Sunday, April 6, 2008

Mod Squares

The other day in Homological Algebra we started talking about group (co)homology. I didn't bring my notes home (so I probably have some bits wrong below), but I noticed a strange thing pop up in class that I recognized in a few other contexts. We were, at the time, talking about computing $H_1(G;\mathbb{Z})$. From the topological perspective (which we aren't using in class), we know this is $H_1(BG;\mathbb{Z})$, which is the abelianization of $\pi_1(G)$. The way we identified it in class was as follows: Consider $\mathbb{Z}[G]\rightarrow \mathbb{Z}$ given on generators $g\in G$ by $g\mapsto 1$ (so the map just adds up the coefficients). Let $J$ be the kernel. In class, if I remember correctly, we saw that $H_1(G;Z)$ was the same as $J/J^2$.

This 'mod squares' bit I have seen before, in two other contexts. Originally, I thought I had seen them when we talked about the Dieudonne determinant in Algebra 4. Looking back, it looks like we defined the determinant as a map $GL_n(D)\rightarrow D^*/[D^*,D^*]$ (looks like $D^*$ should be referring to the units of $D$), so at least that looks similar to the above, it's an abelianization. In fact, 'mod squares' did show up in algebra 4, when I was learning about Clifford algebras and the spin group. I just dug up my notes, and it looks like I had a spin norm, a map $SO(V)\rightarrow F^*/(F^*)^2$. The kernel of this map then had a double cover by $Spin(V)$.

The other place I think I've seen this 'mod squares' bit was in the very small bit of algebraic geometry I remember. Something about an $\mathfrak{m}$ (probably a maximal ideal somewhere), and $\mathfrak{m}/\mathfrak{m}^2$ having a useful interpretation.

Anyway, I know the above was all quite vague, and I apologize. I should put in the energy to go back and look at these three things together, and see if there is some general principle I should be noticing. It feels like there should be, but I'm not sure what. Anybody have any idea what I'm talking about?

While I'm at it, anybody know if there is some formalization about the similarity between Sylow subgroups of finite groups and maximal tori of Lie groups? "They look similar" always seems to hint at something in mathematics.

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