Given a shifted order ideal $U$, we associate to it a family of simplicial complexes ${({\mathrm{\Delta }}_t(U))}_{t⩾0}$ that we call squeezed complexes. In a special case, our construction gives squeezed balls that were defined and used by Kalai to show that there are many more simplicial spheres than boundaries of simplicial polytopes. We study combinatorial and algebraic properties of squeezed complexes. In particular, we show that they are vertex decomposable and characterize when they have the weak or the strong Lefschetz property. Moreover, we define a new combinatorial invariant of pure simplicial complexes, called the singularity index, that can be interpreted as a measure of how far a given simplicial complex is from being a manifold. In the case of squeezed complexes ${({\mathrm{\Delta }}_t(U))}_{t⩾0}$, the singularity index turns out to be strictly decreasing until it reaches (and stays) zero if $t$ grows.

中文翻译：

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挤压复合物

给定转移订单理想 $ü$，我们将其与简单复合体关联 ${（{\mathrm{\Delta }}_{Ť}（ü））}_{Ť⩾0}$我们称之为挤压复合体。在一个特殊情况下，我们的结构给出了由Kalai定义并使用的压缩球，以显示比简单多面体的边界多得多的简单球体。我们研究挤压复合物的组合和代数性质。特别是，我们证明了它们具有可分解的顶点，并具有弱或强的Lefschetz性质。此外，我们定义了一个纯单纯形复合体的新组合不变式，称为奇异性指数，可以将其解释为一个给定的单纯形复合体离流形有多远的度量。在挤压复合物的情况下${（{\mathrm{\Delta }}_{Ť}（ü））}_{Ť⩾0}$，则奇异指数严格降低，直到达到（并保持）零 $Ť$ 成长。