Algebraic Shifting and Basic Constructions on Simplicial Complexes

We try to understand the behavior of algebraic shifting with respect to some basic constructions on simplicial complexes, such as union, coning, and (more generally) join. In particular, for the disjoint union of simplicial complexes we prove ∆(K∪̇L) = ∆(∆(K)∪̇∆(L)) (conjectured by Kalai [2]), and for the join we give an example of simplicial complexes K and L for which ∆(K ∗ L) 6= ∆(∆(K) ∗∆(L)) (disproving a conjecture by Kalai [2]), where ∆ denotes the (exterior) algebraic shifting operator. We develop a ’homological’ point of view on algebraic shifting which is used throughout this work.