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

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 integer 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 entry $A_{\sigma,\tau}$ equals to the algebraic intersection of $f\sigma$ and $f\tau$. We prove some analogues of this result (for any parity of $k$), 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. The main novelty of this paper is passing from the cohomology condition of Pat\'ak-Tancer to the simpler extendability of some intersection function to a low-rank matrix (defined in the paper using the idea of Fulek-Kyn\v cl).