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We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibres admits a reduction to a fibre bundle with compact topological manifold fibres. The criteria lead to an unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen's functor  A ( X ). We also give concrete examples of fibrations having a reduction to a fibre bundle with compact topological manifold fibres but which fail to admit a compact fibre smoothing. The examples are detected by algebraic  K ‐theory invariants. We consider a refinement of the Becker–Gottlieb transfer. We show that a version of the axioms described by Becker and Schultz uniquely determines the refined transfer for the class of fibrations, admitting a reduction to a fibre bundle with compact topological manifold fibres. In the Appendix, we sketch a theory of characteristic classes for fibrations. The classes are primary obstructions to finding a compact fibre smoothing.

The refined transfer, bundle structures, and algebraic K ‐theory