Generalized Lefschetz theorem and a fixed point index formula

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 (C1, E1),…, (Cn, En) of proper ENR-pairs with respect to f, Ci ⊂ Ei − 1, such that f has no fixed points in the boundary of C1, in En, and in the complements of the relative interiors of Ci in Ei − 1. The second theorem asserts that under these hypotheses the fixed point index ind (f, int C1) is equal to ∑ni = 1 Λ(f#(Ci,Ei)). 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.