Differentiable approximations to homotopy resolutions and framed cobordism
type: Item
instance of: scholarly article
title: Differentiable approximations to homotopy resolutions and framed cobordism
zbMATH Open document ID: 0584.55013
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: 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 cont…
Mathematics Subject Classification ID: 55Q40
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55S45
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 55T15
zbMATH DE Number: 3935973
zbMATH Keywords: differentiable manifolds
zbMATH Keywords: Lie groups
zbMATH Keywords: homotopy groups of spheres
zbMATH Keywords: homotopy resolutions
zbMATH Keywords: Composition methods
zbMATH Keywords: Adams spectral sequence
zbMATH Keywords: framed cobordism
zbMATH Keywords: Postnikov towers
zbMATH Keywords: filtration
zbMATH Keywords: k-invariant
author: Donald W. Kahn
MaRDI profile type: Publication
full work available at URL: pjm.1984.113.373
OpenAlex ID: W1976751479
DOI: 10.2140/PJM.1984.113.373
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description: scientific article; zbMATH DE number 3935973