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Anick's spaces and the double loops of odd primary Moore spaces

Author(s): Stephen D. Theriault
Journal: Trans. Amer. Math. Soc. 353 (2001), 1551-1566.
MSC (2000): Primary 55P35; Secondary 55Q25
Posted: October 11, 2000
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Abstract:

Several properties of Anick's spaces are established which give a retraction of Anick's $\Omega T_\infty$ off $\Omega^2P^{2np+1}(p^r)$ if $r\ge2$ and $p\ge5$. The proof is alternate to and more immediate than the two proofs of Neisendorfer's.

References:

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D. Anick, Differential Algebras in Topology, AK Peters, (1993). MR 94h:55020

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D. Anick and B. Gray, Small $H$-spaces related to Moore spaces, Topology 34 (1995), 859-881. MR 97a:55011

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F. R. Cohen, and M. E. Mahowald, A remark on the self-maps of $\Omega^2S^{2n+1}$, Indiana Univ. Math. J. 30 (1981), 583-588. MR 82i:55013

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F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in Homotopy Groups, Annals of Math. 109 (1979), 121-168. MR 80e:55024

[CMN2]
F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Annals of Math. 110 (1979), 549-565. MR 81c:55021

[CMN3]
F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Exponents in homotopy theory, Algebraic Topology and Algebraic $K$-theory, W. Browder, ed., Annals of Math. Study 113, Princeton University Press (1987), 3-34. MR 89d:55035

[G1]
B. Gray, Homotopy commutativity and the EHP sequence, Proc. Internat. Conf., 1988, Contemp.Math., Vol. 1370, Amer. Math. Soc., Providence, RI (1989), 181-188. MR 90i:55025

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[J]
I. M. James, Reduced product spaces, Annals of Math. 62 (1955), 170-197. MR 17:396b

[N1]
J. A. Neisendorfer, Primary Homotopy Theory, Memoirs of the Amer. Math. Soc. No. 232 (1980). MR 81b:55035

[N2]
J. A. Neisendorfer, Product decompositions of the double loops on odd primary Moore spaces, Topology 38 (1999), 1293-1311. MR 2000d:55024

[N3]
J. A. Neisendorfer, James-Hopf invariants, Anick's spaces, and the double loops on odd primary Moore spaces, Canad. Math. Bull. 43 (2000), 226-235. CMP 2000:11

[S1]
P. Selick, Odd primary torsion in $\pi_k(S^3)$, Topology 17 (1978), 407-412. MR 80c:55010

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P. Selick, A decomposition of $\pi_*(S^{2p+1};\mathbf{Z}/p\mathbf{Z})$, Topology 20 (1981), 175-177. MR 82c:55017

[S3]
P. Selick, A reformulation of the Arf invariant one $\operatorname{mod} p$ problem and applications to atomic spaces, Pacific J. Math. 108 (1983), 431-450. MR 85d:55017

[T1]
S. D. Theriault, A reconstruction of Anick's fibration, to appear in Topology.

[T2]
S. D. Theriault, Properties of Anick's spaces, to appear in Trans. Amer. Math. Soc. CMP 2000:01

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Additional Information:

Stephen D. Theriault
Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
Address at time of publication: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: st7b@virginia.edu

DOI: 10.1090/S0002-9947-00-02622-2
PII: S 0002-9947(00)02622-2
Keywords: Loop space decomposition, Hopf invariant
Received by editor(s): December 1, 1998
Posted: October 11, 2000
Additional Notes: The author was supported in part by an NSERC Postdoctoral Fellowship
Copyright of article: Copyright 2000, American Mathematical Society

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