Geometry & Topology

Higher torsion and secondary transfer of unipotent bundles

Bernard Badzioch and Wojciech Dorabiała

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Given a unipotent bundle of smooth manifolds we construct its secondary transfer map and show that this map determines the higher smooth torsion of the bundle. This approach to higher torsion provides a new perspective on some of its properties. In particular it yields in a natural way a formula for torsion of a composition of two bundles.

Article information

Source
Geom. Topol., Volume 20, Number 4 (2016), 1807-1857.

Dates
Received: 27 December 2012
Revised: 26 April 2015
Accepted: 28 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859017

Digital Object Identifier
doi:10.2140/gt.2016.20.1807

Mathematical Reviews number (MathSciNet)
MR3548459

Zentralblatt MATH identifier
1375.19012

Subjects
Primary: 19J10: Whitehead (and related) torsion
Secondary: 55R10: Fiber bundles

Keywords
Reidemeister torsion higher torsion unipotent bundle

Citation

Badzioch, Bernard; Dorabiała, Wojciech. Higher torsion and secondary transfer of unipotent bundles. Geom. Topol. 20 (2016), no. 4, 1807--1857. doi:10.2140/gt.2016.20.1807. https://projecteuclid.org/euclid.gt/1510859017


Export citation

References

  • B Badzioch, W Dorabiała, J R Klein, B Williams, Equivalence of higher torsion invariants, Adv. Math. 226 (2011) 2192–2232
  • B Badzioch, W Dorabiała, B Williams, Smooth parametrized torsion: a manifold approach, Adv. Math. 221 (2009) 660–680
  • J-M Bismut, J Lott, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995) 291–363
  • A J Blumberg, M A Mandell, Algebraic $K$–theory and abstract homotopy theory, Adv. Math. 226 (2011) 3760–3812
  • E H Brown, Jr, Twisted tensor products, I, Ann. of Math. 69 (1959) 223–246
  • U Bunke, On the functoriality of Lott's secondary analytic index, $K$–Theory 25 (2002) 51–58
  • W Dwyer, M Weiss, B Williams, A parametrized index theorem for the algebraic $K$–theory Euler class, Acta Math. 190 (2003) 1–104
  • D R Grayson, The additivity theorem in algebraic $K$–theory, Doc. Math. 16 (2011) 457–464
  • K Igusa, Parametrized Morse theory and its applications, from: “Proceedings of the International Congress of Mathematicians”, (I Satake, editor), volume 1, Math. Soc. Japan, Tokyo (1991) 643–651
  • K Igusa, Higher Franz–Reidemeister torsion, AMS/IP Studies in Advanced Mathematics 31, Amer. Math. Soc., Providence, RI (2002)
  • K Igusa, Axioms for higher torsion invariants of smooth bundles, J. Topol. 1 (2008) 159–186
  • K Igusa, Twisting cochains and higher torsion, J. Homotopy Relat. Struct. 6 (2011) 213–238
  • J R Klein, Higher Reidemeister torsion and parametrized Morse theory, from: “Proceedings of the Winter School `Geometry and Physics' ”, (J Bureš, V Souček, editors), 30 (1993) 15–20
  • J R Klein, B Williams, The refined transfer, bundle structures, and algebraic $K$–theory, J. Topol. 2 (2009) 321–345
  • L Lambe, J Stasheff, Applications of perturbation theory to iterated fibrations, Manuscripta Math. 58 (1987) 363–376
  • J Lott, Secondary analytic indices, from: “Regulators in analysis, geometry and number theory”, (A Reznikov, N Schappacher, editors), Progr. Math. 171, Birkhäuser, Boston (2000) 231–293
  • X Ma, Functoriality of real analytic torsion forms, Israel J. Math. 131 (2002) 1–50
  • F P Peterson, E Thomas, A note on non-stable cohomology operations, Bol. Soc. Mat. Mexicana 3 (1958) 13–18
  • F Waldhausen, Algebraic $K$–theory of spaces, a manifold approach, from: “Current trends in algebraic topology, 1”, (R M Kane, S O Kochman, P S Selick, V P Snaith, editors), CMS Conf. Proc. 2, Amer. Math. Soc., Providence, RI (1982) 141–184
  • F Waldhausen, Algebraic $K$–theory of spaces, from: “Algebraic and geometric topology”, (A Ranicki, N Levitt, F Quinn, editors), Lecture Notes in Math. 1126, Springer, Berlin (1985) 318–419
  • F Waldhausen, Algebraic $K$–theory of spaces, concordance, and stable homotopy theory, from: “Algebraic topology and algebraic $K$–theory”, (W Browder, editor), Ann. of Math. Stud. 113, Princeton Univ. Press (1987) 392–417
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)