Axioms for bundle transfers and traces

Abstract

In topology it is often profitable to characterize the algebraic invariants of spaces or morphisms by axioms. Of course, the most basic examples along these lines are the Eilenberg-Steenrod axioms for homology and cohomology [EiS], and in many cases there are axiomatic characterizations for more refined algebraic structures that exist in homology and cohomology. For example, axioms for the Steenrod cohomology operations appear in [EpS], and axioms for various sorts of transfers known in the nineteen sixties were described in unpublished work of J. M. Boardman [Bo]. In this paper we shall describe axioms for the bundle transfers considered by the first named author and D. H. Gottlieb in [BG2–3] and from a different perspective by A. Dold, M. Clapp and D. Puppe in [Cl, D1–3, DP]. Previous work of F. Roush [Rou] and L. G. Lewis [L] describes axioms for the special case of finite coverings and bundles whose fibers are compact Lie groups that act smoothly on the fibers; another axiomatic characterization of finite covering transfers appears in [BeS, Sect. 13]. The characterization in this paper is valid for topological fiber bundles for which the fibers and the bases are finite-dimensional compact ANR’s and finite CW complexes respectively. We shall also describe axioms for the homomorphisms in generalized homology and cohomology theories that are determined by bundle transfers.