Homotopy groups of highly connected Poincare duality complexes

TL;DR

This paper develops methods to analyze the homotopy types of highly connected Poincare duality complexes, focusing on the based loops of specific cell attachments, advancing understanding in algebraic topology.

Contribution

It introduces new techniques relating principal fibrations to Whitehead products for determining homotopy types of complex spaces.

Findings

Homotopy types of certain highly connected Poincare complexes are characterized.

Methods successfully applied to (n-1)-connected Poincare duality complexes of dimension 2n or 2n+1.

Enhanced understanding of the loop space structures of these 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 Poincare Duality complexes of dimension 2n or 2n+1 with minor cohomological conditions.