ar X iv : 0 70 7 . 02 50 v 1 [ m at h . A T ] 2 J ul 2 00 7 THE REFINED TRANSFER , BUNDLE STRUCTURES AND ALGEBRAIC K - THEORY

For the class of fibrations which admit a reduction to a fiber bundle with compact topological manifold fibers, Becker and Schultz showed that the Becker-Gottlieb transfer is uniquely characterized by four axioms. In this paper, we consider a refinement of the transfer, also described by Becker and Gottlieb. We show that a version of the Becker-Schultz axioms uniquely determines the refined transfer for the same class of fibrations considered by Becker and Schultz. We then give elementary homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits such a fiber bundle reduction. A simple consequence of our criteria is that the Becker-Gottlieb trace map is uniquely determined by our refinement of the Becker-Schultz axioms, even when the fibration fails to admit a lifting to a fiber bundle with compact manifold fibers. The main technical tool for these results is a surjective splitting of the assembly map for Waldhausen’s functor A(X). The surjectivity result gives an new and unexpected corollary about homeomorphism groups of manifolds. We also give examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. These examples are detected by algebraic K-theory invariants. In an appendix, we sketch a theory of characteristic classes for fibrations. The classes are primary obstructions to finding a compact fiber smoothing.