Realizing a complex of unstable modules. (English) Zbl 1234.55012

Ho, Hai Nguyen Dang; Schwartz, Lionel

Proc. Japan Acad., Ser. A 87, No. 5, 83-87 (2011).

In an earlier paper [Adv. Math. 225, No. 3, 1431–1460 (2010; Zbl 1200.55026)] the authors, along with Tran Ngoc Nam, constructed a certain minimal injective resolution of the mod 2 cohomology of a Thom spectrum. In this paper a topological realization of this resolution is constructed, in the form of a family of cofibrations which, after applying mod 2 cohomology, give a family of short exact sequences that can be spliced together to give the desired resolution. This suggests that certain dual Brown-Gitler spectra should also be realizable geometrically (not just homotopically).

Reviewer: Martin D. Crossley (Swansea)

MSC:

55S10	Steenrod algebra
55T15	Adams spectral sequences
55P42	Stable homotopy theory, spectra

Keywords:

unstable module; Brown-Gitler spectrum; Adams spectral sequence

Citations:

Zbl 1200.55026

References:

[1]	P. Goerss, J. Lannes and F. Morel, Vecteurs de Witt non commutatifs et représentabilité de l’homologie modulo \(p\), Invent. Math. 108 (1992), no. 1, 163-227. · Zbl 0769.55011 · doi:10.1007/BF02100603
[2]	D. J. Hunter and N. J. Kuhn, Characterizations of spectra with \(\mathcal{U}\)-injective cohomology which satisfy the Brown-Gitler property, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1171-1190. · Zbl 0929.55011 · doi:10.1090/S0002-9947-99-02375-2
[3]	J. Lannes and L. Schwartz, Sur la structure des \(A\)-modules instables injectifs, Topology 28 (1989), no. 2, 153-169. · Zbl 0683.55016 · doi:10.1016/0040-9383(89)90018-9
[4]	J. Lannes and S. Zarati, Sur les \(\mathcal{U}\)-injectifs, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 303-333. · Zbl 0608.18006
[5]	S. A. Mitchell and S. B. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), no. 3, 285-298. · Zbl 0526.55010 · doi:10.1016/0040-9383(83)90014-9
[6]	N. D. H. Hai, L. Schwartz and T. N. Nam, Résolution de certains modules instables et fonction de partition de Minc, C. R. Math. Acad. Sci. Paris 347 (2009), no. 11-12, 599-602. · Zbl 1170.18008 · doi:10.1016/j.crma.2009.04.009
[7]	N. D. H. Hai, L. Schwartz and T. N. Nam, La fonction génératrice de Minc et une “conjecture de Segal” pour certains spectres de Thom, Adv. Math. 225 (2010), no. 3, 1431-1460. · Zbl 1200.55026 · doi:10.1016/j.aim.2010.03.029
[8]	L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture , Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1994. · Zbl 0871.55001
[9]	R. Steinberg, Prime power representations of finite linear groups, Canad. J. Math. 8 (1956), 580-591. · Zbl 0073.01502 · doi:10.4153/CJM-1956-063-3
[10]	S. Takayasu, On stable summands of Thom spectra of \(B(\mathbf{Z}/2)^{n}\) associated to Steinberg modules, J. Math. Kyoto Univ. 39 (1999), no. 2, 377-398. · Zbl 1002.55006