Geometry & Topology

Topological Hochschild homology of Thom spectra and the free loop space

Andrew J Blumberg, Ralph L Cohen, and Christian Schlichtkrull

Full-text: Open access

Abstract

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f:XBF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc, and the Eilenberg–Mac Lane spectra Hp and H.

Article information

Source
Geom. Topol., Volume 14, Number 2 (2010), 1165-1242.

Dates
Received: 4 November 2008
Revised: 1 March 2010
Accepted: 2 April 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732216

Digital Object Identifier
doi:10.2140/gt.2010.14.1165

Mathematical Reviews number (MathSciNet)
MR2651551

Zentralblatt MATH identifier
1219.19006

Subjects
Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55N20: Generalized (extraordinary) homology and cohomology theories
Secondary: 18G55: Homotopical algebra 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55P47: Infinite loop spaces 55R25: Sphere bundles and vector bundles

Keywords
topological Hochschild homology Thom spectra loop space

Citation

Blumberg, Andrew J; Cohen, Ralph L; Schlichtkrull, Christian. Topological Hochschild homology of Thom spectra and the free loop space. Geom. Topol. 14 (2010), no. 2, 1165--1242. doi:10.2140/gt.2010.14.1165. https://projecteuclid.org/euclid.gt/1513732216


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References

  • M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, Units of ring spectra and Thom spectra
  • A Baker, B Richter, Quasisymmetric functions from a topological point of view, Math. Scand. 103 (2008) 208–242
  • M Basterra, André–Quillen cohomology of commutative $S$–algebras, J. Pure Appl. Algebra 144 (1999) 111–143
  • M Basterra, M A Mandell, Homology and cohomology of $E\sb \infty$ ring spectra, Math. Z. 249 (2005) 903–944
  • A J Blumberg, Topological Hochschild homology of Thom spectra which are ${E}_\infty$ ring spectra
  • A J Blumberg, Progress towards the calculation of the $K$–theory of Thom spectra, PhD thesis, University of Chicago (2005)
  • J M Boardman, R M Vogt, Homotopy-everything $H$–spaces, Bull. Amer. Math. Soc. 74 (1968) 1117–1122
  • M Bökstedt, The topological Hochschild homology of $\Z$ and $\Z/p$, Preprint (1985)
  • A K Bousfield, E M Friedlander, Homotopy theory of $\Gamma $–spaces, spectra, and bisimplicial sets, from: “Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II”, Lecture Notes in Math. 658, Springer, Berlin (1978) 80–130
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, Berlin (1972)
  • M Brun, Z Fiedorowicz, R M Vogt, On the multiplicative structure of topological Hochschild homology, Algebr. Geom. Topol. 7 (2007) 1633–1650
  • S Bullett, Permutations and braids in cobordism theory, Proc. London Math. Soc. $(3)$ 38 (1979) 517–531
  • F R Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978) 99–110
  • F R Cohen, J P May, L R Taylor, $K({\bf Z},\,0)$ and $K(Z\sb{2},\,0)$ as Thom spectra, Illinois J. Math. 25 (1981) 99–106
  • W G Dwyer, J Spaliński, Homotopy theories and model categories, from: “Handbook of algebraic topology”, (I M James, editor), North-Holland, Amsterdam (1995) 73–126
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monogr. 47, Amer. Math. Soc. (1997) With an appendix by M Cole
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Errata for “Rings, modules, and algebras in stable homotopy theory (2007) Available at \setbox0\makeatletter\@url http://www.math.uchicago.edu/~may/BOOKS/Addenda.pdf {\unhbox0
  • T G Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985) 187–215
  • P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003)
  • M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149–208
  • N Iwase, A continuous localization and completion, Trans. Amer. Math. Soc. 320 (1990) 77–90
  • I Kriz, J P May, Operads, algebras, modules and motives, Astérisque 233 (1995)
  • L G Lewis, Jr, J P May, M Steinberger, J E McClure, Equivariant stable homotopy theory, Lecture Notes in Math. 1213, Springer, Berlin (1986) With contributions by J E McClure
  • J Lillig, A union theorem for cofibrations, Arch. Math. $($Basel$)$ 24 (1973) 410–415
  • S Mac Lane, Categories for the working mathematician, second edition, Graduate Texts in Math. 5, Springer, New York (1998)
  • I Madsen, Algebraic $K$–theory and traces, from: “Current developments in mathematics, 1995 (Cambridge, MA)”, (R Bott, M Hopkins, A Jaffe, I Singer, D Stroock, S-T Yau, editors), Int. Press, Cambridge, MA (1994) 191–321
  • M Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979) 549–559
  • M Mahowald, D C Ravenel, P Shick, The Thomified Eilenberg-Moore spectral sequence, from: “Cohomological methods in homotopy theory (Bellaterra, 1998)”, (J Aguadé, C Broto, C Casacuberta, editors), Progr. Math. 196, Birkhäuser, Basel (2001) 249–262
  • M Mahowald, N Ray, A note on the Thom isomorphism, Proc. Amer. Math. Soc. 82 (1981) 307–308
  • M A Mandell, Topological Hochschild homology of an ${E}_n$ ring spectrum is ${E}_{n-1}$, Preprint (2004)
  • M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. $(3)$ 82 (2001) 441–512
  • J P May, The geometry of iterated loop spaces, Lectures Notes in Math. 271, Springer, Berlin (1972)
  • J P May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 155 (1975)
  • J P May, $E\sb{\infty }$ ring spaces and $E\sb{\infty }$ ring spectra, Lecture Notes in Math. 577, Springer, Berlin (1977) With contributions by F Quinn, N Ray, and J Tornehave
  • J P May, Fibrewise localization and completion, Trans. Amer. Math. Soc. 258 (1980) 127–146
  • J P May, J Sigurdsson, Parametrized homotopy theory, Math. Surveys and Monogr. 132, Amer. Math. Soc. (2006)
  • J R Munkres, Topology: a first course, second edition, Prentice-Hall Inc., Englewood Cliffs, NJ (2000)
  • S Sagave, C Schlichtkrull, Diagram spaces and symmetric spectra, in preparation
  • C Schlichtkrull, Higher topological Hochschild homology of Thom spectra
  • C Schlichtkrull, Units of ring spectra and their traces in algebraic $K$–theory, Geom. Topol. 8 (2004) 645–673
  • C Schlichtkrull, The homotopy infinite symmetric product represents stable homotopy, Algebr. Geom. Topol. 7 (2007) 1963–1977
  • C Schlichtkrull, Thom spectra that are symmetric spectra, Doc. Math. 14 (2009) 699–748
  • S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. $(3)$ 80 (2000) 491–511
  • G Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973) 213–221
  • G Segal, Categories and cohomology theories, Topology 13 (1974) 293–312
  • B Shipley, Symmetric spectra and topological Hochschild homology, $K$–Theory 19 (2000) 155–183
  • R E Stong, Notes on cobordism theory, Math. notes, Princeton Univ. Press (1968)
  • A Strøm, The homotopy category is a homotopy category, Arch. Math. $($Basel$)$ 23 (1972) 435–441
  • R W Thomason, Uniqueness of delooping machines, Duke Math. J. 46 (1979) 217–252
  • F Waldhausen, Algebraic $K$–theory of topological spaces. II, from: “Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, 1978)”, (J L Dupont, I Madsen, editors), Lecture Notes in Math. 763, Springer, Berlin (1979) 356–394
  • G W Whitehead, Elements of homotopy theory, Graduate Texts in Math. 61, Springer, New York (1978)