Stratified singular spaces and generalized geometric Poincaré complexes

庞加莱对偶性是经典流形理论的基石。具有奇点的空间的普通上同调不具有这种对偶性。一种解决方案是交上同调，它有一系列的缺点：缺乏空间的普通上同调中可用的许多内部代数结构。它在奇点变形下是不稳定的。它的链论定义使得定义广义交上同调理论变得困难。此外，在IIB弦理论中描述无质量场是错误的理论。在以前的工作中，我们已经将某些奇异空间与它们的交空间联系在一起，使得交空间的普通有理上同调满足互补反常上的Poincaré对偶。特别地，我们得到了分层空间的一个新的上同调理论。我们建议研究这个新理论的内部代数结构，它的自然定义域，它在奇点光滑变形下的稳定性，它与IIB型弦理论的相关性，以及它的相关谱上同调，例如K-理论。

Poincaré duality is a cornerstone of classical manifold theory. The ordinary cohomology of spaces with singularities does not enjoy this duality. One solution is intersection cohomology, which has a series of drawbacks: much of the internal algebraic structure available in the ordinary cohomology of a space is lacking. It is not stable under deformation of singularities. Its chain-theoretic definition makes it hard to define generalized intersection cohomology theories. Furthermore, it is the wrong theory to describe massless fields in type IIB string theory. In previous work we have associated to certain singular spaces their intersection space, in such a way that the ordinary rational cohomology of the intersection space satisfies Poincaré duality across complementary perversities. In particular, one obtains a new cohomology theory for stratified spaces. We propose to investigate the internal algebraic structure of this new theory, its natural domain of definition, its stability under smooth deformation of singularities, its relevance vis-à-vis type IIB string theory, and its associated spectrum cohomology, e.g. K-theory.