THE LEFSCHETZ FIXED POINT THEORY FOR MORPHISMS IN TOPOLOGICAL VECTOR SPACES

Abstract. The Lefschetz Fixed Point Theorem for compact absorbing contraction morphisms (CAC-morphisms) of retracts of open subsets in admissible spaces in the sense of Klee is proved. Moreover, the relative version of the Lefschetz Fixed Point Theorem and the Lefschetz Periodic Theorem are considered. Additionally, a full classification of morphisms with compact attractors in the non-metric case is obtained. 1. Vietoris mappings; admissibility in the sense of Klee We are interested in theory of homology such that Vietoris theorem is satisfied for any topological space. In this paper we use a definition of Čech theory of homology with compact carriers and coefficients in the field of rationals Q given in [15] (see also [18]). A space X is acyclic if: (a) X is non-empty, (b) Hq(X) = 0 for every q ≥ 1 and (c) H0(X) ≈ Q. A continuous mapping f: X → Y of Hausdorff topological spaces X and Y is called perfect if f is closed and for every y ∈ Y a set f −1 (y) is compact. Definition 1.1. A mapping of pair of spaces p: (Γ, Γ0) → (X, X0) is called Vietoris mapping provided it is a perfect surjection such that a set p −1 (x) is acyclic for any x ∈ X and Γ0 = p −1 (X0)