On the multiplicativity of the Euler characteristic

TL;DR

This paper provides two proofs demonstrating that the Euler characteristic is multiplicative in fiber sequences of finitely dominated spaces, linking this property to the functoriality of the Becker-Gottlieb transfer on C0.

Contribution

It offers new proofs establishing the multiplicativity of the Euler characteristic and connects this property to the functoriality of the Becker-Gottlieb transfer.

Findings

Euler characteristic is multiplicative for fiber sequences of finitely dominated spaces

The functoriality of the Becker-Gottlieb transfer on C0 is established

Provides two distinct proofs for the main result

Abstract

In this short paper, we give two proofs that the Euler characteristic is multiplicative, for fiber sequences of finitely dominated spaces. This is equivalent to proving that the Becker-Gottlieb transfer is functorial on $\pi_0$.