Localizations of abelian Eilenberg–Mac Lane spaces of finite type

Casacuberta, Carles¹ ; Rodríguez, José² ; Tai, Jin-yen³

¹Institut de Matemàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
²Departamento de Matemáticas, Universidad de Almería, 04120 Almería, Spain
³Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 2379-2420

Résumé

We prove that every homotopical localization of the circle S1 is an aspherical space whose fundamental group A is abelian and admits a ring structure with unit such that the evaluation map End(A) → A at the unit is an isomorphism of rings. Since it is known that there is a proper class of nonisomorphic rings with this property, and we show that all occur in this way, it follows that there is a proper class of distinct homotopical localizations of spaces (in spite of the fact that homological localizations form a set). This answers a question asked by Farjoun in the nineties.

More generally, we study localizations LfK(G,n) of Eilenberg–Mac Lane spaces with respect to any map f, where n ≥ 1 and G is any abelian group, and we show that many properties of G are transferred to the homotopy groups of LfK(G,n). Among other results, we show that, if X is a product of abelian Eilenberg–Mac Lane spaces and f is any map, then the homotopy groups πm(LfX) are modules over the ring π1(LfS1) in a canonical way. This explains and generalizes earlier observations made by other authors in the case of homological localizations.

Détail

DOI : 10.2140/agt.2016.16.2379

Classification : 55P20, 55P60, 18A40, 16S10
Keywords: homotopy, localization, Eilenberg–Mac Lane space, solid ring, rigid ring

Bibliographie

[1] J Adámek, J Rosický, Locally presentable and accessible categories, 189, Cambridge Univ. Press (1994)

[2] B Badzioch, Recognition principle for generalized Eilenberg–Mac Lane spaces, from: "Cohomological methods in homotopy theory" (editors J Aguadé, C Broto, C Casacuberta), Progr. Math. 196, Birkhäuser (2001) 21

[3] B Badzioch, M Feshbach, A note on localizations of perfect groups, Proc. Amer. Math. Soc. 133 (2005) 693

[4] M Blomgren, W Chachólski, E D Farjoun, Y Segev, Idempotent transformations of finite groups, Adv. Math. 233 (2013) 56

[5] A K Bousfield, Types of acyclicity, J. Pure Appl. Algebra 4 (1974) 293

[6] A K Bousfield, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1976/77) 207

[7] A K Bousfield, On homology equivalences and homological localizations of spaces, Amer. J. Math. 104 (1982) 1025

[8] A K Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994) 831

[9] A K Bousfield, Unstable localization and periodicity, from: "Algebraic topology: new trends in localization and periodicity" (editors C Broto, C Casacuberta, G Mislin), Progr. Math. 136, Birkhäuser (1996) 33

[10] A K Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997) 1321

[11] A K Bousfield, D M Kan, The core of a ring, J. Pure Appl. Algebra 2 (1972) 73

[12] R A Bowshell, P Schultz, Unital rings whose additive endomorphisms commute, Math. Ann. 228 (1977) 197

[13] G Braun, R Göbel, E–algebras whose torsion part is not cyclic, Proc. Amer. Math. Soc. 133 (2005) 2251

[14] K S Brown, Cohomology of groups, 87, Springer (1982)

[15] J Buckner, M Dugas, Co-local subgroups of abelian groups, from: "Abelian groups, rings, modules, and homological algebra" (editors P Goeters, O M G Jenda), Lect. Notes Pure Appl. Math. 249, Chapman Hall/CRC (2006) 29

[16] J Buckner, M Dugas, Co-local subgroups of nilpotent groups of class 2, from: "Models, modules and abelian groups" (editors R Göbel, B Goldsmith), Walter de Gruyter (2008) 351

[17] C Casacuberta, Recent advances in unstable localization, from: "The Hilton Symposium 1993" (editor G Mislin), CRM Proc. Lecture Notes 6, Amer. Math. Soc. (1994) 1

[18] C Casacuberta, Anderson localization from a modern point of view, from: "The Čech centennial" (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 35

[19] C Casacuberta, On structures preserved by idempotent transformations of groups and homotopy types, from: "Crystallographic groups and their generalizations" (editors P Igodt, H Abels, Y Félix, F Grunewald), Contemp. Math. 262, Amer. Math. Soc. (2000) 39

[20] C Casacuberta, J J Gutiérrez, Homotopical localizations of module spectra, Trans. Amer. Math. Soc. 357 (2005) 2753

[21] C Casacuberta, J J Gutiérrez, I Moerdijk, R M Vogt, Localization of algebras over coloured operads, Proc. Lond. Math. Soc. 101 (2010) 105

[22] C Casacuberta, O Raventós, A Tonks, Comparing localizations across adjunctions, preprint (2014)

[23] C Casacuberta, J L Rodríguez, On towers approximating homological localizations, J. London Math. Soc. 56 (1997) 645

[24] W Chachólski, E Damian, E D Farjoun, Y Segev, The A–core and A–cover of a group, J. Algebra 321 (2009) 631

[25] W Dicks, W Stephenson, Epimorphs and dominions of Dedekind domains, J. London Math. Soc. 29 (1984) 224

[26] A Dold, R Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. 67 (1958) 239

[27] M Dugas, Localizations of torsion-free abelian groups, J. Algebra 278 (2004) 411

[28] M Dugas, Localizations of torsion-free abelian groups, II, J. Algebra 284 (2005) 811

[29] M Dugas, Co-local subgroups of abelian groups, II, J. Pure Appl. Algebra 208 (2007) 117

[30] M Dugas, A Mader, C Vinsonhaler, Large E–rings exist, J. Algebra 108 (1987) 88

[31] W G Dwyer, J H Palmieri, Ohkawa’s theorem : there is a set of Bousfield classes, Proc. Amer. Math. Soc. 129 (2001) 881

