Let $k$ be an algebraically closed field and $G_m$ an algebraic one-dimensional torus. Let $X$ be a smooth irreducible variety over $k$, $Y$ an affine scheme of finite type over $k$ and $f\colon X\rightarrow Y$ a proper morphism.

Assume that there exist actions of $G_m$ on $X$ and $Y$ such that $f$ is $G_m$-equivariant. Moreover, the fixed point locus $Y^{G_m}$ of $Y$ consists of only one point $y.$

Edit (after Allen's answer): Moreover, we assume that for any point $x\in X$ there exists $\lim_{t\rightarrow 0} t\cdot x.$

**Question:** Are $X$ and $f^{-1}(y)$ homotopy equivalent?

`$\mathbb{G}_m \amalg \mathbb{A}^1$`

, with the action on each factor by multiplication, is a counterexample. – Tyler Lawson Apr 22 '12 at 12:49