and his students have had somesuccess in applying discrete Morse theory to perform such computations [Wan11, Meh11]; it isquite possible that SD(g, k, `) is is also well-adapted to these techniques.

Further, the parameters of SD(g, k, `) which are redundant for string topology exist only on thepseudomanifold boundary of SD(g, k, `). Forgetting them would also describe a compactificationof moduli space.

Conjecture 3. The space SD(g, k, `)/∼ is homeomorphic to the compactificationM(g, k, `) givenby LD(g, k, `)/∼ above.

The characterization of M(g, k, `) using SD(g, k, `)/∼, while still combinatorial, gives a moreconcrete description than the one given by LD(g, k, `)/∼; the redundant parameters that areforgotten correspond to collapsing curves in the surface.

4.4. String topology and unstable homology of moduli space. Moduli space provides infor-mation about the string topology of M via the action of its homology on the homology of the freeloop space of M . String topology may provide information about moduli space as well. Tamanoihas shown that homology classes in the images of Harer’s stability maps

H∗(M(g, k, `))→ H∗(M(g + 1, k, `))

yield trivial string topology operations [Har85, Tam09]. Very little is known about the unsta-ble homology of M(g, k, `). If we were able to find a manifold M and a homology class inH∗(M(g, k, `)) that gave a nontrivial operation, then we would know that the homology class isunstable. While compactifying moduli space changes its topology, it may still be possible to usechain-level string topology to detect unstable classes in open moduli space. Indeed, operationscoming from stable classes may be trivial because they vanish in the compactification. Wahl hassome promising preliminary computations of nontrivial operations for M = S3. With this inmind, I am very interested in performing more computations to understand how compactifyingmoduli space affects the stable and unstable classes and how this relates to the vanishing ornon-vanishing of string topology operations.

References

[AP] David Ayala and Kate Poirier, String topology for mapping spaces, In progress.[Bas11] Somnath Basu, Transversal string topology and invariants of manifolds, Ph.D. thesis, Stony Brook Uni-

versity, 2011.[CG04] Ralph L. Cohen and Veronique Godin, A polarized view of string topology, Topology, geometry and quan-

tum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge,2004, pp. 127–154. MR 2079373 (2005m:55014)

[Cha05] David Chataur, A bordism approach to string topology, Int. Math. Res. Not. (2005), no. 46, 2829–2875.MR 2180465 (2007b:55009)


[CJ02] Ralph L. Cohen and John D. S. Jones, A homotopy theoretic realization of string topology, Math. Ann.324 (2002), no. 4, 773–798. MR 1942249 (2004c:55019)

[Cos07] Kevin Costello, Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007),no. 1, 165–214. MR 2298823 (2008f:14071)

[CS99] Moira Chas and Dennis Sullivan, String topology, arXiv preprint, 1999, math.GT/9911159v1, To appearin the Annals of Mathematics.

[CS04] Moira Chas and Dennis Sullivan, Closed string operators in topology leading to Lie bialgebras and higherstring algebra, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 771–784. MR MR2077595(2005f:55007)

[DCP] Gabriel C. Drummond-Cole and Kate Poirier, String topology and the master equation, In progress.[DCPR] Gabriel C. Drummond-Cole, Kate Poirier, and Nathaniel Rounds, Combinatorial compactified string topol-

ogy, In progress.[Gad11] Siddhartha Gadgil, The Goldman bracket characterizes homeomorphisms, arXiv preprint, 2011,

math.GT/109.1395v2.[God07] Veronique Godin, Higher string topology operations, arXiv preprint, 2007, math.AT/0711.4859v2.[Gol86] William M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group repre-

sentations, Invent. Math. 85 (1986), no. 2, 263–302. MR 846929 (87j:32069)[Har85] John L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math.

(2) 121 (1985), no. 2, 215–249. MR 786348 (87f:57009)[Kup11] Sander Kupers, String topology operations, M.Sc. thesis, Utrecht University, 2011.[Meh11] Stefan Mehner, Homologieberechnungen von Modulraumen Riemannscher Flachen durch diskrete Morse-

Theorie, Diploma thesis, Universitat Bonn, 2011.[Poi] Kate Poirier, String diagrams and moduli space, In progress.[Poi10] , String topology and compactified moduli spaces, Ph.D. Thesis, City University of New York, 2010.[PR11] Kate Poirier and Nathaniel Rounds, Compactifying string topology, arXiv preprint, 2011,

math.GT/1111.3635v1.[Tam09] Hirotaka Tamanoi, Stable string operations are trivial, Int. Math. Res. Not. IMRN (2009), no. 24, 4642–

4685. MR 2564371 (2010k:55015)

[Tur91] Vladimir G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Ecole Norm.Sup. (4) 24 (1991), no. 6, 635–704. MR MR1142906 (94a:57023)

[Wan11] Rui Wang, Homology computations for mapping class groups, in particular for Γ03,1, Ph.D. Thesis, Uni-

versitat Bonn, 2011.[WW11] Nathalie Wahl and Craig Westerland, Hochschild homology of structured algebras, arXiv preprint, 2011,

math.AT/arXiv:1110.0651.

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