Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices

TL;DR

This paper establishes a new criterion linking the embeddability of $k$-complexes into $2k$-manifolds with the existence of certain skew-symmetric matrices and intersection properties, extending previous graph embedding results.

Contribution

It introduces a novel algebraic-topological criterion for embedding $k$-complexes into $2k$-manifolds using matrix rank and intersection data, generalizing prior graph embedding criteria.

Findings

Characterization of embeddability via skew-symmetric matrices.

Extension of graph embedding criteria to higher-dimensional complexes.

Analogues for $ ext{Z}_2$- and $ ext{Z}$-embeddability.

Abstract

Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension $k$, and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are $\bullet$ a skew-symmetric $n\times n$-matrix $A$ with $\mathbb Z$-entries whose rank over $\mathbb Q$ does not exceed $rk H_k(M;\mathbb Z)$, $\bullet$ a general position PL map $f:K\to\mathbb R^{2k}$, and $\bullet$ orientations on $k$-faces of $K$ such that for any nonadjacent $k$-faces $\sigma,\tau$ of $K$ the element $A_{\sigma,\tau}$ equals to the algebraic intersection of $f\sigma$ and $f\tau$. We prove some analogues of this result including those for $\mathbb Z_2$- and $\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kyn\v cl criteria for the $\mathbb Z_2$- and $\mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Pat\'ak-Tancer criteria for the embeddability of $k$-complexes into $2k$-manifolds.