Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Exotic smooth structures on topological fiber bundles I


Authors: Sebastian Goette, Kiyoshi Igusa and Bruce Williams
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
Published electronically: July 12, 2013
MathSciNet review: 3130316
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. Bernhard Badzioch, Wojciech Dorabiala, John R. Klein, and Bruce Williams, Equivalence of higher torsion invariants, Adv. Math., 226 (2011), 2192-2232. MR 2739777 (2012d:57040)
  • 2. Jean-Michel Bismut and Sebastian Goette, Families torsion and Morse functions, Astérisque (2001), no. 275, x+293. MR 1867006 (2002h:58059)
  • 3. 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. MR 1303026 (96g:58202)
  • 4. 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.
  • 5. Dan Burghelea and Richard Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1-50. MR 0356103 (50:8574)
  • 6. -, Stability of concordances and the suspension homomorphism, Ann. of Math. (2) 105 (1977), no. 3, 449-472. MR 0438365 (55:11280)
  • 7. 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 (2004d:19004)
  • 8. W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980), no. 2, 239-251. MR 569072 (81b:18006)
  • 9. 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), Part 1, Amer. Math. Soc., Providence, R.I., 1978, 325-337. MR 520509 (80g:57043)
  • 10. Sebastian Goette, Torsion invariants for families, Astérisque 328 (2009), 161-206. MR 2674876 (2011d:58081)
  • 11. Sebastian Goette and Kiyoshi Igusa, Exotic smooth structures on topological fiber bundles II, ArXiv:1011.4653.
  • 12. M. L. Gromov and Ja. M. Èliašberg, Elimination of singularities of smooth mappings, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 600-626. MR 0301748 (46:903)
  • 13. Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242-276. MR 0119214 (22:9980)
  • 14. Kiyoshi Igusa, Higher Franz-Reidemeister Torsion, AMS/IP Studies in Advance Mathematics, vol. 31, International Press, 2002.
  • 15. -, Axioms for higher torsion invariants of smooth bundles, J. Topol. 1 (2008), no. 1, 159-186. MR 2365656 (2009h:58070)
  • 16. 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 (95c:19005b)
  • 17. J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190-199. MR 0180986 (31:5216)
  • 18. J. Lees, Immersions and surgeries on topological manifolds, Bull. Amer. Math. Soc., 75 (1969), p. 529-534. MR 0239602 (39:959)
  • 19. Bruce Williams, Stable smoothings of fiber bundles, handwritten notes, April 2006.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57R22, 57R10, 57Q10

Retrieve articles in all journals with MSC (2010): 57R22, 57R10, 57Q10


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
Email: igusa@brandeis.edu

Bruce Williams
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: williams.4@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05857-6
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
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society