Algebra and Topology Seminar 

23/06/2006, 15:00 — 16:00 — Room P4.35, Mathematics Building
John Harper, University of Rochester

Finite $H$-spaces with retractile generating complex

Let $p$ be an odd prime. We study simply connected finite complexes having $p$-torsion free homology which are rationally equivalent to a product of odd dimensional spheres (=rational $H$-spaces). We ask whether the p-localization is an $H$-space. If the rank $r\le p-2$, an answer has been available for about 25 years. In particular, if the $p$-localization has a retractile generating complex, then the p-localization is an $H$-space. When the rank $r\ge p-1$, the statement is not necessarily true. Recent work (with L. Fernandez-Suarez, M. Mimura and J. Wu) has gained information for the case $r=p-1$ with detailed results for $r=2$, $p=3$ in terms of classical homotopy invariants.

We describe the main result for $r=2$, $p=3$. Let $\alpha \in {\pi }_{q-1}(S^n)$ with $n,q$ odd and localization at $3$ understood. Let $C=S^k{\cup }_{\alpha }{e}^q$ be the two cell complex with attaching map $\alpha$. Let $w\in {\pi }_{3n}{S}^n$ generate the kernel of the double suspension map ($\simeq Z/3Z$). Let $j:S^n\to C$ be the inclusion of the bottom cell. The space $\Lambda (C)$, with $C$ retractile and rationally equivalent to $S^n\times S^q$, exists. It is a $3$-local $H$-space if and only if $j\circ w\circ D(\alpha )=0$ where $D(\alpha ):{\Sigma }^{n+q-1}C\to C$ is a certain map constructed from a splitting of $\Sigma C\wedge C$.