The character group is an algebraic group.

Question

Let $X$ be a finite dimensional connected CW complex. Let $G=\pi_1(X)$. Let $\hat G=Hom(G,\mathbb C^*)$ the character group of $G$. Then $\hat G$ is an affine algebraic group. Could someone explain why this is true, i cannot relate $\hat G$ to any set of zeros of polynomials. Thank you for your help!

Answer 1

Assume $X$ is compact, else it's false (typically $Hom(\pi_1(X), \Bbb C^*)$ is infinite dimensional if $X = \Bbb R^2 \backslash \Bbb Z$).

Now by standard topology $\pi_1(X)$ is finitely generated, say by $r$ elements : it follows that $\widehat G$ is a quotient of $(\Bbb C^*)^r$ by some relations, so it's a product between a torus and a finite group, in particular this is an affine algebraic group.