On the $p$-primary obstructions to finding a cross-section

Abstract:

Let $q:T \to B$ be a (weak) fibration, and let $p$ be an odd prime. In this paper, we show that the existence of a fiber-preserving map $A:T \to T$ having certain properties implies that the $p$-primary obstructions to a cross-section of $q$ vanish. Assuming that $q:T \to B$ has a cross-section, we prove a related theorem which bears on the problem of enumerating the homotopy classes of cross-sections of $q$.