Questions tagged [string-topology]
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28 questions
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String cobracket and co-Hochschild homology
Let $M$ be a closed oriented manifold and take a field of char. zero to be the ground ring. String Topology gives, to the homology $H_\bullet(LM)$ of the free loop space of $M$, the structure of ...
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vote
1
answer
179
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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 ...
- 113
6
votes
1
answer
455
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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 $n$-...
- 512
7
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0
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189
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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 ...
- 448
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217
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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 ...
- 383
2
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0
answers
143
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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 ...
- 745
7
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174
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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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7
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200
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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 ...
- 71
8
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1
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378
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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 ...
- 661
10
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1
answer
548
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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 ...
- 3,357
5
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1
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163
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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) ...
- 1,981
8
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1
answer
264
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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 ...
- 11.9k
7
votes
3
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377
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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 ...
- 35.9k
12
votes
2
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520
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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 ...
- 235
4
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1
answer
157
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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 ...
- 745
3
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1
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396
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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 ...
- 307
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301
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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.
- 131
6
votes
3
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569
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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) ...
- 1,974
11
votes
2
answers
1k
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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) \...
- 12.8k
9
votes
1
answer
768
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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(...
- 19.8k
15
votes
3
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1k
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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 ...
- 8,167
4
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468
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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})...
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11
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1
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964
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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 $\...
- 8,167
7
votes
1
answer
710
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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;\...
- 8,167
5
votes
1
answer
910
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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 ...
- 8,167
27
votes
6
answers
3k
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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 ...
- 8,167
8
votes
2
answers
2k
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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 ...
- 21k
9
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2
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778
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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 ...
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