Abstract
For a given odd prime number p, in this paper we construct a minimal generating set for the mod-p cohomology of the Steinberg summand of a family of Thom spectra over the classifying space of an elementary p-abelian group. This resolves the remaining cases for odd prime numbers of a problem studied previously by M. Inoue (Contemporary Mathematics, vol. 293, pp. 125–139, 2002) and (J. Lond. Math. Soc. 75: 317–329, 2007) and by the author (J. Algebra 381: 164–175, 2013).
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Arone, G., Mahowald, M.: The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres. Invent. Math. 135, 743–788 (1999)
Arone, G.Z., Dwyer, W.G.: Partition complexes, Tits buildings and symmetric products. Proc. Lond. Math. Soc. 82, 229–256 (2001)
Chon, P.H., Ha, L.M.: On the action of the Iwahori–Hecke algebra on modular invariants. Commun. Algebra 47, 735–748 (2019)
Dickson, L.E.: A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. Amer. Math. Soc. 12, 75–98 (1911)
Hai, N.D.H.: Un complexe de Koszul de modules instables et cohomotopie d’un spectre de Thom. Bull. Soc. Math. Fr. 140, 257–308 (2012)
Hai, N.D.H.: Generators for the mod 2 cohomology of the Steinberg summand of Thom spectra over \(b(\mathbb {Z}/2)^{n}\). J. Algebra 381, 164–175 (2013)
Hai, N.D.H., Schwartz, L., Nam, T.N.: La fonction génératrice de Minc et une “conjecture de Segal” pour certains spectres de Thom. Adv. Math. 225, 1431–1460 (2010)
Hung, N.H.V., Sum, N.: On Singer’s invariant-theoretic description of the lambda algebra: a mod p analogue. J. Pure Appl. Algebra 99, 297–329 (1995)
Inoue, M.: \(\mathcal {A}\)-generators of the cohomology of the steinberg summand M(n). In: Davis, D.M., Morava, J., Nishida, G., Wilson, W. S., Yagita, N (eds.) Recent Progress in Homotopy Theory (Baltimore, MD, 2000). Contemporary Mathematics, vol. 293, pp 125–139. American Mathematical Society, Providence (2002)
Inoue, M.: Generators of the cohomology of M(n) as a module over the odd primary Steenrod algebra. J. Lond. Math. Soc. (2) 75, 317–329 (2007)
Kameko, M.: Products of Projective Spaces as Steenrod Modules. Ph.D. thesis, Johns Hopkins University (1990)
Kuhn, N.J.: The modular Hecke algebra and Steinberg representation of finite Chevalley groups. J. Algebra 91, 125–141 (1984). With an appendix by Peter Landrock
Kuhn, N.J.: The rigidity of L(n). In: Miller, H. R., Ravenel, D. C. (eds.) Algebraic Topology. Lecture Notes in Mathematics, vol. 1286, pp 286–292. Springer, Berlin (1987)
Kuhn, N.J.: The whitehead conjecture, the tower of S1, conjecture, and Hecke algebras of type A. arXiv:1308.4441 (2013)
Kuhn, N.J., Priddy, S.B.: The transfer and Whitehead’s conjecture. Math. Proc. Camb. Philos. Soc. 98, 459–480 (1985)
Milnor, J.: The Steenrod algebra and its dual. Ann. Math. (2) 67, 150–171 (1958)
Mitchell, S.A., Priddy, S.B.: Stable splittings derived from the Steinberg module. Topology 22, 285–298 (1983)
Mui, H.: Modular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22, 319–369 (1975)
Schwartz, L.: Unstable Modules over the Steenrod Algebra and Sullivan’S Fixed Point Set Conjecture. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1994)
Steinberg, R.: Prime power representations of finite linear groups. Canad. J. Math. 8, 580–591 (1956)
Sum, N.: The action of the primitive Steenrod–Milnor operations on the modular invariants. In: Proceedings of the School and Conference in Algebraic Topology. Geometry & Topology Monographs, vol. 11, pp 349–367. Geom. Topol. Publ., Coventry (2007)
Takayasu, S.I.: On stable summands of Thom spectra of B(Z/2)n associated to Steinberg modules. J. Math. Kyoto Univ. 39, 377–398 (1999)
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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.10. The author is very grateful to the referees for their valuable comments and suggestions which helped to improve the quality of the manuscript.
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Hai, N.D.H. Generators for the Mod-p Cohomology of the Steinberg Summand of Thom Spectra Over \(\mathrm {B}(\mathbb {Z}/p)^{n}\)-Odd Primary Cases. Vietnam J. Math. 50, 229–248 (2022). https://doi.org/10.1007/s10013-021-00501-y
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DOI: https://doi.org/10.1007/s10013-021-00501-y