Upper bound theorem for odd-dimensional flag triangulations of manifolds

We prove that among all flag triangulations of manifolds of odd dimension 2r-1 with sufficiently many vertices the unique maximizer of the entries of the f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses methods from extremal graph theory.Comment: Clarifications and new references, title has change

Gamma-polynomials of flag homology spheres

Chapter 1 contains the main definitions used in this thesis. It also includes some basic theory relating to these fundamental concepts, along with examples. Chapter 1 includes an original result, Theorem 1.5.4, answering a question of Postnikov-Reiner-Williams, which characterises the normal fans of nestohedra. Chapter 2 contains the content of the paper [2], of which Theorem 2.0.6 is the main result. As mentioned, [2] shows that the Nevo and Petersen conjecture holds for simplicial complexes in sd(Σd−1). . Chapter 3 includes the content of the paper [1], where we show that the Nevo and Petersen conjecture holds for the dual simplicial complexes to nestohedra in Theorem 3.0.4. Chapter 4 contains the content of the paper [3] in which we prove Conjecture 0.0.4 in Theorem 4.1.2 by showing that tree shifts lower the γ-polynomial of graph-associahedra. Chapter 4 also includes Theorem 4.2.1, which shows that flossing moves also lower the γ-polynomial of graph-associahedra. In Chapter 5 we include smaller results that have been made. This chapter includes a result proving Gal’s conjecture for edge subdivisions of the order complexes of Gorenstein* complexes, and shows that this result can be attributed to the work of Athanasiadis in [4]. Chapter viii INTRODUCTION 5 also includes some work we have done towards answering Question 14.3 of [26] for interval building sets