Differentiable approximations to homotopy resolutions and framed cobordism
From the author's summary and introduction: ''The determination of homotopy groups of spheres remains a central problem in algebraic topology. The various methods for addressing this problem cover some considerable ground, from intricate algebra to the structure of manifolds. Our general purpose here is to show that the gap between some of these methods can in fact be closed, and that one may find geometric structures (manifolds) which reflect the filtrations arising from algebraic methods. ''To be specific, the different methods for studying homotopy groups of spheres include the construction of homotopy resolutions [\textit{H. Toda}, Composition methods in homotopy groups of spheres (1962; Zbl 0101.407)], the Adams spectral sequence [\textit{J. F. Adams}, Comment. Math. Helv. 32, 180-214 (1958; Zbl 0083.178)], and its variants, and the theory of framed cobordism [\textit{L. Pontryagin}, Tr. Mat. Inst. Steklova 45 (1955; Zbl 0064.174)], which is historically the first general method. Naturally, there are now many extensions of these methods, and a great many results have been obtained. But even in the area where the groups are now well- known, the relationship between the different methods is often not clear... An early attempt to analyze the relationship between the methods of homotopy resolutions and the Adams spectral sequence is the paper of \textit{H. Gershenson} [Math. Z. 81, 223-259 (1963; Zbl 0118.184)]. Since that time, there has been some work on analyzing what sort of manifolds, for example Lie groups, can carry framings which represent certain classes in homotopy groups of spheres... ''The present paper originates in my idea that it should be possible, at least in theory, to bridge the gap between framed cobordism and the theory of homotopy resolutions (Postnikov towers). Specifically, one should be able to build a ''filtration'' in framed cobordism, which is some sort of reflection of the basic homotopy resolution of the space. This is roughly the content of Theorem 2... One looks for manifolds and maps, which approximate in the sense of k-type, a given, finite piece of a homotopy resolution. That this can be done in all reasonable cases is our Theorem 1. We also indicate, in passing, how one might interpret the k- invariant in this differentiable setting. Some examples and applications follow Theorem 2.'' This programmatic paper closes with some interesting relevant remarks and problems.
scientific article; zbMATH DE number 3935973