C∗-algebras, positive scalar curvature, and the novikov conjecture—III
C∗-algebras, positive scalar curvature, and the novikov conjecture—III
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C*-代数、正标量曲率和诺维科夫猜想—III
DOI：
10.1016/0040-9383(86)90047-9
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发表时间：
1986
期刊：
影响因子：
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通讯作者：
Jonathan Rosenberg
中科院分区：
文献类型：
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作者：
Jonathan Rosenberg;Jonathan Rosenberg
正如标题所示，这篇论文是我早期论文[35]和[36]的延续。和以前一样，我们研究微分几何中的以下问题：给定一个闭合的，连通的，光滑流形M ' '， M何时允许一个正标量曲率的黎曼度规？[23],[lS],[17],[17],[37],[30],[31]，[35]和[36]的证据表明，这个问题既有趣又深刻，并且证明了M的微分拓扑和黎曼几何之间的显著相互作用。从刚才引用的参考文献中可以清楚地看出，我们问题的答案在很大程度上取决于基本群IL= rri (M)和最低的Stiefel-Whitney类wi (M)和w*(M)。这里我们将重点讨论wr和w2消失的情况，因此M允许自旋结构s（[26],[27]）。这种结构不一定是唯一的，但不同的选择只不同于H'（M， E,）的一个元素。设f: M-+ BIL为M的泛盖fi-+ M的分类图，则mikhail Gromov和Blaine Lawson（在[151，[16]和[17]）提出了如下猜想：
As the title indicates, this paper is a continuation of my earlier papers [35] and [36]. As before, we study the following question in differential geometry: given a closed, connected, smooth manifold M", when does M admit a Riemannian metric of positive scalar curvature? Evidence from [23],[lS],[16],[17],[37],[30],[31],[35] and [36] shows that thisquestion is both interesting and deep, and demonstrates a remarkable interplay between the differential topology and Riemannian geometry of M. As is clear from the references just cited, the answer to our question depends in a vital way on the fundamental group IL= rri (M) and on the lowest Stiefel-Whitney classes wi (M) and w*(M). We shall focus here on the case when wr and w2 vanish, so that M admits a spin structure s ([26],[27]). This structure is not necessarily unique, but the various choices differ only by an element of H'(M, E,). Let f: M-+ BIL be the classitying map for the universal cover fi-+ M of M. Then Mikhael Gromov and Blaine Lawson (in [151,[16] and [17]) have made the following conjecture: