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Let $X$ be a smooth, affine complex variety, which, considered as a complex manifold, has the singular $\mathbb{Z}$-cohomology of a point. Suppose that $G$ is a complex algebraic group acting algebraically on $X$. Our main results are the following: if $G$ is semi-simple, then the generic fiber of the quotient map $\pi :X\to X//G$ contains a dense orbit. If $G$ is connected and reductive, then the action has fixed points if $\dim X//G\le 3$.