  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 $|f\sigma \cap f\tau|$ is even for any non-adjacent $k$-faces
$\sigma,\tau$. 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-Kyn\v{c}l-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 K\"uhnel
problem on embeddings to a purely algebraic problem.
  Our proof is interplay between geometric topology, combinatorics and linear
algebra.
