The Lefschetz number and Borsuk-Ulam theorems

Daniel H. Gottlieb

Vol. 103 (1982), No. 1, 29–37

DOI: 10.2140/pjm.1982.103.29

Abstract

Let M be a manifold, with or without boundary, which is dominated by a finite complex. Let G be a finite group which acts faithfully and freely on M. Let f : M → M be a G-map. Let Λ_f denote the Lefschetz number of f and let o(G) denote the order of G. The main result states, under the conditions above, that o(G) divides Λ_f. Even in the case of compact M this result was not widely known. We use Wall’s finiteness obstruction theory to extend the result from compact M to finitely dominated M.

Mathematical Subject Classification 2000

Primary: 55M20

Secondary: 55M35, 57S17

Milestones

Received: 25 October 1980

Revised: 20 May 1981

Published: 1 November 1982