Vol. 21, No. 2, 1967

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On maps with identical fixed point sets

Robert F. Brown

Vol. 21 (1967), No. 2, 215–217

If two maps on a space X, which admits a fixed point index, have identical sets of fixed points and agree on an open subset of X which contains the fixed point set, then the maps have the same Lefschetz number. If the subset is closed, the con clusion is no longer true in general. However, a theorem of Leray implies that some kinds of maps on cartesian products of convexoid spaces which agree on a certain closed subset of their common fixed point set do have the same Lefschetz number, even though the maps may not be homotopic and may not agree on any open set containing the fixed point set. The purpose of this note is to prove a very general form of Leray‘s theorem for maps on ANR’s.

Mathematical Subject Classification
Primary: 54.85
Secondary: 55.00
Published: 1 May 1967
Robert F. Brown
Department of Mathematics
University of California, Los Angeles
Los Angeles CA 90095-1555
United States