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While going through Matrix groups , the problem I am facing is that how to determine different topological aspects of them, to be specific, Compactness is not a big problem due to Heine-Borel theorem , but dealing with other properties such as Connectedness and Density is not as easy.

Firstly, I tried to think geometrically (like $SO(2) \cong S^1$ , $SU(2) \cong S^3$ , here $S^n$ denotes the group of the n-sphere in ${\Bbb R}^{n+1}$ , so $S^1$ denotes the circle group etc. ). But I am not able to do it for all the groups there on the table.

Next, for connectedness, I tried to do it with by considering continuous functions from that group to $\{1,-1\}$ , but still am facing difficulty.

And for density in particular, I have no clue.

Please help me by providing some strategies on how to learn and solve problems on Connectedness and Density of Matrix groups.

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    $\begingroup$ For connectedness, one strategy is to have a transitive proper action on a connected space with connected point stabilizers. For density there are no good general strategies one needs more specifics. $\endgroup$ – Moishe Kohan Dec 29 '17 at 15:40
  • $\begingroup$ @MoisheCohen would you like to explain your comment in details and post it as an answer. $\endgroup$ – reflexive Jan 5 '18 at 5:21
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    $\begingroup$ See Smooth Manifolds by Lee, propositions 21.33-21.35. Or Semi-Riemannian geometry by O'Neill, chapter 9 lemma 6 for a differential geometry approach. I'm not sure if there's easy way to prove these theorems without using some fancier technology like theorems on quotient manifolds, connections, etc. $\endgroup$ – juan arroyo Feb 20 '18 at 19:32
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    $\begingroup$ [Connectedness] It is rather easy to prove that a classical matrix group is NOT connected (using determinant function). However, it is not simple to prove connectedness. If you want to find a direct way of proving connectedness, I recommend to read the lecture note by Mats Boij & Dan Laksov. (Section 3.8, p.82) $\endgroup$ – ChoF Feb 21 '18 at 0:49
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As requested, here is an extended form of my comment.

  1. For connectedness of a compact Lie group $G$ acting transitively on a space $X$ (say, a manifold), you have a fiber bundle $$ G_x\to G\to X, x\in X. $$ Hence, if $X$ is connected and $G_x$ is connected, then $G$ is also connected. This can be viewed as a special case of the long exact sequence of homotopy groups of a fibration (see e.g. Hatcher's "Algebraic Topology").

  2. For density, one needs much more information.

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