复丛及其连接的特征标
Characters for Complex Bundles and their Connections
摘要：
本文结合了几个幸运的小奇迹，以实现其两个目标。这些是在几年的努力中编织在一起的，以回答Iz Singer十年前提出的问题。我们的答案是访问的拓扑学家，微分几何和分析家谁赞赏声明的指数定理的阿蒂亚，帕托迪，辛格流形的边界。迷你奇迹是这些：[a]证明复边界张紧在Todd亏格上的Conner Floyd奇迹和证明稳定复向量丛分别满足广义同调论和广义余同调论公理的Bott奇迹。b]这些理论，协变和逆变几何表示表明，稳定几乎复（SAC）流形模积关系和稳定的复丛，不仅与亚历山大对偶，但他们也与庞特里亚金对偶。c] b]的抽象推论，稳定复丛具有完整的数值不变量系统，并且由于APS指标定理，这些数值不变量可以通过边界约化模整数的流形上的Chern Weil特征形式的积分来计算。d]最后一节附录中的绝热极限论元表明了在具有连通的黎曼流形的黎曼族的全空间上的直和连通，它在极限上等价于直和度量的Levi Civita连通。这使得不变量可以由奇SAC流形约化模整数的eta不变量来描述。
英文摘要：
The paper combines several fortunate mini miracles to achieve its two objectives. These were woven together in a several year's effort to answer a question raised by Iz Singer a decade ago. Our answer is accessible to the topologist, to the differential geometer and to the analyst who appreciates the statement of the Index theorem of Atiyah,Patodi,Singer for manifolds with boundary. The mini miracles are these: a] The Conner Floyd miracle that complex bordism tensored over the Todd genus and the Bott miracle that stable complex vector bundles respectively satisfy the axioms of a generalized homology theory and of a generalized cohomology theory. b] That these theories, with the covariant and contravariant geometric representations indicated, stably almost complex (SAC) manifolds modulo product relations and stable complex bundles, are not only related by Alexander duality but they are also related by Pontryagin duality. c] The abstract corollary of b] that stable complex bundles have a complete system of numerical invariants and that these can be computed by integrals of chern weil characteristic forms over manifolds with boundary reduced modulo integers, thanks to the APS Index Theorem. d] The adiabatic limit argument of the appendix to the last section showing a direct sum connection on the total space of a riemannian family of Riemannian manifolds with connection is Chern Simons equivalent in the limit to the Levi Civita connection of the direct sum metric. This allows the invariants to be described by the eta invariants of odd SAC manifolds reduced mod integers.
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