An Euler Characteristic for Unbounded Chain Complexes

TL;DR

This paper introduces a new way to define an Euler characteristic for unbounded chain complexes by using a limiting process on the ranks of homology modules, revealing a large Grothendieck group.

Contribution

It proposes a novel limiting-based Euler characteristic for unbounded complexes and analyzes the structure of the associated category and its Grothendieck group.

Findings

Defined an Euler characteristic for unbounded complexes via a limiting process.

Established a category with cofibrations and weak equivalences where this Euler characteristic applies.

Showed the Grothendieck group of this category is uncountably large.

Abstract

We propose a definition of an Euler characteristic for unbounded chain complexes by taking the (usual) Euler characteristics of successively longer parts of the complex, weighted inversely proportional to the length, and passing to the limit. This amounts to taking the limit of the sequence of ranks of homology modules with alternating signs in the sense of the H\"older summation method. We establish the structure of a category with cofibrations and weak equivalences on unbounded complexes for which the infinite Euler characteristic is defined, and show that its Grothendieck group is unusually large (viz., uncountable).