Hostname: page-component-6bb9c88b65-6vlrh Total loading time: 0 Render date: 2025-07-26T10:04:11.539Z Has data issue: false hasContentIssue false
  • English
  • Français

Commutativity properties of Quinn spectra

Published online by Cambridge University Press:  25 November 2024

Gerd Laures Open the ORCID record for Gerd Laures [Opens in a new window]  and
James E McClure
Gerd Laures*
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, NA1/66, D-44780 Bochum, Germany (gerd@laures.de) (corresponding author)
James E McClure
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA (mcclurej@purdue.edu)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a simple sufficient condition for Quinn’s ‘bordism-type’ spectra to be weakly equivalent to commutative symmetric ring spectra. We also show that the symmetric signature is (up to weak equivalence) a monoidal transformation between symmetric monoidal functors, which implies that the Sullivan–Ranicki orientation of topological bundles is represented by a ring map between commutative symmetric ring spectra. In the course of proving these statements, we give a new description of symmetric L theory which may be of independent interest.

Keywords

L-theorybordism theoriesad theorieshighly structured ring spectraE∞-spectra

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 6 , December 2024 , pp. 1848 - 1936
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Andrade, R.. (mathoverflow.net/users/21095), Multisimplicial geometric realization, MathOverflow, http://mathoverflow.net/questions/124837 (version: 2013-03-19).Google Scholar
Baas, N. A. and Laures, G.. Singularities and Quinn spectra. MüNster J. Math. 10 (2017), .Google Scholar
Banagl, M., Laures, G. and McClure, J. E.. The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture, Selecta Math. (N.S.) 25 (2019), .Google Scholar
Elmendorf, A. D. and Mandell, M. A.. Rings, modules, and algebras in infinite loop space theory. Adv. Math. 205 (2006), (2007g:19001).CrossRefGoogle Scholar
Freedman, M. H. and Quinn, F.. Topology of 4-manifolds, Princeton Mathematical Series, Vol. 39, (Princeton University Press, Princeton, NJ, 1990) (94b:57021).Google Scholar
Friedman, G. and McClure, J.. The symmetric signature of a Witt space, http://arxiv.org/abs/1106.4798.Google Scholar
Goerss, P. G. and Jardine, J. F.. Simplicial homotopy theory, Progress in Mathematics, Vol. 174 (Birkhäuser Verlag, Basel, 1999) MR 1711612 (2001d:55012).Google Scholar
Hirschhorn, P. S.. Model categories and their localizations, Mathematical Surveys and Monographs, Vol. 99 (American Mathematical Society, Providence, RI, 2003) MR 1944041 (2003j:18018).Google Scholar
Hovey, M., Shipley, B. and Smith, J.. Symmetric spectra. J. Amer. Math. Soc. 13 (2000), (2000h:55016).Google Scholar
Kuehl, P., Macko, T. and Mole, A.. The total surgery obstruction revisited, http://arxiv.org/abs/1104.5092.Google Scholar
Laures, G. and McClure, J. E.. Multiplicative properties of Quinn spectra. Forum Math. 26 (2014), .Google Scholar
Leonard Reedy, C.. Homotopy theory of model categories, available at http://www-math.mit.edu/∼psh.Google Scholar
Lurie, J.. Algebraic L-theory and surgery, notes for a course, available at http://www.math.harvard.edu/lurie/287x.html.Google Scholar
Mac Lane, S.. Natural associativity and commutativity. Rice Univ. Studies 49 (1963), (30 #1160).Google Scholar
Madsen, I. and James Milgram, R.. The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, Vol. 92, (Princeton University Press, Princeton, NJ, 1979) (81b:57014).Google Scholar
McClure, J. E.. On semisimplicial objects satisfying the Kan condition, http://arxiv.org/abs/1210.5650, to appear in Homology, Homotopy and Applications.Google Scholar
Peter May, J.. The geometry of iterated loop spaces, Lectures Notes in Mathematics, Vol. 271, (Springer-Verlag, Berlin, 1972) (54 #8623b).Google Scholar
Peter May, J.. Simplicial Objects in Algebraic Topology, Chicago Lectures in Mathematics. (University of Chicago Press, Chicago, IL, 1992) (93m:55025).Google Scholar
Quinn, F.. Assembly maps in bordism-type theories, Novikov conjectures, index theorems and rigidity, (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., Vol. 226, (Cambridge University Press, Cambridge, 1995) (97h:57055).Google Scholar
Ranicki, A.. The algebraic theory of surgery. I. Foundations, Proc. London Math. Soc. (3) 40 (1980), (82f:57024a).Google Scholar
Ranicki, A.. The algebraic theory of surgery. II. Applications to topology. Proc. London Math. Soc. (3) 40 (1980), (82f:57024b).Google Scholar
Ranicki, A.. Algebraic L-theory and topological manifolds, Cambridge Tracts in Mathematics, Vol. 102, (Cambridge University Press, Cambridge, 1992) (94i:57051).Google Scholar
Schlichtkrull, C.. Thom spectra that are symmetric spectra. Doc. Math. 14 (2009), (2011f:55017).Google Scholar
Segal, G.. Categories and cohomology theories. Topology 13 (1974), (50 #5782).Google Scholar
Sullivan, D. P.. The 1970 MIT notes, Edited and with a preface by Andrew Ranicki, K-Monographs in Mathematics, Vol. 8, (Springer, Dordrecht, 2005) (2006m:55002)Google Scholar
Weibel, C. A.. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 1994).Google Scholar