I need to prove that $O(3,19)/SO(2)\times O(1,19)$ is simply connected. In particular $O(n_{+},n_{-})$ denotes the orthogonal group of $\mathbb{R}^{n_{+}+n_{-}}$ endowed with the diagonal quadratic form $diag(1,\cdots,1,-1,\cdots,-1)$ of signature $(n_{+},n_{-})$. I don't know much about the subject, and i don't know how to prove the statement
How to prove that a lie group is simply connected
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algebraic-topology
lie-groups
homotopy-theory
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0Do you know what simply connected means? I assume so. I haven't had a long think about the problem, but I imagine you might want to compute the fundamental group of the manifold using [Seifert-van Kampen](http://en.wikipedia.org/wiki/Seifert%E2%80%93van_Kampen_theorem) and show it equals zero. Just out of curiousity, why are you interested in this Lie group? – 2012-12-14
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0I know very little Lie group theory. However, I Can at least offer the following theorem which breaks down your problem slightly. For topological spaces $X$ and $Y$, $\pi_1(X\times Y,(x_0,y_0))\cong\pi_1(X,x_0)\times\pi_1(Y,y_0)$. And so, you need only prove that $O(3,19)/SO(2)$ and $O(1,19)$ are separately simply connected. – 2012-12-15
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2Isn't it more natural to assume he means to mod out by the subgroup $SO(2) \times O(1,19)$? – 2014-02-14