Lannes, Jean [Zisman, Michel] On function spaces whose source is the classifying space of an elementary abelian \(p\)-group. (Sur les espaces fonctionnels dont la source est le classifiant d’un \(p\)-group abélien élémentaire. (Appendice par Michel Zisman).) (French) Zbl 0857.55011 Publ. Math., Inst. Hautes Étud. Sci. 75, 135-244 (1992). In this long paper, the author studies function spaces \(\text{hom} (BV,Y)\), consisting of all continuous mappings from \(BV\) to a topological space \(Y\), where \(BV\) denotes the classifying space of an elementary abelian \(p\)-group \(V\) for a fixed prime \(p\). His success in this project depends on several remarkable properties of \(H^* (BV)\), the \(\text{mod }p\) cohomology of \(BV\), as both unstable module and unstable algebra over the \(\text{mod }p\) Steenrod algebra \({\mathcal A}\). To state these properties, let \({\mathcal U}\) denote the category of unstable \({\mathcal A}\)-modules, and let \({\mathcal K}\) denote the category of unstable \({\mathcal A}\)-algebras. If \({\mathcal T}\) denotes either of these categories, and \(K\) is an object of \({\mathcal T}\) having finite dimension in each degree, then the functor \({\mathcal T}\to {\mathcal T}\), \(N\mapsto K\otimes N\), admits a left adjoint denoted by \(M\mapsto(M:K)_{\mathcal T}\). (One can view such functors as analogous to functors \(Y\mapsto\text{hom}(X,Y)\) in the category of spaces.) For example, if \(H^*X\) is finite dimensional in each degree, then there is a natural map \((H^*Y:H^*X)_{\mathcal K}\to H^*\text{hom}(X,Y)\); a main result of this paper asserts that this is often an isomorphism in the category \({\mathcal K}\) when \(X=BV\).For an elementary abelian \(p\)-group, \(T_V\) denotes the functor \({\mathcal U}\to {\mathcal U}\), \(M\mapsto(M:H^*(BV))_{\mathcal U}\). It is proved that (a) the functor \(T_V\) is exact; (b) the functor \(T_V\) commutes with tensor products; and (c) if \(M\) is an unstable \({\mathcal A}\)-algebra, then \(T_VM\) can be given the structure of an unstable \({\mathcal A}\)-algebra so that \(T_VM\) coincides with \((M:H^*(BV))_{\mathcal K}\) in the category \({\mathcal K}\). The property (a) generalizes the result of Gunnar Carlsson and Haynes Miller that \(H^*(B\mathbb{Z}/p)\) is injective in \({\mathcal U}\).The remaining task is to deduce properties of \(\text{hom} (BV,Y)\) and the set of homotopy classes \([BV,Y]\) from these algebraic properties of \(H^*(BV)\). For this, it is sometimes helpful to replace \(Y\) by its Bousfield-Kan \(p\)-completion \(\widehat Y\). One notable result asserts that the natural map \([BV,Y]\to\operatorname{Hom}_{\mathcal K} (H^*Y, H^*BV)\) is a bijection provided that \(Y\) is simply connected and \(H^*Y\) is finite dimensional in each degree. The principal result of the paper asserts that the natural map \(T_VH^*Y\to H^*\text{hom}(BV,\widehat Y)\) is an isomorphism of unstable \({\mathcal A}\)-algebras provided that \(H^*Y\) and \(T_VH^*Y\) are finite dimensional in each degree and \(T_VH^*Y\) is zero in degree 1.A final chapter concerns homotopy fixed point spaces for actions of elementary abelian \(p\)-groups. A number of the results of this paper were announced by the author in his paper [Homotopy theory, Proc. Symp. Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 117, 97-116 (1987; Zbl 0654.55013)]. An excellent account of the context in which the results of this paper play a major clarifying rôle is given by Lionel Schwartz in his book [Unstable modules over the Steenrod algebra and Sullivan’s fixed point conjecture, Chicago Lectures in Mathematics, Chicago, IL: Univ. of Chicago Press, (1994)]. Reviewer: P.Landweber (New Brunswick) Cited in 6 ReviewsCited in 75 Documents MSC: 55P99 Homotopy theory 55S10 Steenrod algebra Keywords:Lannes’ \(T\)-functor; elementary abelian \(p\)-group; Bousfield-Kan \(p\)-completion; function spaces; classifying space; unstable module; unstable algebra; Steenrod algebra Citations:Zbl 0654.55013 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] J. F. Adams,Two Theorems of J. Lannes, manuscrit, 1985. [2] D. W. Anderson, A generalisation of the Eilenberg-Moore spectral sequence,Bull. Amer. Math. Soc.,78 (1972), 784–786. · Zbl 0255.55012 · doi:10.1090/S0002-9904-1972-13034-9 [3] M. F. Atiyah etG. Segal,Equivariant cohomology and localization, lecture notes, Warwick, 1965. [4] A. K. Bousfield etD. M. Kan, Homotopy limits, Completions and Localizations,Springer L.N.M.,304, 1972. · Zbl 0259.55004 [5] A. K. Bousfield etD. M. Kan, The homotopy spectral sequence of a space with coefficients in a ring,Topology,11 (1972), 79–106. · doi:10.1016/0040-9383(72)90024-9 [6] A. Borel et al., Seminar on Transformation Groups,Ann. of Math. Studies,46, Princeton, 1960. · Zbl 0091.37202 [7] A. K. Bousfield, Nice homology coalgebras,Trans. Amer. Math. Soc.,148 (1970), 473–489. · Zbl 0199.33302 · doi:10.1090/S0002-9947-1970-0258919-6 [8] A. K. Bousfield, On the homology spectral sequence of a cosimplicial space,Amer. J. Math.,109 (1987), 361–394. · Zbl 0623.55009 · doi:10.2307/2374579 [9] A. K. Bousfield, Homotopy spectral sequences and obstructions,Israël J. Math.,66 (1989), 54–104. · Zbl 0677.55020 · doi:10.1007/BF02765886 [10] G. Carlsson, G. B. Segal’s Burnside ring conjecture for (Z/2) k ,Topology,22 (1983), 83–103. · Zbl 0504.55011 · doi:10.1016/0040-9383(83)90046-0 [11] G. Carlsson, Segal’s Burnside ring conjecture and the homotopy limit problem, Proc. Durham Symposium on Homotopy Theory, 1985, London,Math. Soc. L.N.S.,117, Camb. Univ. Press, 1987, 6–34. [12] G. Carlsson, Equivariant stable homotopy and Sullivan’s conjecture,Invent. math.,103 (1991), 497–527. · Zbl 0736.55008 · doi:10.1007/BF01239524 [13] H. Cartan etS. Eilenberg,Homological algebra, Princeton Univ. Press, 1956. [14] E. Dror, W. G. Dwyer etD. M. Kan, An arithmetic square for virtually nilpotent spaces,Ill. J. of Math.,21 (1977), 242–254. · Zbl 0355.55015 [15] M. Demazure, Lectures onp-Divisible Groups,Springer L.N.M.,302, 1972. · Zbl 0247.14010 [16] W. G. Dwyer, H. R. Miller etJ. A. Neisendorfer, Fibrewise completion and unstable Adams spectral sequence,Israël J. Math.,66 (1989), 160–178. · Zbl 0686.55014 · doi:10.1007/BF02765891 [17] W. G. Dwyer, H. R. Miller etC. W. Wilkerson, The Homotopic Uniqueness of BS3, Algebraic Topology, Barcelona, 1986 (proceedings),Springer L.N.M.,1298 (1987), 90–105. [18] W. G. Dwyer, H. R. Miller etC. W. Wilkerson, Homotopical Uniqueness of Classifysing Spaces,Topology,31 (1992), 29–45. · Zbl 0748.55005 · doi:10.1016/0040-9383(92)90062-M [19] E. Dror, Pro-nilpotent representation of homology types,Proc. Amer. Math. Soc.,38 (1973), 657–660. · Zbl 0275.55017 · doi:10.1090/S0002-9939-1973-0314041-X [20] E. Dror Farjoun etJ. Smith, A Geometric Interpretation of Lannes’ Functor T,Théorie de l’homotopie, Astérisque,191 (1990), 87–95. [21] W. G. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence,Topology,13 (1974), 255–265. · Zbl 0303.55012 · doi:10.1016/0040-9383(74)90018-4 [22] W. G. Dwyer etC. W. Wilkerson, Smith theory and the functor T,Comm. Math. Helv.,66 (1991), 1–17. · Zbl 0726.55011 · doi:10.1007/BF02566633 [23] W. G. Dwyer etA. Zabrodsky, Maps between classifying spaces, Algebraic Topology, Barcelona, 1986 (proceedings),Springer L.N.M.,1298 (1987), 106–119. · Zbl 0646.55007 [24] P. Freyd,Abelian Categories: An introduction to the theory of functors, New York, Harper & Row, 1964. · Zbl 0121.02103 [25] P. Gabriel etM. Zisman, Calculus of fractions and homotopy theory,Ergebnisse der Math.,35, Springer, 1967. · Zbl 0186.56802 [26] H.-W. Henn, J. Lannes etL. Schwartz, Analytic functors unstable algebras and cohomology of classifying spaces, Algebraic Topology, Northwestern University, 1988 (proceedings),Cont. Math.,96 (1989), 197–220. [27] H.