Stratified singular spaces and generalized geometric Poincaré complexes

分层奇异空间和广义几何庞加莱复形

• 批准号: 196751617
• 负责人: Professor Dr. Markus Banagl
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2011
• 资助国家: 德国
• 起止时间: 2010-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/196751617?language=en
• 关键词: Stratified; singular; spaces; generalized; geometric

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.

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。一种解决方案是交交上同调，它具有一系列缺点：缺乏空间的普通上同调中可用的许多内部代数结构。它在奇点变形下不稳定。它的链理论定义使得定义广义交交上同调理论变得困难。此外，用IIB型弦理论来描述无质量场是错误的理论。在之前的工作中，我们将某些奇异空间与它们的交集空间关联起来，使得交集空间的普通有理上同调满足跨互补反常的庞加莱对偶性。特别是，我们获得了一种新的分层空间上同调理论。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、它在奇点平滑变形下的稳定性、它与 IIB 型弦理论的相关性，以及它相关的谱上同调，例如K理论。

项目成果

Isometric Group Actions and the Cohomology of Flat Fiber Bundles

等距群作用和扁平纤维丛的上同调

• DOI: 10.4171/ggd/183
• 发表时间: 2013
• 期刊: arXiv: Algebraic Topology
• 作者: Markus Banagl

Deformation of Singularities and the Homology of Intersection Spaces

奇点的变形与交空间的同调

• DOI: 10.1142/s1793525312500185
• 发表时间: 2012
• 期刊: arXiv: Algebraic Topology
• 作者: Markus Banagl;L. Maxim
• 通讯作者: L. Maxim

Intersection spaces, perverse sheaves and type IIB string theory

交叉空间、反常滑轮和 IIB 型弦理论

• DOI: 10.4310/atmp.2014.v18.n2.a3
• 发表时间: 2014
• 期刊: Advances in Theoretical and Mathematical Physics
• 影响因子: 1.5
• 作者: Markus Banagl;N. Budur;L. Maxim
• 通讯作者: L. Maxim