Transfer maps for fibrations

Abstract

Let f: E → B be a fibration with fibre F over a connected space B. If F is homotopy equivalent to a finite complex, Becker and Gottlieb [2, 3] and others have constructed a transfer mapwhere for simplicity X+ denotes the suspension spectrum of the space obtained from X adding a disjoint basepoint. One key property of τ(f) is the fact that the composite map f+. τ(f): B+ → B+ induces a map on integral homology which is multiplication the Euler characteristic X(F).

String topology of classifying spaces

“…Dwyer's transfer. At some point we will need to use W. Dwyer's more general version of the transfer [18]. It has the same properties as M. Clapp's transfer (or any other classical version) and is equal to it in the case of the sphere spectrum S 0 .…”

The K-theory of p -compact groups

“…The principal ingredients are the "main theorem" of [15] and Dwyer's transfer ( [14] Before going farther, a well-known observation is in order. Suppose that there exists a monomorphism h : X → U (n)p, where U(n)p denotes the p-compact group obtained by p-completing the unitary group U (n).…”

“…Using the naturality and the product formulae for the transfer (see Theorem 2.6 and Theorem 2.8 in [14]) and applying p-adic K-theory, we obtain, for any ξ ∈ K * (BT ; Zp):…”

The fundamental group of a p -compact group

“…It seems virtually certain that the transfer map we construct can be derived by duality from the homological transfer of [7, 1.1]. However, we need a property (2.4) of the transfer that is not easy to check from the point of view of [7], and so for the sake of economy of exposition we develop an ad hoc transfer that serves our immediate purposes.…”