Vol. 103, No. 1, 1982

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The Lefschetz number and Borsuk-Ulam theorems

Daniel H. Gottlieb

Vol. 103 (1982), No. 1, 29–37

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 Λf 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 Λf. 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
Received: 25 October 1980
Revised: 20 May 1981
Published: 1 November 1982
Daniel H. Gottlieb