## Algebraic & Geometric Topology

### Spectra of units for periodic ring spectra and group completion of graded $E_{\infty}$ spaces

Steffen Sagave

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 1203-1251.

Dates
Revised: 11 July 2015
Accepted: 13 July 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841164

Digital Object Identifier
doi:10.2140/agt.2016.16.1203

Mathematical Reviews number (MathSciNet)
MR3493419

Zentralblatt MATH identifier
06577078

#### Citation

Sagave, Steffen. Spectra of units for periodic ring spectra and group completion of graded $E_{\infty}$ spaces. Algebr. Geom. Topol. 16 (2016), no. 2, 1203--1251. doi:10.2140/agt.2016.16.1203. https://projecteuclid.org/euclid.agt/1510841164

#### References

• M Ando, A,J Blumberg, D Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, preprint (2015)
• M Ando, A,J Blumberg, D Gepner, M,J Hopkins, C Rezk, An $\infty$–categorical approach to $R$–line bundles, $R$–module Thom spectra, and twisted $R$–homology, J. Topol. 7 (2014) 869–893
• 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) 1077–1117
• C Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010) 245–320
• J,M Boardman, R,M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer, Berlin (1973)
• F Borceux, Handbook of categorical algebra, $2$, Encyclopedia of Mathematics and its Applications 51, Cambridge Univ. Press (1994)
• A,K Bousfield, E,M Friedlander, Homotopy theory of $\Gamma$–spaces, spectra, and bisimplicial sets, from: “Geometric applications of homotopy theory”, Lecture Notes in Math. 658, Springer, Berlin (1978) 80–130
• B,I Dundas, T,G Goodwillie, R McCarthy, The local structure of algebraic $K\!$–theory, Algebra and Applications 18, Springer, London (2013)
• P,G Goerss, J,F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser, Basel (1999)
• P,S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)
• M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc. (1999)
• J,A Lind, Diagram spaces, diagram spectra and spectra of units, Algebr. Geom. Topol. 13 (2013) 1857–1935
• M,A Mandell, An inverse $K\!$–theory functor, Doc. Math. 15 (2010) 765–791
• M,A Mandell, J,P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
• J,P May, The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer, Berlin (1972)
• J,P May, $E\sb{\infty }$ ring spaces and $E\sb{\infty }$ ring spectra, Lecture Notes in Mathematics 577, Springer, Berlin (1977)
• J Rognes, Topological logarithmic structures, from: “New topological contexts for Galois theory and algebraic geometry”, (A Baker, B Richter, editors), Geom. Topol. Monogr. 16, Coventry (2009) 401–544
• J Rognes, Algebraic $K\!$–theory of strict ring spectra, from: “Proceedings of the International Congress of Mathematicians, Vol. II”, Kyung Moon Sa, Seoul (2014) 1259–1283
• J Rognes, S Sagave, C Schlichtkrull, Localization sequences for logarithmic topological Hochschild homology, Math. Ann. 363 (2015) 1349–1398
• J Rognes, S Sagave, C Schlichtkrull, Logarithmic topological Hochschild homology of topological $K$–theory spectra, preprint (2015)
• S Sagave, Logarithmic structures on topological $K\!$–theory spectra, Geom. Topol. 18 (2014) 447–490
• S Sagave, C Schlichtkrull, Graded units of ring spectra and graded Thom spectra In preparation
• S Sagave, C Schlichtkrull, Diagram spaces and symmetric spectra, Adv. Math. 231 (2012) 2116–2193
• S Sagave, C Schlichtkrull, Group completion and units in $\mathscr I$–spaces, Algebr. Geom. Topol. 13 (2013) 625–686
• S Sagave, C Schlichtkrull, Virtual vector bundles and graded Thom spectra, preprint (2014)
• C Schlichtkrull, Units of ring spectra and their traces in algebraic $K\!$–theory, Geom. Topol. 8 (2004) 645–673
• S Schwede, Stable homotopical algebra and $\Gamma$–spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999) 329–356
• S Schwede, Symmetric spectra, Book project in progress (2007) Available at \setbox0\makeatletter\@url http://www.math.uni-bonn.de/people/schwede/SymSpec.pdf {\unhbox0
• G Segal, Categories and cohomology theories, Topology 13 (1974) 293–312
• N Shimada, K Shimakawa, Delooping symmetric monoidal categories, Hiroshima Math. J. 9 (1979) 627–645