Title:  Non-Vanishing Homology of the Matching Complex

Abstract: A matching on a graph G is any subgraph where the maximum vertex degree is 1.  Since edge-deletion preserves the property of being a matching, the set of all matchings on G forms a simplicial complex M(G).  We will survey results on the lowest non-vanishing homology group for M(K_n) and discuss the extension of these results to more general graphs, specifically the r-stable ones.  Prior familiarity with simplicial complexes and homology is assumed.