Takayasu cofibrations revisited. (English) Zbl 1347.55015

Ho, Hai Nguyen Dang; Schwartz, Lionel

Proc. Japan Acad., Ser. A 91, No. 8, 123-127 (2015).

Summary: This short note gives a new proof for the existence of the cofibrations constructed by S. Takayasu [J. Math. Kyoto Univ. 39, No.2, 377–398 (1999; Zbl 1002.55006)], using techniques in the category of unstable modules over the mod two Steenrod algebra.

MSC:

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

Keywords:

Steinberg modules; Takayasu cofibrations; unstable modules

Citations:

Zbl 1002.55006

References:

[1]	G. Z. Arone and W. G. Dwyer, Partition complexes, Tits buildings and symmetric products, Proc. London Math. Soc. (3) 82 (2001), no. 1, 229-256. · Zbl 1028.55008
[2]	G. Arone and M. Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), no. 3, 743-788. · Zbl 0997.55016
[3]	G. Arone, A note on the homology of \(\Sigma_{n}\), the Schwartz genus, and solving polynomial equations, in An alpine anthology of homotopy theory , Contemp. Math., 399, Amer. Math. Soc., Providence, RI, 2006, pp. 1-10. · Zbl 1098.55013
[4]	M. Behrens, The Goodwillie tower and the EHP sequence, Mem. Amer. Math. Soc. 218 (2012), no. 1026, xii+90 pp. · Zbl 1330.55012
[5]	D. P. Carlisle and N. J. Kuhn, Smash products of summands of \(B(\mathbf{Z}/p)^{n}_{+}\), in Algebraic topology (Evanston, IL, 1988) , 87-102, Contemp. Math., 96, Amer. Math. Soc., Providence, RI, 1989. · Zbl 0686.55009
[6]	N. D. H. Hai, Generators for the mod 2 cohomology of the Steinberg summand of Thom spectra over \(B(\mathbf{Z}/2)^{n}\), J. Algebra 381 (2013), 164-175. · Zbl 1279.55010
[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
[8]	J. C. Harris and R. J. Shank, Lannes’ \(T\) functor on summands of \(H^{*}(B(\mathbf{Z}/p)^{s})\), Trans. Amer. Math. Soc. 333 (1992), no. 2, 579-606.
[9]	M. Inoue, \(\mathcal{A}\)-generators of the cohomology of the Steinberg summand \(M(n)\), in Recent progress in homotopy theory (Baltimore, MD, 2000) , 125-139, Contemp. Math., 293, Amer. Math. Soc., Providence, RI, 2002. · Zbl 1022.55007
[10]	I. M. James, The topology of Stiefel manifolds , Cambridge Univ. Press, Cambridge, 1976. · Zbl 0337.55017
[11]	J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un \(p\)-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135-244. · Zbl 0857.55011
[12]	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
[13]	J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Z. 194 (1987), no. 1, 25-59. · Zbl 0627.55014
[14]	S. A. Mitchell and S. B. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), no. 3, 285-298. · Zbl 0526.55010
[15]	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
[16]	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