Chromatic splitting for the K(2)–local sphere at p = 2

Beaudry, Agnès¹ ; Goerss, Paul G² ; Henn, Hans-Werner³

¹ Department of Mathematics, University of Colorado Boulder, Boulder, CO, United States
² Department of Mathematics, Northwestern University, Evanston, IL, United States
³ Institut de Recherche Mathématique Avancée, CNRS, Université de Strasbourg, Strasbourg, France

Geometry & topology, Tome 26 (2022) no. 1, pp. 377-476.

Résumé

We calculate the homotopy type of $L_1{L}_{K(2)}{S}^0$ and $L_{K(1)}{L}_{K(2)}{S}^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$–theory and $L_1$ localization with respect to $2$–local $K$–theory. In $L_1{L}_{K(2)}{S}^0$ we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology $H^{\ast }({\mathbb{𝔾}}_2,E_0)$, where ${\mathbb{𝔾}}_2$ is the Morava stabilizer group and $E_0=\mathbb{𝕎}[[u_1]]$ is the ring of functions on the height $2$ Lubin–Tate space. We show that the inclusion of the constants $\mathbb{𝕎}\to E_0$ induces an isomorphism on group cohomology, a radical simplification.

Détail

Keywords: chromatic splitting conjecture, chromatic homotopy theory, Morava K–theory localization of the sphere

Bibliographie

[1] A Adem, R J Milgram, Cohomology of finite groups, 309, Springer (1994) | DOI

[2] T Barthel, A Beaudry, Chromatic structures in stable homotopy theory, from: "Handbook of homotopy theory" (editor H Miller), CRC (2020) 163

[3] T Bauer, Computation of the homotopy of the spectrum tmf, from: "Groups, homotopy and configuration spaces" (editors N Iwase, T Kohno, R Levi, D Tamaki, J Wu), Geom. Topol. Monogr. 13, Geom. Topol. (2008) 11 | DOI

[4] A Beaudry, The algebraic duality resolution at p = 2, Algebr. Geom. Topol. 15 (2015) 3653 | DOI

[5] A Beaudry, The chromatic splitting conjecture at n = p = 2, Geom. Topol. 21 (2017) 3213 | DOI

[6] A Beaudry, Towards the homotopy of the K(2)–local Moore spectrum at p = 2, Adv. Math. 306 (2017) 722 | DOI

[7] M Behrens, The homotopy groups of SE(2) at p ≥ 5 revisited, Adv. Math. 230 (2012) 458 | DOI

[8] I Bobkova, P G Goerss, Topological resolutions in K(2)–local homotopy theory at the prime 2, J. Topol. 11 (2018) 918 | DOI

[9] R Bruner, Algebraic and geometric connecting homomorphisms in the Adams spectral sequence, from: "Geometric applications of homotopy theory, II" (editors M G Barratt, M E Mahowald), Lecture Notes in Math. 658, Springer (1978) 131 | DOI

[10] C Bujard, Finite subgroups of extended Morava stabilizer groups, preprint (2012)

[11] E S Devinatz, A Lyndon–Hochschild–Serre spectral sequence for certain homotopy fixed point spectra, Trans. Amer. Math. Soc. 357 (2005) 129 | DOI

[12] E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1 | DOI

[13] C L Douglas, J Francis, A G Henriques, M A Hill, editors, Topological modular forms, 201, Amer. Math. Soc. (2014) | DOI

[14] P G Goerss, H W Henn, M Mahowald, The rational homotopy of the K(2)–local sphere and the chromatic splitting conjecture for the prime 3 and level 2, Doc. Math. 19 (2014) 1271

[15] P Goerss, H W Henn, M Mahowald, C Rezk, A resolution of the K(2)–local sphere at the prime 3, Ann. of Math. 162 (2005) 777 | DOI

[16] P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: "Structured ring spectra" (editors A Baker, B Richter), Lond. Math. Soc. Lect. Note Ser. 315, Cambridge Univ. Press (2004) 151 | DOI

[17] H W Henn, A mini-course on Morava stabilizer groups and their cohomology, from: "Algebraic topology" (editors N H V Hung, L Schwartz), Lecture Notes in Math. 2194, Springer (2017) 149 | DOI

