2009/02/27 by Johannes Ebert, Ebert, Johannes · 1 citation
Mathematics · #55R40 #57R90 #58J20 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55R40 #msc:57R90 #msc:58J20
paper · pdf · doi:10.48550/arxiv.0902.4719
27 pages, expository sections on Thom spectra tightened. The section on nonvanishing results is removed; the author's preprint 0910.1030 contains a stronger result. New section on manifold bundles with boundary added.
openalex publication_date 2009/02/27 · arxiv created 2010/03/09 · arxiv updated 2010/03/10 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We show how the Atiyah-Singer family index theorem for both, usual and self-adjoint elliptic operators fits naturally into the framework of the Madsen-Tillmann-Weiss spectra. Our main theorem concerns bundles of odd-dimensional manifolds. Using completely functional-analytic methods, we show that for any smooth proper oriented fibre bundle E → X with odd-dimensional fibres, the family index \ind (B) ∈ K1 (X) of the odd signature operator is trivial. The Atiyah-Singer theorem allows us to draw a topological conclusion: the generalized Madsen-Tillmann-Weiss map α: B \Diff+ (M2m-1) → \loopinf \MTSO(2m-1) kills the Hirzebruch \cL-class in rational cohomology. If m=2, this means that α induces the zero map in rational cohomology. In particular, the three-dimensional analogue of the Madsen-Weiss theorem is wrong. For 3-manifolds M, we also prove the triviality of α: B \Diff+ (M) → \MTSO (3) in mod p cohomology in many cases. We show an appropriate version of these results for manifold bundles with boundary.