流形上的质点空间与广义Poincaré对偶
Particle Spaces on Manifolds and Generalized Poincaré Dualities
摘要：
流形M上的位形空间与M的某些映射空间有内在联系，这是拓扑学中一个很有趣的现象。在本文中，我们解释并大大扩展了这种关系。建筑（主要）的工作西格尔，我们介绍了一类新的配置空间;粒子空间，这些包括经典的配置空间的不同点，对称产品，截断产品，除数空间，积极和消极的粒子麦克达夫，以及许多其他。我们表明，有“扫描”的地图，这些空间的粒子在光滑流形适当的束的部分空间。我们证明了这些对应总是同调等价的，并且在某些可识别的情况下是同伦等价。应用这两个代数几何和代数拓扑给出。我们恢复并证明了关于位形空间、有界重数多项式和闭路空间的一些新结果。我们的方法提供了有趣的洞察到八十年代初发现的全纯映射空间的稳定性现象，并断言，一些空间的合理的映射给第二闭路空间良好的拓扑逼近。最后，我们重新发现了Poincaré，Alexander和Lefshetz（首先由Gajer给出）的经典对偶定理的一个美丽的空间级描述，并进一步将其扩展到包括Spanier-Whitehead对偶。
英文摘要：
It is quite an interesting phenomenon in Topology that configuration spaces on a manifold M are intrinsically related to certain mapping spaces from M. In this paper we interpret and greatly expand on this relationship. Building (mainly) on work of Segal, we introduce a new class of configuration spaces; the particle spaces, and these include the classical configuration spaces of distinct points, symmetric products, truncated products, divisor spaces, positive and negative particles of McDuff, as well as many others. We show that there are "scanning" maps that relate these spaces of particles on smooth manifolds to section spaces of appropriate bundles. These correspondences we prove are homology equivalences always and in some identifiable cases homotopy equivalences. Applications of this to both algebraic geometry and algebraic topology are given. We recover old as well as prove some new results on configuration spaces, bounded multiplicity polynomials and loop spaces. Our methods provide interesting insight into the stability phenomenon of holomorphic mapping spaces discovered in the early eighties and asserting that some spaces of rational maps give good topological approximations to second loop spaces. Finally, we rediscover a beautiful space level description of the classical duality theorems of Poincaré, Alexander and Lefshetz (first given by Gajer), and further extend it to include Spanier-Whitehead duality.
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。
