Lie Hopf algebras and splittings of suspension spaces

Jelena Grbic (University of Manchester)

Monday 29th November, 2010 16:00-17:00 204


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 $X$ that takes values in the isomorphism class (over the integers) of the Hopf algebra $H_*(\Omega\Sigma X; \mathbb{Z})$. We show that these invariants are obstructions to the existence of a space $Y$ for a given $X$ such that there is a homotopy equivalence $\Sigma X\simeq \Sigma^2Y$. Time permitting, I'll discuss a generalisation of the James stable splitting of $\Sigma\Omega\Sigma X$ to a functorial decomposition of $\Sigma A$ where $A$ is any functorial retract if a looped co-$H$-space.

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