# Questions tagged [string-topology]

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### Goresky-Hingston product on cohomology of the relative free loop space on $S^1$

I'm interested in the computations of the Goresky-Hingston product (defined https://arxiv.org/abs/0707.3486) on the cohomology of the relative free loop space on the circle (or better yet, their ...
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### Homology and cohomology of free loop spaces

String topology, as well as Hochschild (co)homology, give a rich perspective on the homology and cohomology of a free loop space $LM$ of a manifold $M$. Let $k$ be a field and let $M$ be $n$-...
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### THH and string topology

There is an equivalence $THH(S^{\infty}_+ LX) = S^{\infty}_+ FX$ where $FX$ is the free loop space (I already used the letter $L$). The circle action on $THH$ gives rise to a degree $1$ cyclic ...
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### Loop spaces of manifolds with boundary

I would like to ask the same question that was already asked here: loop homology product for oriented compact manifolds with boundary. I hope it is okay to open a new question. The original question ...
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### 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 ...
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### 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 ...
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### 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 ...
368 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### loop space homology and lens spaces

Is the homology of free loop space of lens spaces known? Thanks in advance for your help.
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### 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) ...
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### 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) \...
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### 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 ...
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### 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 ...
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 ...