2007/07/31 by John R. Klein, Bruce Williams
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic number #Algebraic structures and combinatorial models #Bundle #Composite material #Computer science #Homotopy and Cohomology in Algebraic Topology #Materials science #Mathematical analysis #Mathematics #Parallel computing #Pure mathematics #Transfer (computing) #math.AT #math.KT #msc:19D10 #msc:55R10 #msc:55R12 #msc:55R15 #msc:57N65 #msc:57S05
paper · pdf · doi:10.1112/jtopol/jtp010
This version contains mostly minor revisions
arxiv created 2008/12/18 · openalex publication_date 2009/01/01 · arxiv updated 2014/02/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.