-W. Henn, J. Lannes etL. Schwartz, The Categories of unstable modules and unstable algebras modulo nilpotent objects, à paraître dansAmerican Journal of Mathematics. · Zbl 0805.55011 [28] W. Y. Hsiang,Cohomology Theory of Topological Transformation Groups, New York, Springer Verlag, 1975. · Zbl 0429.57011 [29] S. Jackowski, J. E. McClure etR. Oliver, Self-maps of classifying spaces of compact simple Lie groups,Bull. Amer. Math. Soc.,22 (1990), 65–72. · Zbl 0692.55013 · doi:10.1090/S0273-0979-1990-15841-0 [30] J. Lannes, Sur la cohomologie modulop desp-groupes abéliens élémentaires, Proc. Durham Symposium on Homotopy Theory, 1985, London,Math. Soc. L.N.S.,117, Camb. Univ. Press, 1987, 97–116. [31] J. Lannes,Cohomology of groups and function spaces, preprint 1986. [32] J. Lannes etL. Schwartz, Sur la structure des A-modules instables injectifs,Topology,28 (1989), 153–169. · Zbl 0683.55016 · doi:10.1016/0040-9383(89)90018-9 [33] J. Lannes etL. Schwartz, A propos de conjectures de Serre et Sullivan,Invent. math.,83 (1986), 593–603. · Zbl 0563.55011 · doi:10.1007/BF01394425 [34] P. S. Landweber etR. E. Stong, The depth of rings of invariants over finite fields, Number Theory New York, 1984–1985,Springer L.N.M.,1240 (1987), 259–274. [35] J. Lannes etS. Zarati, Sur les U-injectifs,Ann. Scient. Ec. Norm. Sup.,19 (1986), 1–31. · Zbl 0608.18006 [36] J. Lannes etS. Zarati, Foncteurs dérivés de la déstabilisation,Math. Z.,194 (1987), 25–59. · Zbl 0627.55014 · doi:10.1007/BF01168004 [37] S. Mac Lane, Categories for the Working Mathematician,Graduate Texts in Math.,5, Springer, 1971. · Zbl 0232.18001 [38] J. P. May, Simplicial objects in algebraic topology,Van Nostrand Math. Studies,11, 1967. [39] H. R. Miller, The Sullivan conjecture on maps from classifying spaces,Annals of Math.,120 (1984), 39–87, et corrigendum,Annals of Math.,121 (1985), 605–609. · Zbl 0552.55014 · doi:10.2307/2007071 [40] H. R. Miller, The Sullivan Conjecture and Homotopical Representation Theory,Proc. Internat. Congr. Math. Berkeley, 1986, 580–589. [41] H. R. Miller, Massey-Peterson towers and maps from classifying spaces, Algebraic Topology, Aarhus, 1982 (proceedings),Springer L.N.M.,1051 (1984), 401–417. [42] J. W. Milnor, On axiomatic homology theory,Pac. J. Math.,12 (1962), 337–341. · Zbl 0114.39604 [43] F. Morel, Quelques remarques sur la cohomologie modulo-p continue des pro-p-espaces et les résultats de J. Lannes concernant les espaceshom(BV, X), preprint 1991, à paraître dans lesAnnales Scientifiques de l’Ecole Normale Supérieure. [44] D. G. Quillen, Homotopical Algebra,Springer L.N.M.,43, 1967. [45] D. G. Quillen, The spectrum of an equivariant cohomology rung, I, II,Ann. of Math.,94 (1971), 549–572 et 573–602. · Zbl 0247.57013 · doi:10.2307/1970770 [46] D. L. Rector, Steenrod operations in the Eilenberg-Moore spectral sequence,Comm. Math. Helv.,45 (1970), 540–552. · Zbl 0209.27501 · doi:10.1007/BF02567352 [47] J.-P. Serre, Sur la dimension cohomologique des groupes profinis,Topology,3 (1965), 413–420. · Zbl 0136.27402 · doi:10.1016/0040-9383(65)90006-6 [48] N. E. Steenrod etD. B. A. Epstein,Cohomology operations, Princeton Univ. Press, 1962. [49] D. Sullivan,Geometric topology, part I :Localization, periodicity and Galois symmetry, M.I.T. Press, 1970. [50] C. E. Watts, Intrinsic Characterization of Some Additive Functors,Proc. Amer. Math. Soc.,11 (1960), 5–8. · Zbl 0093.04101 · doi:10.1090/S0002-9939-1960-0118757-0 [51] S. Zarati, Quelques propriétés du foncteur HomUp(, H* V), Algebraic Topology Göttingen, 1984,Springer L.N.M.,1172 (1985), 204–209. [52] S. Zarati,Dérivés du foncteur de déstabilisation en caractéristique impaire et applications, thèse de doctorat d’Etat, Orsay, 1984. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.