Lie-Hopf algebras and loop homology of suspension spaces

Details Lie-Hopf algebras and loop homology of suspension spaces Lie-Hopf algebras and loop homology of suspension spaces

2010-11-11

arxiv.org/abs/1011.2549v2

Mathematics

Algebraic Topology

11 pages; typos corrected; new examples added

Scientific paper

For an arbitrary topological space X, the loop space homology $H_*(\Omega\Sigma X; \mathbb{Z})$ 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 $H_*(\Omega\Sigma X; \mathbb{Z})$. This invariant is trivial if and only if the Hopf algebra $H_*(\Omega\Sigma X; \mathbb{Z})$ 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. We further investigate relations between this new invariant and well known classical invariants and constructions in homotopy theory.

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