论文信息 - Single loop space decompositions

Single loop space decompositions

The method of single loop space decompositions, in which CIX 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 QX 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. 1. Background and summary of the major known decomposition theorems In this introduction we provide a summary of the major known theorems of a general nature regarding single loop space decompositions. The original loop space decomposition was discovered by Hilton (H) in 1955 by building upon work of George Whitehead; it was later generalized by Milnor. Theorem 1 (Hilton). Let W be a simply-connected finite type wedge of spheres, i.e., W = V/S1"'. ni>2, and «, —> oo if the index set is infinite. Then

D. Anick

[1] D. Anick. Hopf algebras up to homotopy , 1989 .

[2] J. Neisendorfer. II. The Exponent of a Moore Space , 1988 .

[3] D. Anick. A loop space whose homology has torsion of all orders. , 1986 .

[4] C. Wilkerson,et al. Loop spaces of finite complexes at large primes , 1986 .

[5] C. Wilkerson. Genus and cancellation , 1975 .

[6] Larry Smith. Split Extensions of Hopf Algebras and Semi-Tensor Products. , 1970 .

[7] J. Adams,et al. On the chain algebra of a loop space , 1956 .

[8] I. James. REDUCED PRODUCT SPACES , 1955 .

[9] Peter Hilton,et al. On the Homotopy Groups of the Union of Spheres , 1955 .

[10] Jean-Pierre Serre,et al. Homologie Singuliere Des Espaces Fibres , 1951 .

[11] D. Anick. Homotopy exponents for spaces of category two , 1989 .

[12] C. Löfwall. On the subalgebra generated by the one-dimensional elements in the Yoneda ext-algebra , 1986 .

[13] R. Fröberg,et al. Koszul algebras, Veronese subrings, and rings with linear resolutions , 1985 .

[14] Hans-Joachim Baues,et al. Commutator calculus and groups of homotopy classes , 1981 .

[15] J. Neisendorfer. Primary Homotopy Theory , 1980 .

[16] J. C. Moore,et al. Torsion in Homotopy Groups , 1979 .