Ed Swartz (Cornell)

Higher Cohen-Macaulay Connectivity

Abstract for the Combinatorics Seminar
2004 October 15

The combined work of Hochster, Reisner, and Stanley on Cohen-Macaulay complexes resulted in a complete characterization of the f-vectors of such complexes. Baclawski introduced k-CM complexes, Cohen-Macaulay complexes which remain CM after the removal of any set of k-1 vertices. These include triangulations of spheres, finite buildings, independence complexes of coloop-free matroids, and order complexes of geometric lattices and of supersolvable lattices whose Möbius function is nonzero on every interval. Much less is understood about enumeration in these spaces. We will take a tour of what is known, guessed at, and hoped for.