On the Homological Stability of Orthogonal and Spin Groups

在本项目中，我们证明了分裂正交群O_{n，n}，特殊分裂正交群SO_{n，n}和Spin群Spin_{n，n}的同调稳定性结果。通过以相对同调群的形式排除进一步同调稳定性的障碍，我们猜想我们的结果是最优的。我们有低维计算支持我们的猜想。具体地说，这些障碍与MilnorK理论和Milnor-Witt K理论有关。因此，如果这些假设是真的，那么这些群的同调与K-理论之间就存在着联系。特别是，对于自旋群，这将提供一个美丽的连接之间的概念起源于粒子物理学和米尔诺-维特K理论;一个概念起源于动机同伦理论。

In this project, we have proven homological stability results for the split orthogonal group O_{n,n}; the special split orthogonal group SO_{n,n} and the Spin group Spin_{n,n}. We conjecture that our results are optimal, by conjecturing the obstruction to further homological stability, in the form of the relative homology groups. We have low dimensional computations that support our conjecture. Specifically, it is conjectured that these obstructions are related to Milnor K-Theory and Milnor-Witt K-Theory. Therefore, if these conjectures are true, there would exist a connection between the homology of these groups and K-Theory. In particular, for the Spin groups, this would provide a beautiful connection between a concept originating from Particle Physics and Milnor-Witt K-Theory; a concept originating from Motivic Homotopy Theory.