ETD

• cohen-macaulay
• shellability
• Simon's conjecture
• vertex decomposability

This thesis focuses on some combinatorial, topological and algebraic properties of abstract simplicial complexes. The main properties it introduces are shellability, Cohen-Macaulayness and vertex decomposability and shows the general relationships between them.

Some of the most studied families of simplicial complexes, such as matroid complexes, broken circuit complexes, shifted complexes and order complexes of geometric lattices are addressed, and the properties listed above are established for them.

At the end, we address a powerful extension of shellability with the aim of understanding the state of the art around Simon's conjecture. This conjecture is at the forefront of current research in algebraic and topological combinatorics. We investigate several recent partial results towards this conjecture.