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A note on the Chas-Sullivan product. (English) Zbl 1229.55004

Let \(M\) be an \(n\)-dimensional closed oriented manifold. In 1999 Chas and Sullivan constructed a commutative and associative product on the integer homology of the free loop space \(LM = {\mathcal C}^\infty (S^1,M)\),
\[ H_i(LM;\mathbb Z) \otimes H_j(LM;\mathbb Z)\to H_{i+j-n}(LM;\mathbb Z)\,. \]
This product was defined in the same spirit as the usual intersection product on a manifold.
A different approach has been considered later by Cohen and Jones using the Thom spectrum of a virtual bundle. The work of Laudenbach comes back to the original approach and gives for the first time a good and precise basis for an intersection product on \(LM\).
More precisely Laudenbach defines transversal bi-cycles \(\xi\times \eta\) and associates to them cycles \(\xi\times_{CS}\eta\). He proves that for each \([\xi]\in H_p(LM)\) and \([\eta] \in H_q(LM)\) there is a transversal bi-cycle \(\xi\times \eta\) and that the product \([\xi\times_{CS}\eta]\in H_{p+q-n}(LM)\) depends only on the classes of \([\xi]\) and \([ \eta]\).
Laudenbach does not suppose that the manifold is orientable. He considers at the same time the case of an oriented manifold (coefficients in \(\mathbb Z\)) and the case of a non-orientable manifold (local coefficients).
The last section of the paper shows that, by this approach, we can recover results of Goresky-Hingston, Chataur and Le Borgne in string homology.

MSC:

55P50 String topology
55N45 Products and intersections in homology and cohomology
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