Questions tagged [string-topology]

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2
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0answers
108 views

Splitting of chains of loop space

Evaluation at base point induces a splitting of homology of free loop space $LM$ of a compact manifold $M$, i.e. $H_*(LM)\cong H_*(M) \oplus H_*(LM, M)$. Can such splitting be realised on cellular ...
5
votes
0answers
73 views

How does the $C^\ast$ algebra of an orbifold grupoid relate to the corresponding orbifold?

My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is ...
5
votes
0answers
109 views

Equivariant L-infinity structure associated to a DGBV algebra

Let $V$ be a differential graded Batalin-Vilkovisky (DGBV) algebra over say $\mathbb{C}$. This means that $V$ is a commutative differential graded algebra (CDGA), with differential $\partial$ of ...
5
votes
1answer
243 views

Algebraic models of non-simply connected spaces in string topology

I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space $X$ is simply-connected and we are working over rational numbers, we can use Sullivan ...
9
votes
1answer
284 views

String cobracket from TFT

Let $M$ be a closed oriented manifold. Chas and Sullivan (https://arxiv.org/abs/math/0212358) introduced a Lie bialgebra structure on $H_\bullet^{S^1}(LM, M)$, $S^1$-equivariant homology of the loop ...
5
votes
1answer
134 views

Goldman Lie algebra of a bordered surface vs. a closed surface?

How are the Goldman Lie algebra of a closed surface $\overline{S}$ and the bordered surface $S$ obtained by taking $\overline{S}$ and removing an open disc (or more generally, $n$ disjoint discs) ...
8
votes
1answer
247 views

Rational cohomology of $LBSO(n)$

For which $n$ has the $S^1$-equivariant rational cohomology of $LBSO(n)$ been computed? Here, $SO(n)$ means the isometry group of the round sphere (preserving the orientation), $B$ stands for ...
7
votes
3answers
298 views

Classification of sections of free loop fibration over the two-sphere

For any space $X$ there is a fibration $$ \Omega X\to LX\stackrel{ev}{\to} X $$ where $LX=Map(S^1,X)$ is the free loop space, $\Omega X = Map_*(S^1,X)$ is the based loop space, and $ev:LX\to X$ is the ...
12
votes
2answers
427 views

Gerstenhaber conjecture for free loop space

I- Is the following statement still a conjecture see this article ? Conjecture (?) Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild ...
4
votes
1answer
147 views

Hochschild chain model for the evaluation map at half

Let M be a manifold and $\Lambda(M)$ its free loop space, $A= C^*(M)$ denotes the cochain algebra of $M$. We know that Hochschild chain model for the evaluation $ ev_0: \Lambda(M) \rightarrow M$ is ...
3
votes
1answer
305 views

Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...
0
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0answers
228 views

loop space homology and lens spaces

Is the homology of free loop space of lens spaces known? Thanks in advance for your help.
6
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3answers
482 views

String topology for a Lie group

My question is very naive maybe, I don't have a deep knowledge about string topology. I wanted to ask (explanation or a reference) for the geometric interpretation of free loop space (continues maps) ...
10
votes
2answers
947 views

Relation between Gerstenhaber bracket and Connes differential

Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known: A degree-0 product on the Hochschild cohomology $HH^*(C)$ $$ HH^*(C) \otimes HH^*(C) \...
9
votes
1answer
700 views

Pull-push in Godin's HCFT for string topology

I am reading Veronique Godin's famous article "Higher string topology operations" (http://arxiv.org/abs/0711.4859) that demonstrates that the string topology operations on $(H_\bullet(L X), H_\bullet(...
15
votes
3answers
866 views

Are there graph models for other moduli spaces?

Recall that a ribbon graph is a graph with a cyclic ordering at each vertex and such that each vertex has valence greater than or equal to 3. This cyclic ordering exactly gives one the information to ...
4
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0answers
424 views

Does a Dehn twist in the mapping class group of an cobordism give a BV-operator in string topology?

In her article Higher string topology operations, Godin in particular construct for each surface with $n$ incoming and $m \geq 1$ outgoing boundary circles an operation $H_\ast(BMod(S);det^{\otimes d})...
11
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1answer
692 views

What does this naive attempt at $S^1$-equivariant homology describe?

After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map $\...
6
votes
1answer
657 views

Can string topology be a open-closed TCFT with the full set of branes?

String topology studies the algebraic structure of the homology of the free loop space $LM = Map(S^1,M)$ of a oriented closed manifold. One aspect of this structure is that the pair $(H_\ast(LM;\...
5
votes
1answer
757 views

What is the Gromov-Witten potential associated to String Topology?

Kevin Costello's article on the Gromov-Witten potential associated to a TCFT constructs for each TCFT, i.e. a functor from chains on Riemann surfaces with boundary ...
24
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6answers
2k views

Applications of string topology structure

Chas and Sullivan constructed in 1999 a Batalin-Vilkovisky algebra structure on the shifted homology of the loop space of a manifold: $\mathbb{H}_*(LM) := H_{*+d}(LM;\mathbb{Q})$. This structure ...
8
votes
2answers
2k views

Chas-Sullivan string topology

I recently read the original paper by Chas-Sullivan on string topology, in which they introduce some operations on homology of free loopspace LM, where M is a compact oriented manifold, giving it the ...
8
votes
2answers
683 views

Is the cohomology of a topological operad a cooperad?

For cohomology with coefficients in a field $F$ the map $H^\cdot(X;F) \otimes H^\cdot(Y;F) \to H^\cdot(X \times Y;F)$ of the Kunneth theorem is an isomorphism of algebras over $F$. I am correct in ...