Pontryagin multiplication and a relation with the Whitehead product.

Moore, Eric J. (1971) Pontryagin multiplication and a relation with the Whitehead product. Masters thesis, Memorial University of Newfoundland.

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Abstract

The space Ω of loops over a 1-connected space X possesses a natural multiplication, namely the composition of loops. This multiplication induces a Pontryagin product in homology, while the properties of the multiplication induce, in homology, the structure of an associative algebra with identity. -- The first three chapters of this thesis are introductory: The first describes the concept of cubic homology, the second, spectral sequences, and the third, the spectral sequence of a fiber space. -- In Chapter 4, the Pontryagin product is defined for P, a path space over a 1-connected space X. Since P is well known to be a fiber space over X with fiber Ω, the results of the previous chapters may be used to determine certain properties of the Pontryagin product in P these results are summarized by Theorem (4.45). -- Finally in Chapters 5 and 6, the two main theorems are presented and proven in complete detail. Theorem A, due to Bott and Samelson, determines the Pontryagin algebra of the loop space Ω, where the elements of the homology groups of X are transgressive (in the spectral sequence of P), and its corollaries determine the Pontryagin algebra of the loop space over a sphere, the loop space over the one point union of spheres, and the loop space over the suspension of a 0-connected space. Theorem B, due to Samelson, gives a relationship between the Whitehead and the Pontryagin products. Two proofs are given for Theorem B : the first establishes the relationship up to a factor of ±1, the second determines the sign. -- Both Theorem B and the second corollary of Theorem A are valuable in the determination of the homotopy groups of the one point union of spheres.

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