Filtering the fiber of the pinch map

TL;DR

This paper introduces a new Hopf invariant for the fiber of a pinch map, using it to analyze boundary maps in specific fibrations and achieve a filtered splitting of loop spaces.

Contribution

It develops a novel Hopf invariant for the fiber of pinch maps and applies it to study boundary maps and loop space splittings in certain fibrations.

Findings

New Hopf invariant for pinch map fibers

Compatibility of boundary maps with Hopf invariants

Filtered splitting of loop spaces

Abstract

A new type of Hopf invariant is described for the fiber of the pinch map from the mapping cone of a map from A to X onto to the suspension of A; this is then used to study the boundary map in the fibration sequence of Cohen, Moore and Neisendorfer in the case that the mapping cone is an odd dimensional Moore space. The components of the boundary map are then shown to be compatible with Hopf invariants and a filtered splitting of the loops on the fiber is obtained.