Fundamental groups of character varieties: surfaces and tori

Abstract

We compute the fundamental group of moduli spaces of Lie group valued representations of surface and torus groups.

Appendix: Serre fibrations

Recall that a map $f: E\rightarrow B$ has the left lifting property with respect to a map $i: W\rightarrow Z$ if every commutative diagram

admits a lift $Z\rightarrow E$ making the diagram commute. The map f is a Serre fibration if it has the left lifting property with respect to the inclusions $[0,1]^{n-1}\times \{0\} \hookrightarrow [0,1]^n$ for all $n\ge 1$. It is a well-known fact that Serre fibrations satisfy the left lifting property not just for the inclusions $[0,1]^{n-1}\times \{0\} \hookrightarrow [0,1]^n$, but also for all inclusions $A\hookrightarrow B$ where B is a CW-complex, A is a subcomplex, and the inclusion is a homotopy equivalence (this is one part of a model category structure on topological spaces, as constructed in many places). We need only a very simple special case of this, namely the case of the inclusion $\{\vec {0}\} \hookrightarrow [0,1]^n$, where $\vec {0} = (0, \ldots , 0)$. This special case can be proved by a simple induction on n: the case $n=1$ is already part of the definition of a Serre fibration, and assuming the result for $n-1$, we factor the inclusion $\{\vec {0}\} \hookrightarrow [0,1]^n$ through $[0,1]^{n-1}\times \{0\}$ and apply the left lifting property first to $\{\vec {0}\} \hookrightarrow [0,1]^{n-1}\times \{0\}$, and then to $[0,1]^{n-1}\times \{0\} \hookrightarrow [0,1]^n$.

Proposition 3.1

Let $X\mathop {\longrightarrow }\limits ^{f} Y\mathop {\longrightarrow }\limits ^{g} Z$ be maps between topological spaces, and assume f is surjective. If f and gf are Serre fibrations, then so is g.

Proof

Given a commutative diagram

we must produce a map $[0,1]^{n}\rightarrow Y$ making the diagram commute. Since f is surjective, we may choose a point $x_0\in X$ such that $f(x_0) = \widetilde{H}_0 (\vec {0})$, and since $X\rightarrow Y$ has the left lifting property with respect to $\{\vec {0}\}\hookrightarrow [0,1]^{n-1}$, there exists a map G making the diagram

commute (where $c_{x_0} (\vec {0}) = x_0$). Since gf is a Serre fibration and $g\circ f \circ G = g\circ \widetilde{H}_0 = H\circ i$, there exists a commutative diagram

The desired lift $[0,1]^n \rightarrow Y$ of H is given by $f\circ \widetilde{H}$. $\square$

Corollary 3.2

Let F be a finite group acting freely on Hausdorff spaces E and B, and let $f: E\rightarrow B$ be an equivariant map that is also a Serre fibration. Then the induced map $E/F\rightarrow B/F$ is a Serre fibration.

Proof

We will apply Proposition 3.1 to the composition

$$\begin{aligned} E \longrightarrow E/F \longrightarrow B/F. \end{aligned}$$

Quotient maps for free finite group actions on Hausdorff spaces are covering maps, and covering maps are Serre fibrations, so the first map in this sequence is a (surjective) Serre fibration. The composite map equals the composite map

$$\begin{aligned} E \longrightarrow B \longrightarrow B/F, \end{aligned}$$

which is a composition of Serre fibrations, hence a Serre fibration. $\square$