Fixed point homomorphisms for parameterized maps

Abstract

Let X be an ANR (absolute neighborhood retract), \({\Lambda}\) a k-dimensional topological manifold with topological orientation \({\eta}\) , and \({f : D \rightarrow X}\) a locally compact map, where D is an open subset of \({X \times \Lambda}\) . We define Fix(f) as the set of points\({{(x, \lambda) \in D}}\) such that \({x = f(x, \lambda)}\) . For an open pair (U, V) in \({X \times \Lambda}\) such that \({{\rm Fix}(f) \cap U \backslash V}\) is compact we construct a homomorphism \({\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}\) in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a \({C^{\infty}}\) -manifold \({\Lambda}\) , these properties uniquely determine \({\Sigma}\) . By passing to the direct limit of \({\Sigma_{(f,U,V)}}\) with respect to the pairs (U, V) such that \({K = {\rm Fix}(f) \cap U \backslash V}\) , we define a homomorphism \({\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}\) in the Cech cohomologies. Properties of \({\Sigma}\) and \({\sigma}\) are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.