Rigidity in equivariant stable homotopy theory

Patchkoria, Irakli¹

¹Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 2159-2227

Résumé

For any finite group G, we show that the 2–local G–equivariant stable homotopy category, indexed on a complete G–universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all “higher-order structure” of the 2–local G–equivariant stable homotopy category, such as the equivariant homotopy types of function G–spaces. Our result can be seen as an equivariant version of Schwede’s rigidity theorem at the prime 2.

Détail

DOI : 10.2140/agt.2016.16.2159

Classification : 55P42, 55P91, 18G55
Keywords: equivariant stable homotopy category, model category, rigidity

Bibliographie

[1] J F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960) 20

[2] V Angeltveit, Topological Hochschild homology and cohomology of A∞ ring spectra, Geom. Topol. 12 (2008) 987

[3] D Barnes, C Roitzheim, Rational equivariant rigidity, from: "An alpine expedition through algebraic topology" (editors C Ausoni, K Hess, B Johnson, W Lück, J Scherer), Contemp. Math. 617, Amer. Math. Soc. (2014) 13

[4] H J Baues, F Muro, Toda brackets and cup-one squares for ring spectra, Comm. Algebra 37 (2009) 56

[5] A J Blumberg, Continuous functors as a model for the equivariant stable homotopy category, Algebr. Geom. Topol. 6 (2006) 2257

[6] J M Boardman, On stable homotopy theory and some applications, PhD thesis, University of Cambridge (1964)

[7] A K Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001) 2391

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

[9] J M Cohen, The decomposition of stable homotopy, Ann. of Math. 87 (1968) 305

[10] T Tom Dieck, Transformation groups, 8, de Gruyter (1987)

[11] D Dugger, B Shipley, K–theory and derived equivalences, Duke Math. J. 124 (2004) 587

[12] D Dugger, B Shipley, A curious example of triangulated-equivalent model categories which are not Quillen equivalent, Algebr. Geom. Topol. 9 (2009) 135

[13] W G Dwyer, J Spaliński, Homotopy theories and model categories, from: "Handbook of algebraic topology" (editor I M James), North-Holland (1995) 73

[14] A D Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983) 275

[15] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, 47, Amer. Math. Soc. (1997)

[16] S I Gelfand, Y I Manin, Methods of homological algebra, Springer (2003)

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

[18] J P C Greenlees, J P May, Generalized Tate cohomology, 543, Amer. Math. Soc. (1995)

[19] B Guillou, J P May, Enriched model categories and presheaf categories, preprint (2013)

[20] M Hausmann, G-symmetric spectra, semistability and the multiplicative norm, preprint (2014)

[21] M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. (2) 184 (2016) 1

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

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

[24] M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149

[25] K Hutschenreuter, On rigidity of the ring spectra Pm§(p) and ko, PhD thesis, Universität Bonn (2012)

[26] S Illman, The equivariant triangulation theorem for actions of compact Lie groups, Math. Ann. 262 (1983) 487

[27] D M Kan, Semisimplicial spectra, Illinois J. Math. 7 (1963) 463

[28] G M Kelly, Basic concepts of enriched category theory, 64, Cambridge Univ. Press (1982) 245

[29] L G Lewis Jr., The category of Mackey functors for a compact Lie group, from: "Group representations: cohomology, group actions and topology" (editors A Adem, J Carlson, S Priddy, P Webb), Proc. Sympos. Pure Math. 63, Amer. Math. Soc. (1998) 301

[30] L G Lewis Jr., J P May, M Steinberger, J E Mcclure, Equivariant stable homotopy theory, 1213, Springer (1986)

[31] M Lydakis, Simplicial functors and stable homotopy theory, unpublished manuscript (1998)

[32] M A Mandell, Equivariant symmetric spectra, from: "Homotopy theory : relations with algebraic geometry, group cohomology, and algebraic K–theory" (editors P Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 399

[33] M A Mandell, J P May, Equivariant orthogonal spectra and S–modules, 755, Amer. Math. Soc. (2002)

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

[35] J P May, Equivariant homotopy and cohomology theory, 91, Amer. Math. Soc. (1996)

[36] J P May, The Wirthmüller isomorphism revisited, Theory Appl. Categ. 11 (2003) 132

[37] A Neeman, Triangulated categories, 148, Princeton Univ. Press (2001)

[38] I Patchkoria, On the algebraic classification of module spectra, Algebr. Geom. Topol. 12 (2012) 2329

[39] D G Quillen, Homotopical algebra, 43, Springer (1967)

[40] C Roitzheim, Rigidity and exotic models for the K–local stable homotopy category, Geom. Topol. 11 (2007) 1855

[41] M Schlichting, A note on K–theory and triangulated categories, Invent. Math. 150 (2002) 111

[42] S Schwede, The stable homotopy category has a unique model at the prime 2, Adv. Math. 164 (2001) 24

[43] S Schwede, The stable homotopy category is rigid, Ann. of Math. 166 (2007) 837

[44] S Schwede, Algebraic versus topological triangulated categories, from: "Triangulated categories" (editor Cambridge), London Math. Soc. Lecture Note Ser. 375, Cambridge Univ. Press (2010) 389

[45] S Schwede, Lectures on equivariant stable homotopy theory, lecture notes (2014)

[46] S Schwede, B Shipley, A uniqueness theorem for stable homotopy theory, Math. Z. 239 (2002) 803

[47] S Schwede, B Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103

[48] M Stolz, Equivariant structures on smash powers of commutative ring spectra, PhD thesis, University of Bergen (2011)

[49] H Toda, Composition methods in homotopy groups of spheres, 49, Princeton Univ. Press (1962)

[50] C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994)