Vol. 83, No. 2, 1979

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Fix-finite homotopies

Helga Schirmer

Vol. 83 (1979), No. 2, 531–542

A well-known result by H. Hopf states that every selfmap f of a polyhedron |K| can be deformed into a selfmap f which has only a finite number of fixed points and is arbitrarily close to the given one. In addition one can locate all fixed points of fin maximal simplexes. A map which has a finite fixed point set is here called a fix-finite map, and a homotopy F : |KI →|K| is called a fix-finite homotopy if the map ft = F(,t) is fix-finite for every t I. We extend Hopf’s result to homotopies, and show that two homotopic selfmaps f0 and f1 of a polyhedron |K| which are fix-finite and have all their fixed points located in maximal simplexes can be related by a homotopy which is fix-finite and arbitrarily close to the given one. All fixed points of F can again be located in as high-dimensional simplexes as possible. Some simple properties are derived from the fact that the fix-finite homotopy is constructed in such a way that its fixed point set is a one-dimensional polyhedron in |KI.

Mathematical Subject Classification 2000
Primary: 54H25
Secondary: 55M20
Received: 23 August 1978
Revised: 14 January 1979
Published: 1 August 1979
Helga Schirmer