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Algebraic Shifting and Basic Constructions on Simplicial Complexes

Eran Nevo
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Abstract

We try to understand the behavior of exterior algebraic shifting with respect to basic constructions on simplicial complexes, like union and join. In particular we give a complete combinatorial description of the shifting of a disjoint union, and more generally of a union along a simplex, in terms of the shifting of its components. As a corollary, we prove the following, conjectured by Kalai: $Δ(K \cup L) = Δ(Δ(K) \cup Δ(L))$, where $K,L$ are complexes, $\cup$ means disjoint union, and $Δ$ is the exterior shifting operator. We give an example showing that replacing the operation 'union' with the operation 'join' in the above equation is wrong, disproving a conjecture made by Kalai. We adopt a homological point of view on the algebraic shifting operator, which is used throughout this work.

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Eran Nevo. 2006-02-23. Algebraic Shifting and Basic Constructions on Simplicial Complexes. https://arxiv.org/abs/math/0303233

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