Hochschild cohomology is topological (Q5954119)

There is an isomorphism of graded vector spaces, \(HH_*(C^*(X,k))\cong H^*(X^{S^1})\), due to \textit{J. D. S. Jones} [Invent. Math. 87, No. 2, 403-423 (1987; Zbl 0644.55005)], between the Hochschild homology of the cochain algebra of a space \(X\) and the cohomology of \(X^{S^1}\), the free loop space on \(X\). The right-hand side is a graded algebra with the cup product. If \(C^*(X,k)\) is equivalent to a commutative cochain algebra, the left-hand side may be replaced by the Hochschild homology of this algebra, which is then a graded algebra with the shuffle product. In this situation it is natural to ask if the map is an isomorphism of algebras. It is known, by recent work of the authors [Commutative free loop space models at large primes, Math. Z., to appear] that this is not always the case. Over the rationals, however, \textit{M. Vigué-Poirrier} proves it is an algebra isomorphism [J. Algebra 207, No. 1, 333-341 (1998; Zbl 0909.13006)]. Working over a field of positive characteristic an odd prime \(p\), this paper extends that result to \(p\)-Anick spaces. (A \(p\)-Anick space is a space \(E\) such that \(C_*(\Omega E;{\mathbb F}_p)\) is equivalent as a Hopf algebra to the universal envelopping algebra of a free differential Lie algebra.) The method used is an explicit construction of an appropriate quasi-isomorphism of cochain algebras.