Timeline for Nearby homomorphisms from compact Lie groups are conjugate

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Mar 7, 2013 at 23:01	comment	added	Misha		Charles: Yes, you are absolutely right. I lost track of the fact that in your setting $Hom(K, G)$ is a real-analytic variety (as a homomorphism for connected $K$ is determined by the homomorphism of Lie algebras), so things are easier than I thought. Thus, everything reduces to the fact that $H^1_{cont}(K, {\mathfrak g})=0$, as in Weil's paper. There is one issue you need to check though: Uniform convergence of representations implies $C^1$-convergence (in order to use Lie algebras). But, if you use the topology of $C^1$-convergence, this will not be a problem.
Mar 7, 2013 at 14:22	comment	added	Charles Rezk		This is very helpful! Though I'm a little confused by the penultimate paragraph, since it seems that by this point we've shown that $Z^1=B^1$. Looking at Weil's paper, it seems that the deformation theory already tells us that $\mathrm{Hom}(K,G)$ is a manifold (assuming $K$ finitely generated), and so we are done once we have $H^1=0$.
Mar 6, 2013 at 5:39	history	answered	Misha	CC BY-SA 3.0