Abstract:

Several kinds of connectivity will be reviewed: $k$-connectivity of graphs, topological $k$-connectivity of spaces, and $k$-Cohen-Macaulay connectivity of simplicial complexes. Also, we introduce yet another form of connectivity, called $k$-rigidity. It applies to finite posets, and (via the face poset) to regular cell complexes. It is proved that if any set of at most $k$ vertices is removed from the boundary complex of a $d$-dimensional convex polytope, then the remaining induced subcomplex is topologically $(d-k-1)$-connected. This extends a result on polytope graphs of Balinski (the $k=d-1$ case) and sharpens a result of Fløystad (the homology version).

Results of this nature are obtained via rigidity of homotopy Cohen-Macaulay lattices. The technique applies also to face lattices of some other cell complexes, to basis graphs of matroids and buildings, and to geometric lattices. Finally, an application to finite Coxeter groups is given.