Generalized Lefschetz theorem and a fixed point index formula

Abstract

Let X be a metrizable space and let ( C, E) be a pair of compact ANRs contained in X. For a given continuous map f : C → X, we call ( C, E) proper with respect to f if C ∩ f( E) ⊂ E and C ∩ f(C)⧹C ⊂ E . For such a pair we introduce the so-called transfer endomorphism f # ( C, E) of H ∗(C, E) . The first main theorem of this note asserts that if the Lefschetz number Λ( f # ( C, E) ) is nonzero then f has a fixed point in C⧹E . In order to state the second main result we assume that X is an ENR and there exists a finite family ( C 1, E 1),…, ( C n , E n ) of proper ENR-pairs with respect to f, C i ⊂ E i − 1 , such that f has no fixed points in the boundary of C 1, in E n , and in the complements of the relative interiors of C i in E i − 1 . The second theorem asserts that under these hypotheses the fixed point index ind ( f, int C 1) is equal to ∑ n i = 1 Λ( f # ( C i , E i ) ). We provide an example indicating how to apply that theorem in order to determine the fixed point index of a Poincaré map of a nonautonomous ordinary differential equations, and how to use that index to a result on bifurcation of periodic solutions.