A Topological Proof for a Version of Artin's Induction Theorem

TL;DR

This paper introduces a topological approach to Artin's induction theorem by defining a group-equivariant Euler characteristic for cell complexes and relating it to fixed-point sets and representation theory.

Contribution

It provides a topological proof of a version of Artin's induction theorem using a novel Euler characteristic for group actions on cell complexes.

Findings

Defined a $G$-equivariant Euler characteristic for cell complexes.

Proved the Euler characteristic can be computed via homology representations.

Established a topological proof of a version of Artin's induction theorem.

Abstract

We define a Euler characteristic $\chi(X,G)$ for a finite cell complex $X$ with a finite group $G$ acting cellularly on it. Then, each $K_{i}(X)$ (a complex vector space with basis the $i$-cells of $X$) is a representation of $G$, and we define $\chi(X,G)$ to be the alternating sum of the representations $K_{i}(X)$, as elements of the representation ring $R(G)$ of $G$. By adapting the ordinary proof that the alternating sum of the dimensions of the chain complexes is equal to the alternating sum of the dimensions of the homology groups, we prove that there is another definition of $\chi(X,G)$ with the alternating sum of the representations $H_i(X)$, again as elements of the representation ring $R(G)$. We also show that the character of this virtual representation $\chi(X,G)$, with respect to a given element $g$, is just the ordinary Euler characteristic of the fixed-point set by this element. Finally, we give a topological proof of a version of Artin's induction theorem. More precisely, we show that, if $G$ is a group with an irreducible representation of dimension greater than 1, then each character of $G$ is a linear combination with rational coefficients of characters induced up from characters of proper subgroups of $G$.