Decompositions of Simplicial Complexes

In this thesis we study the interplay between various combinatorial, algebraic, and topological properties of simplicial complexes. We focus on when these properties imply the existence of decompositions of the face poset. In Chapter 2, we present a counterexample to Stanley's partitionability conjecture, we give a characterization of the h-vectors of Cohen-Macaulay relative complexes, and we construct a family of disconnected partitionable complexes. In Chapter 3, we introduce colorated cohomology, which aims to combine the theories of color shifting and iterated homology. Colorated cohomology gives rise to certain decompositions of balanced complexes that preserve the balanced structure. We give conditions that would guarantee the existence of a weaker form of Stanley's partitionability conjecture for balanced Cohen-Macaulay complexes. We consider Stanley's conjecture on k-fold acyclic complexes in Chapter 4, and we show that a relaxation of this conjecture holds in general. We also show that the conjecture holds in the case when k is the dimension of a given complex, and we present a framework that may lead to a counterexample to the original version of this conjecture