Higher Euler characteristics. I
type: Item
instance of: scholarly article
title: Higher Euler characteristics. I
zbMATH Open document ID: 0842.55002
author: Ross Geoghegan
author: Andrew Nicas
published in: L'Enseignement Mathématique. 2e Série
publication date: 1995-11-22T00:00:00Z
review text: In a series of papers the authors have been investigating a series of invariants related to the Euler characteristic. In this paper, the authors study the ``first order'' Euler characteristic. It is a homomorphism \(\chi_1(X) : \Gamma \to H_1(X)\) where \(X\) is a finite complex and \(\Gamma\) is the fundamental group of the space of self homotopy equivalences of \(X\). They give four definitions of \(\chi_1(X)\), which they show to be equivalent over suitable classes of spaces. They show \(\chi_1(X)\) is a homotopy invariant. The definitions involve one on the chain level, one on the homology level, one using intersection theory, and one using the fixed point transfer. The easiest to describe here is the intersection definition. An element \(\gamma \in \Gamma\) corresponds to a map \(F : X \times S^1 \to X\). The coincidence set of \(F\) and the projection \(p : X \times S^1 \to X\), made transverse, is a one cycle in \(X \times S^1\). Project it down to \(X\) by \(p\) and that is \(\chi_1 (X) (\gamma)\). The authors calculate \(\chi_1(X)\) for \(X\) a circle, a 2-torus, certain aspherical complexes, lens spaces, and \(S^1\) fibrations. The main application involves a higher order analogue of Gottlieb's theorem. Gottlieb's theorem states that if \(\chi(K) \neq 0\), then the center \(\pi_1 K\) is trivial when \(K\) is a finite aspherical complex. The analogue states that if \(\chi_1(K) \neq 0\), then the center of \(\pi_1(K)\) is infinite cyclic, provided \(K\) is a finite aspherical complex satisfying a possibly vacuous extra condition.
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 819165
zbMATH Keywords: Euler characteristic
zbMATH Keywords: fundamental group
zbMATH Keywords: space of self homotopy equivalences
zbMATH Keywords: Gottlieb's theorem
reviewed by: Daniel H. Gottlieb
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 819165