We prove that the symmetrized deleted join SymmDelJoin(\(\mathcal{K}\) ) of a “balanced family” \(\mathcal{K}\) = 〈K_i〉 ^r_i=1 of collectively r-unavoidable subcomplexes of 2^[m] is (m−r−1)-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result which is more conceptual and more general than previously known results. Already the case r = 2 of this result seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman’s discrete Morse theory.

中文翻译：

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集体不可避免的复合体的Tverberg型定理

我们证明 了一个“平衡族”的对称删除的连接SymmDelJoin（\（\ mathcal {K} \））\（\ mathcal {K} \） = 〈K _i〉^r _{i = 1}，共同是r的不可避免的子复合体2 ^{[ m ]}连接（m - r -1）。结果，我们获得了Tverberg-Van Kampen-Flores类型的结果，该结果比以前已知的结果更具概念性且更笼统。已经是这样了[R这个结果= 2似乎是新的一样经典的凡坎彭-弗洛雷斯定理的扩展。本文使用的主要工具是R.Forman的离散莫尔斯理论。