Question

Let $G$ be $SL_2({\mathbb C})$ and for $a,b\in G$ let $[a,b]=aba^{-1}b^{-1}$ be the commutator bracket. Let $n$ be a natural number $\ge 2$ and let $X\subset G^{2n}$ be the set of all $g\in G^{2n}$ such that $$ [g_1,g_2]\cdots[g_{2n-1},g_{2n}]=1. $$ The first question is, whether $X$ is connected. If not, can one give a list of the connected components? Finally, does the subset $SU(2)^{2n}\cap X$ meet every connected component?

If the last question has an affirmative answer, every $G$-valued representation of the fundamental group $\Gamma$ of a compact Riemann surface of genus $n$ can be deformed to a unitary one, which explains the title of my question.

Answer 1

$X$ is the $SL_2(\mathbb{C})$--representation variety of the surface group, and, by Goldman's thesis, it is irreducible, and so connected.

See

Goldman, Topological components of spaces of representations. Invent. Math. 93 (1988), no. 3, 557–607.

If you take $G$ to be $PSL_2(\mathbb{C})$, then there are two components (see also Goldman), one for each Stiefel-Whitney class.

Edit: I should say that I recall that this is perhaps not so easy to find in Goldman's paper as he doesn't state it explicitly, but at some point he proves that the smooth locus of $X$ is connected, which gives the result.