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Lectures on groups of homotopy spheres. (English) Zbl 0576.57028
Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lect. Notes Math. 1126, 62-95 (1985).
[For the entire collection see Zbl 0553.00007.]
Kervaire and Milnor’s germinal paper, in which they used the newly discovered techniques of surgery to begin the classification of smooth closed manifolds homotopy equivalent to a sphere (homotopy-\(spheres)\), was intended to be the first of two papers in which this classification would be essentially completed (in dimensions \(\geq 5)\). Unfortunately, the second part never appeared. As a result, in order to extract this classification from the published literature it is necessary to submerge oneself in more far-ranging and complicated works, which cannot help but obscure the beautiful ideas contained in the more direct earlier work of Kervaire and Milnor. This is especially true for the student who is encountering the subject for the first time.
These notes cover the material which I believe would have appeared in ”Groups of homotopy spheres, II. The reader is assumed to be familiar with the papers of Kervaire-Milnor where they define the group \(\theta^ n\) of h-cobordism classes of homotopy n-spheres and the subgroup \(bP^{n+1}\) defined by homotopy spheres which bound parallelizable manifolds. The goal is to compute \(bP^{n+1}\) and \(\theta^ n/bP^{n+1}\) (adapted from the introduction).
Reviewer: Th.Bröcker

57R60 Homotopy spheres, Poincaré conjecture
57R65 Surgery and handlebodies