Homotopy groups of highly connected Poincaré duality complexes

Abstract

Methods are developed to relate the action of a principal fibration to relative Whitehead products in order to determine the homotopy type of certain spaces. The methods are applied to thoroughly analyze the homotopy type of the based loops on certain cell attachments. Key examples are $(n-1)$-connected Poincaré Duality complexes of dimension $2n$ or $2n+1$ with minor cohomological conditions.