On the Homological Stability of Orthogonal and Spin Groups

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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.