双曲流形上的奇异结构：通过射影空间的EO-理论
Exotic structures on hyperbolic manifolds via the EO-theory of Projective Spaces
- Indian Statistical Institute（印度统计研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文利用复射影空间的EO-理论，在多种维度构造了与给定双曲流形同伦等价但非微分同胚的负曲率流形，并证明了奇异球面上自由光滑S^1和S^3作用的存在性。
AI中文摘要：
Farrell和Jones证明了在维度$\geq 5$时，负曲率流形在拓扑上是刚性的，即同伦等价意味着同胚。在光滑范畴中，即使对于双曲流形，该结论也不成立，存在负曲率流形$N$与给定的双曲流形$M$同伦等价但不可微分同胚。对于复双曲流形，类似的构造仅在形如$8n+2$的维度中被证明。在本文中，我们在许多其他维度证明了这一结果。主要方法涉及使用复射影空间的$EO$-理论来构造合适的奇异球面示例。这些理论可视为实$K$-理论的一种版本，并在每个素数$p$处定义。计算还产生了关于奇异球面上自由光滑$S^1$和$S^3$作用存在性的积极结果，这些奇异球面不界定可平行化流形。
英文摘要：
Farrell and Jones showed that negatively curved manifolds in dimension $\geq 5$ are topologically rigid in the sense that homotopy equivalence implies homeomorphism. In the smooth category, the result does not hold even for hyperbolic manifolds, and there are negatively curved manifolds $N$ homotopy equivalent to a given hyperbolic manifold $M$ but not diffeomorphic. For complex hyperbolic manifolds, the analogous construction is demonstrated only in dimensions of the form $8n+2$. In this paper, we prove this result in many other dimensions. The main approach involves the use of $EO$-theory of complex projective spaces to construct suitable examples of exotic spheres. These theories may be viewed as a version of real $K$-theory, and are defined at each prime $p$. The computation also yields positive results about the existence of free smooth $S^1$ and $S^3$ actions on exotic spheres which do not bound parallelizable manifolds.
