Charged Spaces

TL;DR

This paper investigates the conditions under which fiberwise charged spaces over a base space B are fiberwise suspensions, providing an obstruction theory and applications to embedding problems in smooth manifolds.

Contribution

It introduces a necessary and sufficient obstruction for fiberwise charged spaces to be fiberwise suspensions in a metastable range.

Findings

Established an obstruction criterion for fiberwise suspension

Applied the theory to embedding compression problems

Connected charged spaces to classical embedding questions

Abstract

A charged space is a space equipped with two distinct base points. Such spaces arise as the unreduced suspension of an unbased space. More generally, one can work in the fiberwise setting over a fixed space B. A fiberwise charged space over B is a space X equipped with map to B and having two sections. One can ask whether or not a fiberwise charged space over B is a fiberwise suspension. We answer this question in a certain metastable range by producing a necessary and sufficient obstruction. As an application we show how this can be used to study when an embedding into a smooth manifold of the form N x I compresses to an embedding into N .