  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.
