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 ...
Yuan Yao's user avatar
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6 votes
1 answer
380 views

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 $n$-...
skr's user avatar
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7 votes
0 answers
174 views

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 ...
taf's user avatar
  • 448
4 votes
0 answers
207 views

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 ...
Valentin's user avatar
  • 373
2 votes
0 answers
138 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 ...
Arun 's user avatar
  • 725
7 votes
0 answers
163 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 ...
Miguel Moreira's user avatar
7 votes
0 answers
191 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 ...
kyler's user avatar
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8 votes
1 answer
373 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 ...
Yining Zhang's user avatar
10 votes
1 answer
521 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 ...
Pavel Safronov's user avatar
5 votes
1 answer
163 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) ...
Nati's user avatar
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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 ...
Jens Reinhold's user avatar
7 votes
3 answers
367 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 ...
Mark Grant's user avatar
12 votes
2 answers
502 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 ...
lilia's user avatar
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4 votes
1 answer
154 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 ...
Arun 's user avatar
  • 725
3 votes
1 answer
383 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 ...
Felix Y.'s user avatar
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0 answers
287 views

loop space homology and lens spaces

Is the homology of free loop space of lens spaces known? Thanks in advance for your help.
Murat Saglam's user avatar
6 votes
3 answers
552 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) ...
Ilias A.'s user avatar
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11 votes
2 answers
1k 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) \...
Kevin Walker's user avatar
  • 12.3k
9 votes
1 answer
754 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(...
Urs Schreiber's user avatar
15 votes
3 answers
1k 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 ...
skupers's user avatar
  • 7,923
4 votes
0 answers
467 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})...
skupers's user avatar
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11 votes
1 answer
885 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 $\...
skupers's user avatar
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7 votes
1 answer
704 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;\...
skupers's user avatar
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5 votes
1 answer
881 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 ...
skupers's user avatar
  • 7,923
26 votes
6 answers
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 ...
skupers's user avatar
  • 7,923
8 votes
2 answers
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 ...
Kevin H. Lin's user avatar
  • 20.7k
9 votes
2 answers
767 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 ...
skupers's user avatar
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