Partition and Cohen-Macaulay Extenders

The f- and h-vectors of a simplicial complex are two important combinatorial invariants whose study has motivated a great deal of work in algebraic and topological combinatorics. The f-vector records the number of faces of the simplicial complex by dimension, while the h-vector is obtained from the f-vector by an invertible transformation. Although the entries of the f-vector of any complex have obvious combinatorial interpretations, the entries of the h-vector typically do not. However, if we impose stronger combinatorial conditions such as partitionability or the Cohen–Macaulay condition on the complex, then the entries of the h-vector are known to have combinatorial interpretations.

In this talk I will present joint work with Joseph Doolittle (Freie Universit\:{a}t Berlin) and Bennet Goeckner (University of Washington) in which we consider complexes that do not have these two properties. Given any $d$-dimensional simplicial complex $\Delta$, we construct another $d$-dimensional complex $\Gamma$ containing $\Delta$ such that $\Gamma$ and the relative complex $(\Gamma, \Delta)$ are (relatively) partitionable. This allows us to view the h-vector of $\Delta$ as the difference of h-vectors of partitionable complexes, and thus gives a combinatorial intepretation of its entries.

By contrast, there is a large class of complexes $\Delta$ for which there is no Cohen–Macaulay complex $\Gamma$ containing $\Delta$ such that $(\Gamma, \Delta)$ is Cohen–Macaulay. We give a complete characterization of when such $\Gamma$ exist, and give a straightforward description of all such $\Gamma$.

Finally, we consider the possibility of a similar construction for shellable complexes, and give a connection to Simon’s conjecture on extendable shellability.