[32] E D Farjoun, Cellular spaces, null spaces and homotopy localization, 1622, Springer (1996)

[33] E D Farjoun, R Göbel, Y Segev, Cellular covers of groups, J. Pure Appl. Algebra 208 (2007) 61

[34] E D Farjoun, R Göbel, Y Segev, S Shelah, On kernels of cellular covers, Groups Geom. Dyn. 1 (2007) 409

[35] L Fuchs, R Göbel, Cellular covers of abelian groups, Results Math. 53 (2009) 59

[36] R Göbel, B Goldsmith, Classifying E-algebras over Dedekind domains, J. Algebra 306 (2006) 566

[37] R Göbel, J L Rodríguez, S Shelah, Large localizations of finite simple groups, J. Reine Angew. Math. 550 (2002) 1

[38] R Göbel, J L Rodríguez, L Strüngmann, Cellular covers of cotorsion-free modules, Fund. Math. 217 (2012) 211

[39] R Göbel, S Shelah, Constructing simple groups for localizations, Comm. Algebra 30 (2002) 809

[40] R Göbel, L Strüngmann, Almost-free E(R)–algebras and E(A,R)–modules, Fund. Math. 169 (2001) 175

[41] R Göbel, J Trlifaj, Approximations and endomorphism algebras of modules, 41, Walter de Gruyter (2006)

[42] R Göbel, J Trlifaj, Approximations and endomorphism algebras of modules, 2 : Predictions, 41, Walter de Gruyter (2012)

[43] J J Gutiérrez, Localización y conservación de estructuras en homotopía estable, PhD thesis, Universitat de Barcelona (2004)

[44] J J Gutiérrez, Homological localizations of Eilenberg–MacLane spectra, Forum Math. 22 (2010) 349

[45] J J Gutiérrez, Cellularization of structures in stable homotopy categories, Math. Proc. Cambridge Philos. Soc. 153 (2012) 399

[46] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

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

[48] I Kaplansky, Infinite abelian groups, The University of Michigan Press (1969)

[49] A Libman, Cardinality and nilpotency of localizations of groups and G–modules, Israel J. Math. 117 (2000) 221

[50] S Mac Lane, Categories for the working mathematician, 5, Springer (1971)

[51] J P May, Simplicial objects in algebraic topology, 11, D Van Nostrand Co (1967)

[52] J P May, Infinite loop space theory, Bull. Amer. Math. Soc. 83 (1977) 456

[53] H Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984) 39

[54] G Mislin, Localization with respect to K–theory, J. Pure Appl. Algebra 10 (1977/78) 201

[55] R J Nunke, Modules of extensions over Dedekind rings, Illinois J. Math. 3 (1959) 222

[56] T Ohkawa, The injective hull of homotopy types with respect to generalized homology functors, Hiroshima Math. J. 19 (1989) 631

[57] N O’Sullivan, Localizations of free soluble groups, J. Group Theory 4 (2001) 89

[58] C Parker, J Saxl, Localization and finite simple groups, Israel J. Math. 153 (2006) 285

[59] R S Pierce, E–modules, from: "Abelian group theory" (editors L Fuchs, R Göbel, P Schultz), Contemp. Math. 87, Amer. Math. Soc. (1989) 221

[60] R S Pierce, C I Vinsonhaler, Classifying E–rings, Comm. Algebra 19 (1991) 615

[61] D J S Robinson, A course in the theory of groups, 80, Springer (1996)

[62] J L Rodríguez, On homotopy colimits of spaces with a single homology or homotopy group, PhD thesis, Universitat Autònoma de Barcelona (1997)

[63] J L Rodríguez, J Scherer, Cellular approximations using Moore spaces, from: "Cohomological methods in homotopy theory" (editors J Aguadé, C Broto, C Casacuberta), Progr. Math. 196, Birkhäuser (2001) 357

[64] J L Rodríguez, J Scherer, A connection between cellularization for groups and spaces via two-complexes, J. Pure Appl. Algebra 212 (2008) 1664

[65] J L Rodríguez, J Scherer, L Strüngmann, On localization of torsion abelian groups, Fund. Math. 183 (2004) 123

[66] J L Rodríguez, J Scherer, J Thévenaz, Finite simple groups and localization, Israel J. Math. 131 (2002) 185

[67] J L Rodríguez, J Scherer, A Viruel, Preservation of perfectness and acyclicity : Berrick and Casacuberta’s universal acyclic space localized at a set of primes, Forum Math. 17 (2005) 67

[68] J L Rodríguez, J Scherer, A Viruel, Non-simple localizations of finite simple groups, J. Algebra 305 (2006) 765

[69] J L Rodríguez, L Strüngmann, On cellular covers with free kernels, Mediterr. J. Math. 9 (2012) 295

[70] P Schultz, The endomorphism ring of the additive group of a ring, J. Austral. Math. Soc. 15 (1973) 60

[71] B Stenström, Rings of quotients, 217, Springer (1975)

[72] L Strüngmann, Almost-free E(R)–algebras over countable domains, PhD thesis, Universität Essen (1998)

[73] J Y Tai, On f–localization functors and connectivity, from: "Stable and unstable homotopy" (editors W G Dwyer, S Halperin, R Kane, S O Kochman, M E Mahowald, P S Selick), Fields Inst. Commun. 19, Amer. Math. Soc. (1998) 285

[74] J Y Tai, A version of Zabrodsky’s lemma, Proc. Amer. Math. Soc. 126 (1998) 1573

[75] G W Whitehead, Elements of homotopy theory, 61, Springer (1978)

[76] G V Wilson, Additive groups of T–rings, Proc. Amer. Math. Soc. 99 (1987) 219