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June 18th, 2018, 09:59 AM  #1 
Senior Member Joined: Aug 2014 From: United States Posts: 137 Thanks: 21 Math Focus: Learning  De Rham Cohomology of Lorentz Group
Hi, My ultimate goal is to compute the de Rham cohomology groups $H^k(G)$ of the Lorentz group $G=O(3,1)$. I know that $H^0(G)$ is simply given by $\mathbb R^n$ where $n$ is the number of connected components of $G$. To show $n$ is four is it enough to show that the map $f:G\to \mathbb R^2$ given by $X\mapsto (\det X, \text{sgn}X^0_0)$ has an image consisting of a discrete set of four points (namely $(1,1)$, $(1,1)$, $(1,1)$, $(1,1)$)? (In particular, I am wondering what particular theorem or lemma one states if proving it in this manner.) Also, how does one go about calculating the higher order cohomology groups? Thanks. 

Tags 
cohomology, group, lorentz, rham 
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