Spectra of units for periodic ring spectra and group completion of graded E∞ spaces

Sagave, Steffen¹

¹Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands

Algebraic and Geometric Topology, Tome 16 (2016) no. 2, pp. 1203-1251

Résumé

We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its augmentation map is a non-connected delooping of the usual spectrum of units whose bottom homotopy group detects periodicity.

Our approach builds on the graded variant of E∞ spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded E∞ spaces and use it to exhibit our spectrum of units functor as a right adjoint on the level of homotopy categories. The resulting group completion functor is an essential tool for studying ring spectra with graded logarithmic structures.

Détail

DOI : 10.2140/agt.2016.16.1203

Classification : 55P43, 55P48
Keywords: E-infinity space, symmetric spectrum, group completion, units of ring spectra, Gamma-space

Bibliographie

[1] M Ando, A J Blumberg, D Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, preprint (2015)

[2] M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, An ∞–categorical approach to R–line bundles, R–module Thom spectra, and twisted R–homology, J. Topol. 7 (2014) 869

[3] M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory, J. Topol. 7 (2014) 1077

[4] C Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010) 245

[5] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, 347, Springer (1973)

[6] F Borceux, Handbook of categorical algebra, 2, 51, Cambridge Univ. Press (1994)

[7] A K Bousfield, E M Friedlander, Homotopy theory of Γ–spaces, spectra, and bisimplicial sets, from: "Geometric applications of homotopy theory", Lecture Notes in Math. 658, Springer (1978) 80

[8] B I Dundas, T G Goodwillie, R Mccarthy, The local structure of algebraic K–theory, 18, Springer (2013)

[9] P G Goerss, J F Jardine, Simplicial homotopy theory, 174, Birkhäuser (1999)

[10] P S Hirschhorn, Model categories and their localizations, 99, Amer. Math. Soc. (2003)

[11] M Hovey, Model categories, 63, Amer. Math. Soc. (1999)

[12] J A Lind, Diagram spaces, diagram spectra and spectra of units, Algebr. Geom. Topol. 13 (2013) 1857

[13] M A Mandell, An inverse K–theory functor, Doc. Math. 15 (2010) 765

[14] M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441

[15] J P May, The geometry of iterated loop spaces, 271, Springer (1972)

[16] J P May, E∞ ring spaces and E∞ ring spectra, 577, Springer (1977) 268

[17] J Rognes, Topological logarithmic structures, from: "New topological contexts for Galois theory and algebraic geometry" (editors A Baker, B Richter), Geom. Topol. Monogr. 16 (2009) 401

[18] J Rognes, Algebraic K–theory of strict ring spectra, from: "Proceedings of the International Congress of Mathematicians, Vol. II", Kyung Moon Sa (2014) 1259

[19] J Rognes, S Sagave, C Schlichtkrull, Localization sequences for logarithmic topological Hochschild homology, Math. Ann. 363 (2015) 1349

[20] J Rognes, S Sagave, C Schlichtkrull, Logarithmic topological Hochschild homology of topological K–theory spectra, preprint (2015)

[21] S Sagave, Logarithmic structures on topological K–theory spectra, Geom. Topol. 18 (2014) 447

[22] S Sagave, C Schlichtkrull, Graded units of ring spectra and graded Thom spectra,

[23] S Sagave, C Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012) 2116

[24] S Sagave, C Schlichtkrull, Group completion and units in I–spaces, Algebr. Geom. Topol. 13 (2013) 625

[25] S Sagave, C Schlichtkrull, Virtual vector bundles and graded Thom spectra, preprint (2014)

[26] C Schlichtkrull, Units of ring spectra and their traces in algebraic K–theory, Geom. Topol. 8 (2004) 645

[27] S Schwede, Stable homotopical algebra and Γ–spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999) 329

[28] S Schwede, Symmetric spectra, Book project in progress (2007)

[29] G Segal, Categories and cohomology theories, Topology 13 (1974) 293

[30] N Shimada, K Shimakawa, Delooping symmetric monoidal categories, Hiroshima Math. J. 9 (1979) 627