Properties of Anick's spaces

Author:
Stephen D. Theriault

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1009-1037

MSC (2000):
Primary 55P45; Secondary 55Q15

Published electronically:
August 8, 2000

MathSciNet review:
1709780

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Abstract: We prove three useful properties of Anick's space . First, at odd primes a map from into a homotopy commutative, homotopy associative -space can be extended to a unique -map from into . Second, at primes larger than , is itself homotopy commutative and homotopy associative. And third, the first two properties combine to show that the order of the identity map on is .

**[A]**David Anick,*Differential algebras in topology*, Research Notes in Mathematics, vol. 3, A K Peters, Ltd., Wellesley, MA, 1993. MR**1213682****[AG]**David Anick and Brayton Gray,*Small 𝐻 spaces related to Moore spaces*, Topology**34**(1995), no. 4, 859–881. MR**1362790**, 10.1016/0040-9383(95)00001-1**[CMN1]**F. R. Cohen, J. C. Moore, and J. A. Neisendorfer,*Torsion in homotopy groups*, Ann. of Math. (2)**109**(1979), no. 1, 121–168. MR**519355**, 10.2307/1971269**[CMN2]**F. R. Cohen, J. C. Moore, and J. A. Neisendorfer,*The double suspension and exponents of the homotopy groups of spheres*, Ann. of Math. (2)**110**(1979), no. 3, 549–565. MR**554384**, 10.2307/1971238**[CMN3]**F. R. Cohen, J. C. Moore, and J. A. Neisendorfer,*Exponents in homotopy theory*, Algebraic topology and algebraic 𝐾-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 3–34. MR**921471****[Ga]**Tudor Ganea,*Cogroups and suspensions*, Invent. Math.**9**(1969/1970), 185–197. MR**0267582****[Gr1]**Brayton Gray,*Homotopy commutativity and the 𝐸𝐻𝑃 sequence*, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 181–188. MR**1022680**, 10.1090/conm/096/1022680**[Gr2]**B. Gray, EHP spectra and periodicity. I: Geometric constructions,*Trans. Amer. Math. Soc.***340**No. 2 (1993), 595-616. MR**94c:55035****[J]**I. M. James,*Reduced product spaces*, Ann. of Math. (2)**62**(1955), 170–197. MR**0073181****[L]**Arunas Liulevicius,*The factorization of cyclic reduced powers by secondary cohomology operations*, Mem. Amer. Math. Soc. No.**42**(1962), 112. MR**0182001****[N1]**Joseph Neisendorfer,*Primary homotopy theory*, Mem. Amer. Math. Soc.**25**(1980), no. 232, iv+67. MR**567801**, 10.1090/memo/0232**[N2]**Joseph Neisendorfer,*Properties of certain 𝐻-spaces*, Quart. J. Math. Oxford Ser. (2)**34**(1983), no. 134, 201–209. MR**698206**, 10.1093/qmath/34.2.201**[N3]**J.A. Neisendorfer, James-Hopf invariants, Anick's spaces, and the double loops on odd primary Moore spaces, preprint.**[N4]**J.A. Neisendorfer, Product decompositions of the double loops on odd primary Moore spaces,*Topology***38**(1999), 1293-1311. CMP**99:12****[Se]**Paul Selick,*Odd primary torsion in 𝜋_{𝑘}(𝑆³)*, Topology**17**(1978), no. 4, 407–412. MR**516219**, 10.1016/0040-9383(78)90007-1**[St]**James Stasheff,*𝐻-spaces from a homotopy point of view*, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR**0270372****[Th]**S.D. Theriault, A reconstruction of Anick's fibration, to appear in*Topology*.**[To1]**Hirosi Toda,*𝑝-primary components of homotopy groups. II. Mod 𝑝 Hopf invariant*, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math.**31**(1958), 143–160. MR**0105683****[To2]**Hirosi Toda,*Composition methods in homotopy groups of spheres*, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR**0143217****[W]**George W. Whitehead,*Elements of homotopy theory*, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR**516508**

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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 22903

Email:
st7b@virginia.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-00-02623-4

Keywords:
$H$-spaces,
universal Whitehead product,
exponent

Received by editor(s):
December 4, 1998

Published electronically:
August 8, 2000

Additional Notes:
The author was supported in part by an NSERC Postdoctoral Fellowship.

Article copyright:
© Copyright 2000
American Mathematical Society