## Geometry & Topology

### The character of the total power operation

#### Abstract

We compute the total power operation for the Morava $E$–theory of any finite group up to torsion. Our formula is stated in terms of the $GLn(ℚp)$–action on the Drinfel’d ring of full level structures on the formal group associated to $E$–theory. It can be specialized to give explicit descriptions of many classical operations. Moreover, we show that the character map of Hopkins, Kuhn and Ravenel from $E$–theory to $GLn(ℤp)$–invariant generalized class functions is a natural transformation of global power functors on finite groups.

#### Article information

Source
Geom. Topol., Volume 21, Number 1 (2017), 385-440.

Dates
Revised: 26 September 2016
Accepted: 15 January 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859137

Digital Object Identifier
doi:10.2140/gt.2017.21.385

Mathematical Reviews number (MathSciNet)
MR3608717

Zentralblatt MATH identifier
1360.55004

#### Citation

Barthel, Tobias; Stapleton, Nathaniel. The character of the total power operation. Geom. Topol. 21 (2017), no. 1, 385--440. doi:10.2140/gt.2017.21.385. https://projecteuclid.org/euclid.gt/1510859137

#### References

• J F Adams, Maps between classifying spaces, II, Invent. Math. 49 (1978) 1–65
• J F Adams, M F Atiyah, $K$–theory and the Hopf invariant, Quart. J. Math. Oxford Ser. 17 (1966) 31–38
• M Ando, Isogenies of formal group laws and power operations in the cohomology theories $E_n$, Duke Math. J. 79 (1995) 423–485
• M Ando, M J Hopkins, N P Strickland, The sigma orientation is an $H_\infty$ map, Amer. J. Math. 126 (2004) 247–334
• T Barthel, N Stapleton, Centralizers in good groups are good, Algebr. Geom. Topol. 16 (2016) 1453–1472
• R R Bruner, J P May, J E McClure, M Steinberger, $H\sb \infty$ ring spectra and their applications, Lecture Notes in Mathematics 1176, Springer, Berlin (1986)
• H Carayol, Nonabelian Lubin–Tate theory, from “Automorphic forms, Shimura varieties, and $L$–functions, II” (L Clozel, J S Milne, editors), Perspect. Math. 11, Academic Press, Boston (1990) 15–39
• E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1–47
• V G Drinfel'd, Elliptic modules, Mat. Sb. 94(136) (1974) 594–627 In Russian; translated in Math. USSR-Sb. 23 (1974) 561–592
• N Ganter, Global Mackey functors with operations and $n$–special lambda rings, preprint (2013)
• P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from “Structured ring spectra” (A Baker, B Richter, editors), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
• M J Hopkins, N J Kuhn, D C Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000) 553–594
• J Lubin, J Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966) 49–59
• P Nelson, On the Morava $E$–theory of wreath products of symmetric groups, in preparation
• C Rezk, Notes on the Hopkins–Miller theorem, from “Homotopy theory via algebraic geometry and group representations” (M Mahowald, S Priddy, editors), Contemp. Math. 220, Amer. Math. Soc., Providence, RI (1998) 313–366
• C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969–1014
• C Rezk, Power operations for Morava E-theory of height $2$ at the prime $2$, preprint (2008)
• C Rezk, Power operations in Morava $E$–theory: structure and calculations, preprint (2013) Available at \setbox0\makeatletter\@url http://www.math.uiuc.edu/~rezk/power-ops-ht-2.pdf {\unhbox0
• J Rognes, Galois extensions of structured ring spectra: stably dualizable groups, Mem. Amer. Math. Soc. 898, Amer. Math. Soc., Providence, RI (2008)
• T M Schlank, N Stapleton, A transchromatic proof of Strickland's theorem, Adv. Math. 285 (2015) 1415–1447
• B Schuster, Morava $K$–theory of groups of order $32$, Algebr. Geom. Topol. 11 (2011) 503–521
• N Stapleton, An introduction to HKR character theory, preprint (2013)
• N Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013) 171–203
• N P Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997) 161–208
• N P Strickland, Morava $E$–theory of symmetric groups, Topology 37 (1998) 757–779
• A V Zelevinsky, Representations of finite classical groups: a Hopf algebra approach, Lecture Notes in Mathematics 869, Springer, Berlin (1981)
• Y Zhu, The power operation structure on Morava $E$–theory of height $2$ at the prime $3$, Algebr. Geom. Topol. 14 (2014) 953–977