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 α : S q → S p , and let C_X be the cofiber of the diagonal inclusion X → X × X. It is shown that the topological complexity (TC ) of X agrees with the Lusternik-Schnirelmann category (cat ) of C_X in the (almost stable) range q ≤ 2p − 1. In addition, the equality TC (X) = cat (C_X ) is proved in the (strict) metastable range 2p − 1 < q ≤ 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.