More algebraic topology

I have a real question now. Suppose S is a simplicial complex of dimension n with the property that every n-1 face of every n simplex is a face of at least two n simplexes. I want to conclude that H_n(S) is non-zero. Assume that the union of the n simplexes is connected (any connected component would inherit the hypothesis). Then it seems clear geometrically that the union of the faces would enclose one or more holes, but I don't see how to actually prove this. The space would appear to have no boundary, but it also is not a manifold since a point in one of the faces could have a branched neighborhood.