可扩性与指数理论
Enlargeability and index theory
摘要：
设M是闭可扩自旋流形。本文证明了M的基本群的极大$C^*$ -代数的K-理论中泛指数阻塞的非平凡性。我们的证明与M的基本群的Baum-Connes集结映射的内射性无关，而依赖于从M上曲率趋于零的有限维向量丛序列中构造出某个无穷维平坦向量丛.除了众所周知的事实，M不携带一个度量与正的数量曲率，我们的结果意味着，分类映射$M \to B π_1(M)$发送的基本类M的一个非平凡的同调类在$H_n(B π_1(M) ; \Q)$。这回答了Burghelea（1983）的一个问题。
英文摘要：
Let M be a closed enlargeable spin manifold. We show non-triviality of the universal index obstruction in the K-theory of the maximal $C^*$-algebra of the fundamental group of M. Our proof is independent from the injectivity of the Baum-Connes assembly map for the fundamental group of M and relies on the construction of a certain infinite dimensional flat vector bundle out of a sequence of finite dimensional vector bundles on M whose curvatures tend to zero. Besides the well known fact that M does not carry a metric with positive scalar curvature, our results imply that the classifying map $M \to B π_1(M)$ sends the fundamental class of M to a nontrivial homology class in $H_n(B π_1(M) ; \Q)$. This answers a question of Burghelea (1983).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
