Lower Bounds for Buchsbaum* Complexes

The class of $(d-1)$-dimensional Buchsbaum* simplicial complexes is studied. It is shown that the rank-selected subcomplexes of a (completely) balanced Buchsbaum* simplicial complex are also Buchsbaum*. Using this result, lower bounds on the $h$-numbers of balanced Buchsbaum* simplicial complexes are established. In addition, sharp lower bounds on the $h$-numbers of flag $m$-Buchsbaum* simplicial complexes are derived, and the case of equality is treated.