An elementary construction of Anick’s fibration

Gray, Brayton¹ ; Theriault, Stephen²

¹ Department of Math, Stats and Comp Sci, University of Illinois at Chicago, 851 S Morgan Street, Chicago, IL, 60607-7045, USA
² Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

Geometry & topology, Tome 14 (2010) no. 1, pp. 243-275.

Résumé

Cohen, Moore, and Neisendorfer’s work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author’s work on the secondary suspension, predicted the existence of a $p$–local fibration $S^{2n-1}\to T_{2n-1}\to \Omega S^{2n+1}$ whose connecting map is degree $p^r$. In a long and complex monograph, Anick constructed such a fibration for $p\ge 5$ and $r\ge 1$. Using new methods we give a much more conceptual construction which is also valid for $p=3$ and $r\ge 1$. We go on to establish an $H$ space structure on $T_{2n-1}$ and use this to construct a secondary $EHP$ sequence for the Moore space spectrum.

Détail

DOI : 10.2140/gt.2010.14.243

Keywords: Anick's fibration, double suspension, EHP sequence, Moore space

Bibliographie

[1] D Anick, Differential algebras in topology, Research Notes in Math. 3, A K Peters Ltd. (1993)

[2] D Anick, B Gray, Small $H$ spaces related to Moore spaces, Topology 34 (1995) 859

[3] F R Cohen, The unstable decomposition of $\Omega ^{2}\Sigma ^{2}X$ and its applications, Math. Z. 182 (1983) 553

[4] F R Cohen, J C Moore, J A Neisendorfer, Decompositions of loop spaces and applications to exponents, from: "Algebraic topology, Aarhus 1978 (Proc. Sympos.)" (editors J L Dupont, I H Madsen), Lecture Notes in Math. 763, Springer (1979) 1

[5] F R Cohen, J C Moore, J A Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. $(2)$ 110 (1979) 549

[6] F R Cohen, J C Moore, J A Neisendorfer, Torsion in homotopy groups, Ann. of Math. $(2)$ 109 (1979) 121

[7] B Gray, A note on the Hilton–Milnor theorem, Topology 10 (1971) 199

[8] B Gray, On the iterated suspension, Topology 27 (1988) 301

[9] B Gray, $\mathit{EHP}$ spectra and periodicity. I. Geometric constructions, Trans. Amer. Math. Soc. 340 (1993) 595

[10] B Gray, $\mathit{EHP}$ spectra and periodicity. II. $\Lambda$–algebra models, Trans. Amer. Math. Soc. 340 (1993) 617

[11] B Gray, On decompositions in homotopy theory, Trans. Amer. Math. Soc. 358 (2006) 3305

[12] I M James, Reduced product spaces, Ann. of Math. $(2)$ 62 (1955) 170

[13] M Mahowald, On the double suspension homomorphism, Trans. Amer. Math. Soc. 214 (1975) 169

[14] M Mather, Pull-backs in homotopy theory, Canad. J. Math. 28 (1976) 225

[15] J A Neisendorfer, $3$–primary exponents, Math. Proc. Cambridge Philos. Soc. 90 (1981) 63

[16] J A Neisendorfer, The exponent of a Moore space, from: "Algebraic topology and algebraic $K$–theory (Princeton, N.J., 1983)" (editor W Browder), Ann. of Math. Stud. 113, Princeton Univ. Press (1987) 35

[17] J Neisendorfer, Algebraic methods in unstable homotopy theory, New Math. Monogr. 12, Cambridge Univ. Press (2009)

[18] S D Theriault, A reconstruction of Anick's fibration, PhD thesis, University of Toronto (1997)

[19] S D Theriault, Properties of Anick's spaces, Trans. Amer. Math. Soc. 353 (2001) 1009

[20] S D Theriault, The $3$–primary classifying space of the fiber of the double suspension, to appear in Proc. Amer. Math. Soc.