Higher Euler characteristics. I
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.
scientific article; zbMATH DE number 819165