Algebraic shifting of strongly edge decomposable spheres

强边缘可分解球的各向异性

The generic anisotropy of strongly edge decomposable spheres

本文证明了任意域上所有强边缘可分解球均满足各向异性猜想，支持了同调球的强Lefschetz性质与McMullen的$g$-猜想。

A note on Lefschetz spheres and their relatives

Inspired by the works of Adiprasito, Babson, Nevo, and Murai on the $g$-conjecture, we consider different classes of PL-spheres and the relations between them. We focus on a certain class of spheres that is in the intersection of vertex-decomposable spheres (a concept due to Provan and Billera) and strongly edge-decomposable spheres (a concept due to Eran Nevo). The spheres in this class are exactly those vertex-decomposable ones for which Adiprasito's recent proof method works.

Spheres arising from multicomplexes

In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex $Δ$ on the vertex set $V$ with $Δ\ne 2^V$, the deleted join of $Δ$ with its Alexander dual $Δ^\vee$ is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that thes…

Algebraic Shifting and f-Vector Theory

This thesis focuses on algebraic shifting and its applications to f-vector theory of simplicial complexes and more general graded posets. In particular, several approaches and partial results concerning the g-conjecture for simplicial spheres are presented here.

Embeddability and Stresses of Graphs

Gluck (1975) has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that linklessly embeddable graphs are generically 4-stress free. Both of these results are corollaries of the following theorem: every K_{r+2}-minor free graph is generically r-stress free for 0<r<5. (This assertion is false for r>5.) We give an equivalent formulation of this theorem in the language of symmetric algebraic shifting and show that its analogue for exterior algebraic shifting also holds. Some further extensions are detailed.