可嵌入性与部分矩阵的最小秩
Embeddability of joinpowers, and minimal rank of partial matrices
AI总结：
本文提出了高维复形到流形的Z₂嵌入性准则，将Kühnel问题转化为代数问题，结合了几何拓扑、组合学和线性代数。
摘要：
一个一般位置映射f:K→M将k维单纯复形映射到2k维流形（对于k=1，将图映射到表面）是Z₂嵌入，当且仅当任何非相邻的k维面σ,τ的fσ∩fτ的交集大小为偶数。我们为某些k维复形（对于k=1，任何图）到2k维流形的Z₂嵌入性提供了准则。这些准则包括：一种类似于Kuratowski版本的Fulek-Kynčl-Bikeev准则（对于k=1），以及Dzhenzher-Skopenkov必要条件的逆向。我们的高维准则使我们将模2Kühnel问题的嵌入性问题转化为纯粹的代数问题。我们的证明是几何拓扑、组合学和线性代数之间的相互作用。它基于计算某些复形（joinpower）的配置空间（删除积）的同调中的生成元。
英文摘要：
A general position map $f:K\to M$ of a $k$-dimensional simplicial complex to a $2k$-dimensional manifold (for $k=1$, of a graph to a surface) is a $\mathbb Z_2$-embedding if $|fσ\cap fτ|$ is even for any non-adjacent $k$-faces $σ,τ$. We present criteria for $\mathbb Z_2$-embeddability of certain $k$-dimensional complex (for $k=1$, of any graph) to $2k$-dimensional manifolds. These criteria are $\bullet$ a `Kuratowski-type' version of the Fulek-Kynčl-Bikeev criteria (for $k=1$), and $\bullet$ a converse to the Dzhenzher-Skopenkov necessary condition (for $k>1$). Our higher-dimensional criterion allows us to reduce the modulo 2 Kühnel problem on embeddings to a purely algebraic problem. Our proof is interplay between geometric topology, combinatorics and linear algebra. It is based on calculation of generators in the homology of certain configuration space (the deleted product) of certain complex (joinpower).
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