On the Homological Stability of Orthogonal and Spin Groups

在这个项目中，我们证明了分裂正交群O_{n，n}、特殊分裂正交群SO_{n，n}和自旋群自旋_{n，n}的同调稳定性结果。我们猜想我们的结果是最优的，因为我们猜想了以相对同调群的形式进一步同调稳定性的障碍。我们有低维计算来支持我们的猜想。具体地说，推测这些障碍与米尔纳K理论和米尔纳-威特K理论有关。因此，如果这些猜想是真的，那么这些群的同调与K-理论之间就存在联系。特别是，对于自旋群来说，这将在源自粒子物理学的概念和Milnor-Witt K理论之间提供一个美丽的联系；一个源自动机同伦理论的概念。

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.