Lectures on groups of homotopy spheres
Lectures on groups of homotopy spheres
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同伦球群讲座
作者：
J. Levine
在Kervaire和Milnor的萌芽论文[15]中，他们使用新发现的外科技术开始对等效于球的光滑闭流形（同伦-球）进行分类，这是两篇论文中第一个基本完成这种分类的论文（在维度>和5中）。不幸的是，第二部分没有出现。因此，为了从已发表的文献中提取出这种分类，必须将自己淹没在更广泛、更复杂的作品中（例如[7]、[16]、[30]），这不能不掩盖Kervaire和Milnor更直接的早期作品中所包含的美好思想。对于第一次接触这门学科的学生来说尤其如此。准备了这些讲座的油印笔记，以及一些额外的背景资料，这些资料可以从布兰代斯大学获得。这篇文章几乎与这些笔记一模一样。我希望它能填补文学教学上的空白。假设读者熟悉[15]，[20]。在这篇文章中，Kervaire-Milnor定义了同伦n球的h-协类群en和同伦n球所定义的包含可并行流形的子群bP n+l。目的是计算bP n+l和en/bpn+ i。第1节回顾了关于球上矢量束和经典群的同伦的一些著名结果，以及关于嵌入和浸入的惠特尼定理。由于同伦n球~n与nsn（具有标准微分结构的n球）是h-协同的，所以为了计算bP n+l，我们感兴趣的是寻找和实现将可并行流形转换为可压缩流形的“障碍”
Kervaire and Milnor's germinal paper [15], 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 > 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 (e.g. [7], [16], [30]), 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. prepared mimeographed notes from these lectures, with some extra background material, which have been available from Brandeis University. The present article is almost identical with these notes. I hope it will serve to fill a pedagogical gap in the literature. The reader is assumed to be familiar with [15], [20]. In these papers, Kervaire-Milnor define the group en of h-cobordism classes of homotopy n-spheres and the subgroup bP n+l defined by homotopy spheres which bound parallelizable manifolds. The goal is to compute bP n+l and en/bpn+ i Section 1 reviews some well known results on vector bundles over spheres and the homotopy of the classical groups, as well as some theorems of Whitney on embeddings and immersions. Since a homotopy n-sphere ~n is h-cobordant to S n (the n-sphere with its standard differential structure) iff ~n bounds a contractible manifold, in order to calculate bP n+l we are interested in finding and realizing "obstructions" to surgering parallelizable manifolds into contractible