Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-25T23:05:49.724Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZED THOM SPECTRA AND THEIR TOPOLOGICAL HOCHSCHILD HOMOLOGY

Part of: Homology and cohomology theories Homotopy theory Higher algebraic $K$-theory

Published online by Cambridge University Press:  02 November 2017

Samik Basu ,
Steffen Sagave  and
Christian Schlichtkrull
Samik Basu
Affiliation:
Department of Mathematical and Computational Science, Indian Association for the Cultivation of Science, Kolkata - 700032, India (samik.basu2@gmail.com)
Steffen Sagave
Affiliation:
Radboud University Nijmegen, IMAPP, PO Box 9010, 6500 GL Nijmegen, The Netherlands (s.sagave@math.ru.nl)
Christian Schlichtkrull
Affiliation:
Department of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, Norway (christian.schlichtkrull@math.uib.no)
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a theory of $R$-module Thom spectra for a commutative symmetric ring spectrum $R$ and we analyze their multiplicative properties. As an interesting source of examples, we show that $R$-algebra Thom spectra associated to the special unitary groups can be described in terms of quotient constructions on $R$. We apply the general theory to obtain a description of the $R$-based topological Hochschild homology associated to an $R$-algebra Thom spectrum.

Keywords

Thom spectrumstructured ring spectrumtopological Hochschild homology

MSC classification

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology 55N20: Generalized (extraordinary) homology and cohomology theories 55P42: Stable homotopy theory, spectra 55P43: Spectra with additional structure ($E_infty$, $A_infty$, ring spectra, etc.) 55P48: Loop space machines, operads
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 1 , January 2020 , pp. 21 - 64
Copyright
© Cambridge University Press 2017 

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.)

References

Adams, J. F., Stable Homotopy and Generalised Homology, Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1995). Reprint of the 1974 original.Google Scholar
Ando, M., Blumberg, A. J., Gepner, D., Hopkins, M. J. and Rezk, C., An ∞-categorical approach to R-line bundles, R-module Thom spectra, and twisted R-homology, J. Topol. 7(3) (2014), 869–893.Google Scholar
Ando, M., Blumberg, A. J., Gepner, D., Hopkins, M. J. and Rezk, C., Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory, J. Topol. 7(4) (2014), 1077–1117.Google Scholar
Angeltveit, V., Topological Hochschild homology and cohomology of A ∞ ring spectra, Geom. Topol. 12(2) (2008), 987–1032.Google Scholar
Basterra, M. and Mandell, M. A., Homology and cohomology of E ∞ ring spectra, Math. Z. 249(4) (2005), 903–944.Google Scholar
Basu, S., Topological Hochschild homology of K/p as a K p ∧ module, Homology Homotopy Appl. 19(1) (2017), 253–280.Google Scholar
Basu, S. and Schlichtkrull, C., Topological Hochschild homology of quotients via generalized Thom spectra. In preparation.Google Scholar
Blumberg, A. J., Cohen, R. L. and Schlichtkrull, C., Topological Hochschild homology of Thom spectra and the free loop space, Geom. Topol. 14(2) (2010), 1165–1242.Google Scholar
Bousfield, A. K. and Kan, D. M., Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, Volume 304 (Springer, Berlin–New York, 1972).Google Scholar
Brun, M., Dundas, B. I. and Stolz, M., Equivariant structure on smash powers, preprint, 2016, arXiv:1604.05939.Google Scholar
Elmendorf, A. D., Kriz, I., Mandell, M. A. and May, J. P., Rings, Modules, and Algebras in Stable Homotopy Theory, Mathematical Surveys and Monographs, vol. 47 (American Mathematical Society, Providence, RI, 1997). With an appendix by M. Cole.Google Scholar
Fiedorowicz, Z., Classifying spaces of topological monoids and categories, Amer. J. Math. 106(2) (1984), 301–350.Google Scholar
Goerss, P. G. and Hopkins, M. J., Moduli spaces of commutative ring spectra, in Structured Ring Spectra, London Mathematical Society Lecture Note Series, Volume 315, pp. 151–200 (Cambridge University Press, Cambridge, 2004).Google Scholar
Hopkins, M. J. and Lurie, J., On Brauer groups of Lubin–Tate spectra I, preprint, 2017, available at http://www.math.harvard.edu/∼lurie/.Google Scholar
Hovey, M., Model Categories, Mathematical Surveys and Monographs, Volume 63 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Hovey, M., Shipley, B. and Smith, J., Symmetric spectra, J. Amer. Math. Soc. 13(1) (2000), 149–208.Google Scholar
Lewis, L. G. Jr., When is the natural map X→𝛺𝛴X a cofibration? Trans. Amer. Math. Soc. 273(1) (1982), 147–155.Google Scholar
Lewis, L. G. Jr., May, J. P., Steinberger, M. and McClure, J. E., Equivariant Stable Homotopy Theory, Lecture Notes in Mathematics, Volume 1213 (Springer, Berlin, 1986). With contributions by J. E. McClure.Google Scholar
Lind, J. A., Diagram spaces, diagram spectra and spectra of units, Algebr. Geom. Topol. 13(4) (2013), 1857–1935.Google Scholar
Mandell, M. A., May, J. P., Schwede, S. and Shipley, B., Model categories of diagram spectra, Proc. Lond. Math. Soc. (3) 82(2) (2001), 441–512.Google Scholar
May, J. P., The Geometry of Iterated Loop Spaces, Lectures Notes in Mathematics, Volume 271 (Springer, Berlin–New York, 1972).Google Scholar
May, J. P., Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1(1) (1975), 155. xiii+98.Google Scholar
May, J. P., E ∞ Ring Spaces and E ∞ Ring Spectra, Lecture Notes in Mathematics, Volume 577 (Springer, Berlin–New York, 1977). With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave.Google Scholar
Rognes, J., Sagave, S. and Schlichtkrull, C., Localization sequences for logarithmic topological Hochschild homology, Math. Ann. 363(3–4) (2015), 1349–1398.Google Scholar
Sagave, S. and Schlichtkrull, C., Diagram spaces and symmetric spectra, Adv. Math. 231(3–4) (2012), 2116–2193.Google Scholar
Sagave, S. and Schlichtkrull, C., Group completion and units in 𝓘-spaces, Algebr. Geom. Topol. 13 (2013), 625–686.Google Scholar
Sagave, S. and Schlichtkrull, C., Topological André–Quillen homology of generalized Thom spectra. In preparation.Google Scholar
Schlichtkrull, C., Units of ring spectra and their traces in algebraic K-theory, Geom. Topol. 8 (2004), 645–673. (electronic).Google Scholar
Schlichtkrull, C., Thom spectra that are symmetric spectra, Doc. Math. 14 (2009), 699–748.Google Scholar
Schwede, S., Symmetric spectra, 2012, Book project, available at the author’s home page.Google Scholar
Schwede, S. and Shipley, B. E., Algebras and modules in monoidal model categories, Proc. Lond. Math. Soc. (3) 80(2) (2000), 491–511.Google Scholar
Shipley, B., Symmetric spectra and topological Hochschild homology, K-Theory 19(2) (2000), 155–183.Google Scholar
Shipley, B., A convenient model category for commutative ring spectra, in Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Contemporary Mathematics, Volume 346, pp. 473–483 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Steenrod, N. E., A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152.Google Scholar
Whitehead, G. W., Elements of Homotopy Theory, Graduate Texts in Mathematics, Volume 61 (Springer, New York–Berlin, 1978).Google Scholar