Cluster expansions and topology

A fundamental problem in mathematical physics is to perform a sum of terms indexed by connected graphs. Penrose showed that this sum may be rewritten as a sum indexed by tree graphs. This exhibits remarkable cancellations that lead to a proof of convergence. What makes this work? The set of connected graphs defines a simplicial complex. For a simplicial complex such resummation is possible when the topology of the complex is exclusively in top dimension. Penrose had the good fortune to encounter this case. Dedicated to Robert A. Minlos on his 81st birthday.