Algebraic Shifting and f-Vector Theory

This manuscript focusses on algebraic shifting and its applications to f -vector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the g-conjecture for simplicial spheres, which is considered by many as the main open problem in f -vector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in f -vector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds. • f -vector theory for simplicial spheres, and more general complexes. • f -vector theory for (non-simplicial) graded partially ordered sets. • Graph minors. Combinatorially, a (finite) simplicial complex is a finite collection of finite sets which is closed under inclusion. This basic object has been subjected to extensive research. Its elements are called faces. Its f -vector (f0, f1, f2, ...) counts the number of faces according to their dimension, where fi is the number of its faces of size i + 1. f -vector theory tries to characterize the possible f -vectors, by means of numerical relations between the components of the vector, for interesting families of simplicial complexes (and more general objects); for example for simplicial complexes which topologically are spheres. Algebraic shifting associates with each simplicial complex K a shifted simplicial complex, denoted by ∆(K), which is combinatorially simpler. This is an invariant which on the one hand preserves important invariants of K, like its f -vector and Betti numbers, while on the other hand loses other invariants, like the topological, and even homotopical, type of K. A general problem is to understand which invariants of K can be read off from the faces of ∆(K), and how. There are two different variations of this operator: one is based on the exterior algebra, the other on the symmetric algebra; both