Homotopy fibrations with a section after looping

TL;DR

This paper investigates a class of fibrations that admit sections after looping, providing methods to determine their homotopy types and classes, with applications to various topological constructions.

Contribution

It introduces new techniques for analyzing homotopy types of fibrations with sections after looping, applicable to complex topological structures.

Findings

Methods to determine homotopy types of fibres

Techniques for classifying homotopy classes of maps

Applications to two-cones, Poincare complexes, and polyhedral products

Abstract

We analyze a general family of fibrations which, after looping, have sections. Methods are developed to determine the homotopy type of the fibre and the homotopy classes of the map from the fibre to the base. The methods are driven by applications to two-cones, Poincare Duality complexes, the connected sum operation, and polyhedral products.