Exotic smooth structures on topological fiber bundles I
HTML articles powered by AMS MathViewer
- by Sebastian Goette, Kiyoshi Igusa and Bruce Williams PDF
- Trans. Amer. Math. Soc. 366 (2014), 749-790 Request permission
Abstract:
When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and stably, this is a complete invariant. We give a more or less complete and self-contained exposition of this theory.
An important application is the computation of the Igusa-Klein higher Reidemeister torsion invariants of these exotic smooth structures. Namely, the higher torsion invariant is equal to the Poincaré dual of the image of the smooth structure class in the homology of the base. This is proved in the companion paper written by the first two authors.
References
- Bernard Badzioch, Wojciech Dorabiała, John R. Klein, and Bruce Williams, Equivalence of higher torsion invariants, Adv. Math. 226 (2011), no. 3, 2192–2232. MR 2739777, DOI 10.1016/j.aim.2010.09.017
- Jean-Michel Bismut and Sebastian Goette, Families torsion and Morse functions, Astérisque 275 (2001), x+293 (English, with English and French summaries). MR 1867006
- Jean-Michel Bismut and John Lott, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), no. 2, 291–363 (English, with English and French summaries). MR 1303026, DOI 10.1090/S0894-0347-1995-1303026-5
- M. Bökstedt and F. Waldhausen, The map ${BSG}\to {A}(\ast )\to {QS}^0$, Algebraic Topology and Algebraic ${K}$-theory (William Browder, ed.), Annals of Math. Studies, vol. 113, 1987, pp. 418–431.
- Dan Burghelea and Richard Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1–36; ibid. 196 (1974), 37–50. MR 356103, DOI 10.1090/S0002-9947-1974-0356103-2
- D. Burghelea and R. Lashof, Stability of concordances and the suspension homomorphism, Ann. of Math. (2) 105 (1977), no. 3, 449–472. MR 438365, DOI 10.2307/1970919
- W. Dwyer, M. Weiss, and B. Williams, A parametrized index theorem for the algebraic $K$-theory Euler class, Acta Math. 190 (2003), no. 1, 1–104. MR 1982793, DOI 10.1007/BF02393236
- W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980), no. 2, 239–251. MR 569072, DOI 10.2307/1971200
- F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 325–337. MR 520509
- Sebastian Goette, Torsion invariants for families, Astérisque 328 (2009), 161–206 (2010) (English, with English and French summaries). MR 2674876
- Sebastian Goette and Kiyoshi Igusa, Exotic smooth structures on topological fiber bundles II, ArXiv:1011.4653.
- M. L. Gromov and Ja. M. Èliašberg, Elimination of singularities of smooth mappings, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 600–626 (Russian). MR 0301748
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
- Kiyoshi Igusa, Higher Franz-Reidemeister Torsion, AMS/IP Studies in Advance Mathematics, vol. 31, International Press, 2002.
- Kiyoshi Igusa, Axioms for higher torsion invariants of smooth bundles, J. Topol. 1 (2008), no. 1, 159–186. MR 2365656, DOI 10.1112/jtopol/jtm011
- Kiyoshi Igusa and John Klein, The Borel regulator map on pictures. II. An example from Morse theory, $K$-Theory 7 (1993), no. 3, 225–267. MR 1244002, DOI 10.1007/BF00961065
- J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR 180986, DOI 10.2307/1970498
- J. Alexander Lees, Immersions and surgeries of topological manifolds, Bull. Amer. Math. Soc. 75 (1969), 529–534. MR 239602, DOI 10.1090/S0002-9904-1969-12231-7
- Bruce Williams, Stable smoothings of fiber bundles, handwritten notes, April 2006.
Additional Information
- Sebastian Goette
- Affiliation: Mathematisches Institut, Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany
- Email: sebastian.goette@math.uni-freiburg.de
- Kiyoshi Igusa
- Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454
- MR Author ID: 90790
- ORCID: 0000-0003-2780-0924
- Email: igusa@brandeis.edu
- Bruce Williams
- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- Email: williams.4@nd.edu
- Received by editor(s): December 5, 2010
- Received by editor(s) in revised form: March 17, 2012, and April 7, 2012
- Published electronically: July 12, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 749-790
- MSC (2010): Primary 57R22; Secondary 57R10, 57Q10
- DOI: https://doi.org/10.1090/S0002-9947-2013-05857-6
- MathSciNet review: 3130316