Symmetries in topology and algebra

拓扑和代数中的对称性

• 批准号: 527329998
• 负责人: Professor Dr. Markus Land
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/527329998?language=en
• 关键词: Symmetries; topology; algebra

This project proposal is about cohomological and homotopical invariants of certain groups of symmetries appearing in topology and algebra, and in particular their interaction. The main focus is on mapping class groups of surfaces and symplectic groups over the integers. These groups are related by a natural homomorphism, and one aspect of this proposal is to study the induced map on (stable) cohomology with finite coefficients. A combination of a number of results says that this is equivalently described by studying the map induced on cohomology of a map $\alpha$ of spaces associated to the spectra MTSO(2) and GW^{-s}(Z) - the Thom spectrum of the negative of the universal bundle over BSO(2) and the antisymmetric (aka symplectic) Grothendieck--Witt spectrum of the integers. These spaces also carry interesting higher homotopy groups (in contrast to the classifying spaces of the above mentioned symmetry groups) and another aspect of this proposal is to compare these homotopy groups via the above mentioned map $\alpha$. The proposal consists of several individual milestones, for instance that $\alpha$ is such that one obtains an infinite family of elements of degree $8i-3$ in the stable mod 2 cohomology of mapping class group, which should be studied in more detail, but also to calculate the effect of $\alpha$ on homotopy groups (modulo torsion) via its relation to the signature homomorphism on MTSO(2). Several variants of such questions arise naturally, like studying the spin mapping class group and its relation to the quadratic symplectic group, or to study higher dimensional analogs where surfaces are replaced by suitable high dimensional manifolds.

这个专题计划是关于拓扑学和代数学中出现的某些对称群的上同调和同伦不变量，特别是它们之间的相互作用。主要的重点是映射类群的曲面和辛群的整数。这些群通过一个自然同态联系在一起，这个建议的一个方面是研究有限系数（稳定）上同调的诱导映射。一系列结果的组合表明，这是等价描述通过研究映射$\alpha$的空间的上同调诱导的映射MTSO（2）和GW^{-s}（Z）-BSO（2）上的泛丛的负的Thom谱和整数的反对称（又名辛）Grothendieck-Witt谱。这些空间也带有有趣的高阶同伦群（与上面提到的对称群的分类空间相反），这个建议的另一个方面是通过上面提到的映射$\alpha$比较这些同伦群。该提案由几个单独的里程碑组成，例如，$\alpha$使得在映射类群的稳定模2上同调中获得一个次数为8i-3$的元素的无限族，这应该被更详细地研究，但也要通过它与MTSO（2）上的签名同态的关系来计算$\alpha$对同伦群（模挠）的影响。这些问题的几个变体自然出现，如研究自旋映射类群及其与二次辛群的关系，或研究高维类似物，其中表面被适当的高维流形取代。