On the negative cyclic homology of shc-algebras

Abstract

Let ${\mathbb{K}}$ be a field of characteristic ${p\geq 0}$ and S ¹ the unit circle. We prove that the shc-structure on a cochain algebra (A,d _A ) induces an associative product on the negative cyclic homology HC ⁻_* A. When the cochain algebra (A,d _A ) is the algebra of normalized cochains of the simply connected topological space X with coefficients in ${\mathbb{K}}$ , then HC ⁻_* A is isomorphic as a graded algebra to ${H^{-*}_{S^1}(LX;\mathbb{K})}$ the S ¹-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S ¹-equivariant cohomology algebras of the free loop space of the complex projective space ${\mathbb{C}P(n)}$ when n + 1 =  0 [p] and of the even spheres S ²ⁿ when p =  2.