Vol. 17, No. 3, 1966

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On a homotopy converse to the Lefschetz fixed point theorem

Robert F. Brown

Vol. 17 (1966), No. 3, 407–411

Let α be a homotopy class of maps of X, a connected compact metric ANR, into itself and let Lα denote the Lefschetz number of α. A converse to the Lefschetz fixed point theorem is: if Lα = 0 then α contains a fixed point free map. The converse is true if X is a compact connected simply-connected topological n-manifold (Fadell) or if X is a compact connected topological n-manifold, with or without boundary, and α contains the identity map (Brown-Fadell). Let μ(α) denote the fixed point class invariant of α, then every map in α has at least μ(α) fixed points. The purpose of this paper is to generalize the preceding results by proving that if X is a compact connected topological n-manifold, n 3, with or without boundary, then there is a map in α which has exactly μ(α) fixed points. It follows that the converse to the Lefschetz theorem will hold whenever α contains a map all of whose fixed points are in a single fixed point class.

Mathematical Subject Classification
Primary: 54.85
Received: 10 December 1964
Published: 1 June 1966
Robert F. Brown
Department of Mathematics
University of California, Los Angeles
Los Angeles CA 90095-1555
United States