Hopf algebras and homotopy invariants

Abstract

In this paper we explore new relations between Algebraic Topology and the theory of Hopf Algebras. For an arbitrary topological space $X$, the loop space homology $H_*(\Omega\Sigma X; \coefZ)$ is a Hopf algebra. We introduce a new homotopy invariant of a topological space $X$ taking for its value the isomorphism class (over the integers) of the Hopf algebra $H_*(\Omega\Sigma X; \coefZ)$. This invariant is trivial if and only if the Hopf algebra $H_*(\Omega\Sigma X; \coefZ)$ is isomorphic to a Lie-Hopf algebra, that is, to a primitively generated Hopf algebra. We show that for a given $X$ these invariants are obstructions to the existence of a homotopy equivalence $\Sigma X\simeq \Sigma^2Y$ for some space $Y$. Further on, using the notion of Hopf algebras, we establish new structural properties of the cohomology ring, in particular, of the cup product. For example, using the fact that the suspension of a polyhedral product $X$ is a double suspension, we obtain a strong condition on the cohomology ring structure of $X$. This gives an important application in toric topology. For an algebra to be realised as the cohomology ring of a moment-angle manifold $\Z_P$ associated to a simple polytope $P$, we found an obstruction in the Hopf algebra $H_*(\Omega\Sigma \Z_P)$. In addition, we use homotopy decompositions to study particular Hopf algebras.