The connectivity $\eta(X)$ of a simplicial complex $X$ is defined as the $$1+\min_j\{j \mid \tilde{H}_j(X)\neq 0\}$$ (see here for this definition.)
If $M$ is a matroid not having a co-loop (an element in every independent set of $M$), then $\eta(M)=rank(M)$. (See, for example, Lemma 3.2 here) Anyone knows a proof of this theorem? I am not able to find a proof in the reference they cite there, and it seems the reference uses some other notation.
I heard that a matroid looks like a wedge sum of spheres. Not sure if it helps.