The connectivity $\eta(X)$ of a simplicial complex $X$ is defined as the 
$$1+\min_j\{j \mid \tilde{H}_j(X)\neq 0\}.$$
If no such $j$ exists, then $\eta(X):=\infty$. 
(See [here][1] for this definition, which is also related to homological connectivity.) 

I hear that if $M$ is a matroid, then $$\eta(M)\ge rank(M).$$ 

I am wondering if anyone knows some reference of this theorem.
（I heard that a matroid looks like a wedge sum of spheres. Not sure if it helps.）

  [1]: https://en.wikipedia.org/wiki/Homological_connectivity