Fredholm theory for cofinite sets

TL;DR

This paper develops a new index theory for self-maps of infinite sets, exploring their proximity to bijections, and extends the symmetric group quotient by integers.

Contribution

It introduces a $bZ$-valued index theory and extends the symmetric group quotient by $bZ$, providing new tools for analyzing self-maps of infinite sets.

Findings

Generated a $bZ$-valued index theory for self-maps.

Extended the quotient of the symmetric group by its finitary subgroup by $bZ$.

Provided new insights into the structure of self-maps close to bijections.

Abstract

We investigate two ways in which self-maps of an infinite set may be close to bijections; our investigation generates a $\mathbb{Z}$-valued index theory and a corresponding extension by $\mathbb{Z}$ for the quotient of the full symmetric group by its finitary subgroup.