The cohomology ring of free loop spaces

2000/09/18 by Luc Menichi, Menichi, Luc
Mathematics · #16E40 (secondary) #55P35 (primary) #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:16E40 #msc:55P35

paper · pdf · doi:10.48550/arxiv.math/0009168

39 pages

arxiv created 2000/09/18 · openalex publication_date 2000/09/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^S¹). We proved that this isomorphism is compatible with both the cup product on HH^*S_*(ΩX) and on H^*(X^S¹). In particular, we explicit the algebra H^*(X^S¹) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that \frac 1(p-1)!∈ k.

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