Since the bounty is now ended, I will post my comment above as an answer. When the dimension $n$ equals $2$, the quotient of $\mathbb{C}^n$ by a finite group $G$ of holomorphic automorphisms is a topological manifold if and only if $G$ is generated by pseudoreflections. Up to using the positive direction (and much harder direction) of Chevalley-Shephard-Todd, it suffices to consider the case when $G$ contains no pseudoreflections. Thus, the branch locus $B$ of the quotient morphism, $$q : \mathbb{C}^n \to Q,$$ has real codimension $4$. When $n$ equals $2$, this implies that $B$ is isolated. The local fundamental group of $(\mathbb{R}^m,0)$ is trivial for $m\geq 3$, in particular for $m=2n=4$. This contradicts the existence of the covering map, $$ q^*: \mathbb{C}^n\setminus q^{-1}(B)\to Q\setminus B.$$

For all $n$, the natural PL structure on $\mathbb{C}^n/G$ is PL-equivalent to a PL-manifold if and only if $G$ is generated by pseudoreflections. The point here is that there is a triangulation of $Q$, as a PL-space, such that $B$ is a polyhedral subcomplex, cf. the main theorem, p. 170, of Hironaka's classic article.

MR0374131 (51 #10331) Reviewed

Hironaka, Heisuke

Triangulations of algebraic sets.

Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 165–185. Amer. Math. Soc., Providence, R.I., 1975.

14B99 (32B20 57C15)

If $Q$ were PL-equivalent to a PL-manifold, then we prove, inductively on the strata of $B_k$ of $B$ that have dimension $\leq k$, that excising $B_k$ with $k\leq 2n-3$ does not affect the fundamental group of $Q$. Since $\text{dim}_{\mathbb{R}}(B)\leq 2n-4$, finally we conclude that the pushforward homomorphism, $$\pi_1(Q\setminus B) \to \pi_1(Q),$$ is an isomorphism. As above, this contradicts the existence of the covering $q^*$.

For $n>3$, there do exist wild embeddings of $(n-3)$-dimensional PL spaces in $n$-manifolds whose open complement has nontrivial local fundamental groups. If I have properly understood the following article, every such fundamental group $\pi$ is perfect and has trivial $H_2(\pi)$.

MR0994411 (90f:57025) Reviewed

Ferry, Steven C.(1-KY); Pedersen, Erik Kjaer(1-KY); Vogel, Pierre(1-KY)

On complements of codimension-3 embeddings in $S^n$.

Topology Appl. 31 (1989), no. 2, 197–202.

57N35 (57N45 57Q35)

This rules out many cases. If there is a good reason for considering other wild cases, then I recommend "localizing" on the components of $B$ of maximal dimension. There is a relation between the strata of $B$ and the corresponding inertia subgroups of $G$, which seems to distinguish this case from the general case of Ferry-Pedersen-Vogel.

**Edit.** Apparently the topological Chevalley-Shephard-Todd theorem is **FALSE** in dimensions $\geq 3$, and the issue is precisely that an algebraic variety that is not a PL-manifold may nonetheless be a topological manifold. I just learned all of this from Greg Kuperberg's answer (and the articles linked in his answer) for this MO question: Algebraic varieties which are topological manifolds.

In particular, for the icosahedral group $\Gamma \subset \textbf{SL}_2(\mathbb{C})$, extending the action to $\mathbb{C}^2\times \mathbb{C}$ by taking the trivial action on the $\mathbb{C}$-factor, the quotient $\mathbb{C}^3/\Gamma$ is a topological manifold.

notgenerated by pseudo-reflections? If $G$ is generated by pseudo-reflections, then the Chevalley-Shephard-Todd theorem says that the quotient space is a manifold. $\endgroup$ – Jason Starr Jul 8 '15 at 17:49shouldit so? $\endgroup$ – Nico Bellic Jul 9 '15 at 19:369more comments