A problem in commutative algebra is the multiplicity conjecture of Huneke, Herzog, and Srinivasan. I will examine how the face-ring version of this conjecture translates into a problem relating the number of maximal faces of a simplicial complex to topological and combinatorial properties of the various skeleta of the complex. Higher Cohen-Macaulay connectivity, a concept introduced by Baclawski, plays a key role in the analysis. I will not assume any advanced knowledge of commutative algebra.