Let $M^n$ be a complete, open Riemannian manifold with $\Ric \geq 0$. In 1994, Grigori Perelman showed that there exists a constant $\delta_{ n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $\alpha_M := \lim_{ r \to \infty} \frac{ \Vol(B_p(r))}{ \omega_n r^n} \geq 1-\delta_{ n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $\alpha(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial.
Categories: math.DG