  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
  $\bullet$ a collection of 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-Schaefer-Stefankovi\v c 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.
