Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes

Abstract

Let $X$ be a two-cell complex with attaching map $\alpha :S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity ($\mathrm{TC}$) of $X$ agrees with the Lusternik-Schnirelmann category ($\mathrm{cat}$) of $C_X$ in the (almost stable) range $q\le 2p-1$. In addition, the equality $\mathrm{TC}(X)=\mathrm{cat}(C_X)$ is proved in the (strict) metastable range $2p-1<q\le 3(p-1)$ under fairly mild conditions by making use of the Hopf invariant techniques recently developed by the authors in their study of the sectional category of arbitrary maps.

arXiv comment15 pages