The cohomology ring of free loop spaces

Luc Menichi

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arXiv · math/0009168

The cohomology ring of free loop spaces

Luc Menichi

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Abstract

Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH^{*}S_*(ΩX) is isomorphic to the free loop space cohomology H^{*}(X^{S^{1}}). We proved that this isomorphism is compatible with both the cup product on HH^{*}S_*(ΩX) and on H^{*}(X^{S^{1}}). In particular, we explicit the algebra H^{*}(X^{S^{1}}) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that \frac {1}{(p-1)!}\in k.

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Luc Menichi. 2000-09-18. The cohomology ring of free loop spaces. https://arxiv.org/abs/math/0009168

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