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This is my first steeps in string topology and please forgive the basic level of my questions: I reformulate my question

Chas and Sullivan define the loop homology product for closed (=compact with no boundary) and oriented maniflods. Is there such loop homology product for oriented compact manifolds with boundary

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    $\begingroup$ In your question 2: Do you really want your manifold to be contractible? $\endgroup$
    – André Henriques
    Commented Nov 10, 2014 at 10:35
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    $\begingroup$ In Question 1, isomorphic as what? It's elementary as vector spaces since the G action allows you to split the usual free loop based loop fibration sequence. If you want to include some extra structure you should check that the iso you get in this way respects the extra structure. In question 2: why contractible? $\endgroup$
    – Daniel Pomerleano
    Commented Nov 10, 2014 at 10:36
  • $\begingroup$ For question 1: isomorphic as algebra. Question 2 can be surely asked in a general case, however my frameworks require manifolds to be contractible $\endgroup$
    – MyIsmail
    Commented Nov 10, 2014 at 11:26
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    $\begingroup$ I am really puzzled by your requirement that the manifold be contractible. A closed manifold (=compact with no boundary) is never contractible, except if it is zero-dimensional, in which case it is a point. Do you maybe have some different meaning in mind for the words "closed" and "contractible"? $\endgroup$
    – André Henriques
    Commented Nov 10, 2014 at 21:22
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    $\begingroup$ Contractible manifolds (assuming they are not a point) are of two types: (1) compact with boundary, and (2) non-compact without boundary. Which type of contractible manifolds are you studying? $\endgroup$
    – André Henriques
    Commented Nov 10, 2014 at 22:13

1 Answer 1

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Edit 1: In a first version there was a question about string topology of Lie groups.

A very good reference about string topology of Lie groups is "String Topology for Lie Groups." J. Topology (2010) 3(2): 424-442 by Richard Hepworth.

The main point concerning your first question is to understand how to describe the BV-operator (the $S^1$-action) on $\mathbb{H}(G)\otimes H(\Omega G)$. Luc Menichi has also a nice paper on that theme: "A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops", Trans. Amer. Math. Soc. 363 (2011), 4443-4462.

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Edit 1 (continued):

  • You can extend the definition of the Chas-Sullivan loop product to open manifolds even when they are non-orientable (by using a local coefficient system). In order to do so you can use the same techniques as in Chas-Sullivan's original paper. If you want more details look at François Laudenbach's note "A note on the Chas-Sullivan product." Enseign. Math. (2) 57 (2011) In the non-compact case you will loose the unit.

  • Another funny thing to do when you have a manifold with boundary is to consider the space $L_{\partial M} M$ of loops $\gamma$ in $M$ such that $\gamma(0)$ is on the boundary $\partial M$. Then you get an algebra $H_{*+dim(\partial M)}(L_{\partial M} M)$ mixing the intersection product on the boundary and the pontryagin algebra of based loops $H_*(\Omega M)$.

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Edit 2: Concerning your first question, you can compare François Laudenbach's definition (which is so close in spirit to the very first definition of the loop product) with Cohen-Jones' definition in terms of Gysin maps and with Dennis Sullivan's definition given in his paper "String Topology: Background and Present State", they all give isomorphic algebra structures at the homology level. The proof follows the same lines as in the case of the classical intersection product.

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  • $\begingroup$ After reading Laudenbach's notes, I've some more basic questions: Question 1: In the case of closed and oriented maniflods, the Chas-Sullivan loop product and that of Laudenbach are the same?if yes why? Question 2: The loop homology once endowed with the Laudenbach product is just an associative and commutative graded ring. Why not an algebra? maybe because that the cofficients are twisted? $\endgroup$
    – MyIsmail
    Commented Nov 13, 2014 at 10:39
  • $\begingroup$ I am sorry but I do not understand your question 2. $\endgroup$
    – David C
    Commented Nov 13, 2014 at 19:28
  • $\begingroup$ In the framwork of Chas-Sullivan the loop product endows the loop homology $\mathbb{H}_*(LM;\mathbb Z)$ with an associative and commutative graded algebra. In that of Laudenbach, the coefficients system is twisted due to some orientation, and here the loop product endows the loop homology $\mathbb{H}_*(LM;\mathbb Z_{or})$ with an associative and commutative graded ring. Why not an algebra as in the Chas-Sullivan context? $\endgroup$
    – MyIsmail
    Commented Nov 14, 2014 at 11:18
  • $\begingroup$ $\mathbb{H}_*(LM,\mathbb{k}_{or})$ is a $\mathbb{k}$-algebra for any commutative ring $\mathbb{k}$. $\endgroup$
    – David C
    Commented Nov 14, 2014 at 19:53
  • $\begingroup$ Thanks David. Is there any thing known about $\mathbb{H}(LM,\mathbb{k}_{or})$ in the special case when $M$ is contractible. $\endgroup$
    – MyIsmail
    Commented Nov 14, 2014 at 20:02

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