[18] H W Henn, The centralizer resolution of the K(2)–local sphere at the prime 2, from: "Homotopy theory: tools and applications" (editors D G Davis, H W Henn, J F Jardine, M W Johnson, C Rezk), Contemp. Math. 729, Amer. Math. Soc. (2019) 93 | DOI

[19] H W Henn, N Karamanov, M Mahowald, The homotopy of the K(2)–local Moore spectrum at the prime 3 revisited, Math. Z. 275 (2013) 953 | DOI

[20] M J Hopkins, B H Gross, The rigid analytic period mapping, Lubin–Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. 30 (1994) 76 | DOI

[21] M J Hopkins, J H Smith, Nilpotence and stable homotopy theory, II, Ann. of Math. 148 (1998) 1 | DOI

[22] M Hovey, Bousfield localization functors and Hopkins’ chromatic splitting conjecture, from: "The Čech centennial" (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 225 | DOI

[23] M Hovey, Operations and co-operations in Morava E–theory, Homology Homotopy Appl. 6 (2004) 201 | DOI

[24] M Hovey, N P Strickland, Morava K–theories and localisation, 666, Amer. Math. Soc. (1999) | DOI

[25] J Kohlhaase, On the Iwasawa theory of the Lubin–Tate moduli space, Compos. Math. 149 (2013) 793 | DOI

[26] O Lader, Une résolution projective pour le second groupe de Morava pour p ≥ 5 et applications, PhD thesis, Université de Strasbourg (2013)

[27] M Lazard, Groupes analytiques p–adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965) 389

[28] W H Lin, D M Davis, M E Mahowald, J F Adams, Calculation of Lin’s Ext groups, Math. Proc. Cambridge Philos. Soc. 87 (1980) 459 | DOI

[29] M Mahowald, The image of J in the EHP sequence, Ann. of Math. 116 (1982) 65 | DOI

[30] M Mahowald, C Rezk, Topological modular forms of level 3, Pure Appl. Math. Q. 5 (2009) 853 | DOI

[31] H R Miller, D C Ravenel, W S Wilson, Periodic phenomena in the Adams–Novikov spectral sequence, Ann. of Math. 106 (1977) 469 | DOI

[32] J Morava, Noetherian localisations of categories of cobordism comodules, Ann. of Math. 121 (1985) 1 | DOI

[33] D C Ravenel, The cohomology of the Morava stabilizer algebras, Math. Z. 152 (1977) 287 | DOI

[34] D C Ravenel, A novice’s guide to the Adams–Novikov spectral sequence, from: "Geometric applications of homotopy theory, II" (editors M G Barratt, M E Mahowald), Lecture Notes in Math. 658, Springer (1978) 404 | DOI

[35] D C Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984) 351 | DOI

[36] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 121, Academic (1986)

[37] D C Ravenel, Nilpotence and periodicity in stable homotopy theory, 128, Princeton Univ. Press (1992)

[38] C Rezk, Notes on the Hopkins–Miller theorem, from: "Homotopy theory via algebraic geometry and group representations" (editors M Mahowald, S Priddy), Contemp. Math. 220, Amer. Math. Soc. (1998) 313 | DOI

[39] K Shimomura, The Adams–Novikov E2–term for computing π∗(L2V (0)) at the prime 2, Topology Appl. 96 (1999) 133 | DOI

[40] K Shimomura, X Wang, The Adams–Novikov E2–term for π∗(L2S0) at the prime 2, Math. Z. 241 (2002) 271 | DOI

[41] K Shimomura, A Yabe, The homotopy groups π∗(L2S0), Topology 34 (1995) 261 | DOI

[42] J H Silverman, The arithmetic of elliptic curves, 106, Springer (1986) | DOI

[43] N P Strickland, Gross–Hopkins duality, Topology 39 (2000) 1021 | DOI

[44] N Strickland, Level three structures, preprint (2018)

[45] P Symonds, T Weigel, Cohomology of p–adic analytic groups, from: "New horizons in pro- groups" (editors M du Sautoy, D Segal, A Shalev), Progr. Math. 184, Birkhäuser (2000) 349 | DOI