Lannes’ $T$ functor on summands of $H^ *(B(\textbf {Z}/p)^ s)$
HTML articles powered by AMS MathViewer
- by John C. Harris and R. James Shank PDF
- Trans. Amer. Math. Soc. 333 (1992), 579-606 Request permission
Abstract:
Let $H$ be the $\bmod \text {-}p$ cohomology of the classifying space $B({\mathbf {Z}}/p)$ thought of as an object in the category, $\mathcal {U}$, of unstable modules over the Steenrod algebra. Lannes constructed a functor $T:\mathcal {U} \to \mathcal {U}$ which is left adjoint to the functor $A \mapsto A \otimes H$. In this paper we evaluate $T$ on the indecomposable $\mathcal {U}$-summands of ${H^{ \otimes s}}$, the tensor product of $s$ copies of $H$. Our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring ${{\mathbf {F}}_p}[{{\mathbf {M}}_{s,}}_s({\mathbf {Z}}/p)]$. The main application is to determine the homotopy type of the space of maps from $B({\mathbf {Z}}/p)]$ to $X$ when $X$ is a wedge summand of the space $\Sigma (B{({\mathbf {Z}}/p)^s})$.References
- J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian $p$-groups, Topology 24 (1985), no. 4, 435–460. MR 816524, DOI 10.1016/0040-9383(85)90014-X
- A. K. Bousfield, On the $p$-adic completions of nonnilpotent spaces, Trans. Amer. Math. Soc. 331 (1992), no. 1, 335–359. MR 1062866, DOI 10.1090/S0002-9947-1992-1062866-4
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
- H. E. A. Campbell and P. S. Selick, Polynomial algebras over the Steenrod algebra, Comment. Math. Helv. 65 (1990), no. 2, 171–180. MR 1057238, DOI 10.1007/BF02566601 D. Carlisle, The modular representation theory of ${\text {GL}}(n,p)$, and applications to topology, Ph.D. thesis, Univ. of Manchester, 1985.
- David Carlisle and Nicholas J. Kuhn, Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras, J. Algebra 121 (1989), no. 2, 370–387. MR 992772, DOI 10.1016/0021-8693(89)90073-2 —, Smash products of summands of $B({\mathbf {Z}}/p)_ + ^n$, Algebraic Topology, Contemp. Math., vol. 96, Amer. Math. Soc., Providence, R.I., 1989, pp. 87-102.
- David P. Carlisle and Grant Walker, Poincaré series for the occurrence of certain modular representations of $\textrm {GL}(n,p)$ in the symmetric algebra, Proc. Roy. Soc. Edinburgh Sect. A 113 (1989), no. 1-2, 27–41. MR 1025452, DOI 10.1017/S0308210500023933
- Gunnar Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), no. 2, 189–224. MR 763905, DOI 10.2307/2006940
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- Fred Cohen, Splitting certain suspensions via self-maps, Illinois J. Math. 20 (1976), no. 2, 336–347. MR 405412
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013113
- D. J. Glover, A study of certain modular representations, J. Algebra 51 (1978), no. 2, 425–475. MR 476841, DOI 10.1016/0021-8693(78)90116-3
- John C. Harris, On certain stable wedge summands of $B(\textbf {Z}/p)^n_+$, Canad. J. Math. 44 (1992), no. 1, 104–118. MR 1152669, DOI 10.4153/CJM-1992-006-8
- John C. Harris and Nicholas J. Kuhn, Stable decompositions of classifying spaces of finite abelian $p$-groups, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 3, 427–449. MR 932667, DOI 10.1017/S0305004100065038
- Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- Nicholas J. Kuhn, The Morava $K$-theories of some classifying spaces, Trans. Amer. Math. Soc. 304 (1987), no. 1, 193–205. MR 906812, DOI 10.1090/S0002-9947-1987-0906812-8
- J. Lannes, Sur la cohomologie modulo $p$ des $p$-groupes abéliens élémentaires, Homotopy theory (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 97–116 (French). MR 932261
- Jean Lannes and Lionel Schwartz, Sur la structure des $A$-modules instables injectifs, Topology 28 (1989), no. 2, 153–169 (French). MR 1003580, DOI 10.1016/0040-9383(89)90018-9
- Jean Lannes and Saïd Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Z. 194 (1987), no. 1, 25–59 (French). MR 871217, DOI 10.1007/BF01168004
- Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716, DOI 10.2307/2007071
- Stephen A. Mitchell and Stewart B. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), no. 3, 285–298. MR 710102, DOI 10.1016/0040-9383(83)90014-9 R. J. Shank, Polynomial algebras over the Steenrod algebra, summands of ${H^{\ast } }(B{({\mathbf {Z}}/2)^s})$ and Lannes’ division functors, Ph.D. thesis, Univ. of Toronto, 1989. —, Symmetric algebras over the Steenrod algebra and Lannes’ $T$ functor, Preprint, 1989.
- R. M. W. Wood, Splitting $\Sigma (\textbf {C}\textrm {P}^\infty \times \cdots \times \textbf {C}\textrm {P}^\infty )$ and the action of Steenrod squares $\textrm {Sq}^i$ on the polynomial ring $F_2[x_1,\cdots ,x_n]$, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 237–255. MR 928837, DOI 10.1007/BFb0083014
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 333 (1992), 579-606
- MSC: Primary 55S10
- DOI: https://doi.org/10.1090/S0002-9947-1992-1118825-6
- MathSciNet review: 1118825