Single loop space decompositions

Anick, David J.

doi:10.1090/S0002-9947-1992-1145728-3

Single loop space decompositions
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Trans. Amer. Math. Soc. 334 (1992), 929-940 Request permission

Abstract:

The method of single loop space decompositions, in which $\Omega X$ is factored into minimal factors, is an important one for understanding the unstable homotopy of many simply-connected spaces $X$. This paper begins with a survey of the major known theorems along these lines. We then give a necessary and sufficient condition for $\Omega X$ to be decomposable as a product of spaces belonging to a certain list. We conclude with a nontrivial instance of an application of this condition.

References

J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305–330. MR 77929, DOI 10.1007/BF02564350
David J. Anick, A loop space whose homology has torsion of all orders, Pacific J. Math. 123 (1986), no. 2, 257–262. MR 840843
David J. Anick, Homotopy exponents for spaces of category two, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 24–52. MR 1000365, DOI 10.1007/BFb0085216
David J. Anick, Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989), no. 3, 417–453. MR 991015, DOI 10.1090/S0894-0347-1989-0991015-7
Hans Joachim Baues, Commutator calculus and groups of homotopy classes, London Mathematical Society Lecture Note Series, vol. 50, Cambridge University Press, Cambridge-New York, 1981. MR 634675
Jörgen Backelin and Ralf Fröberg, Koszul algebras, Veronese subrings and rings with linear resolutions, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 2, 85–97. MR 789425
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, DOI 10.2307/1971269
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
P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172. MR 68218, DOI 10.1112/jlms/s1-30.2.154
I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, DOI 10.2307/2007107

On the subalgebra generated by the one-dimensional elements in the Yoneda

algebra

C. A. McGibbon and C. W. Wilkerson, Loop spaces of finite complexes at large primes, Proc. Amer. Math. Soc. 96 (1986), no. 4, 698–702. MR 826505, DOI 10.1090/S0002-9939-1986-0826505-X
Joseph Neisendorfer, Primary homotopy theory, Mem. Amer. Math. Soc. 25 (1980), no. 232, iv+67. MR 567801, DOI 10.1090/memo/0232
Joseph A. Neisendorfer, The exponent of a Moore space, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 35–71. MR 921472
J. A. Neisendorfer and P. S. Selick, Some examples of spaces with or without exponents, Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 343–357. MR 686124
Jean-Pierre Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425–505 (French). MR 45386, DOI 10.2307/1969485
Larry Smith, Split extensions of Hopf algebras and semi-tensor products, Math. Scand. 26 (1970), 17–41. MR 260848, DOI 10.7146/math.scand.a-10963
Clarence Wilkerson, Genus and cancellation, Topology 14 (1975), 29–36. MR 367995, DOI 10.1016/0040-9383(75)90032-4