  The fact that the complete graph $K_5$ does not embed in the plane has been
generalized in two independent directions. On the one hand, the solution of the
classical Heawood problem for graphs on surfaces established that the complete
graph $K_n$ embeds in a closed surface $M$ (other than the Klein bottle) if and
only if $(n-3)(n-4)\leq 6b_1(M)$, where $b_1(M)$ is the first $\mathbb
Z_2$-Betti number of $M$. On the other hand, van Kampen and Flores proved that
the $k$-skeleton of the $n$-dimensional simplex (the higher-dimensional
analogue of $K_{n+1}$) embeds in $\mathbb R^{2k}$ if and only if~$n \le 2k+1$.
  Two decades ago, K\"uhnel conjectured that the $k$-skeleton of the
$n$-simplex embeds in a compact, $(k-1)$-connected $2k$-manifold with $k$th
$\mathbb Z_2$-Betti number $b_k$ only if the following generalized Heawood
inequality holds: $\binom{n-k-1}{k+1} \le \binom{2k+1}{k+1}b_k$. This is a
common generalization of the case of graphs on surfaces as well as the van
Kampen--Flores theorem (the special cases $k=1$ and $b_k=0$, respectively), and
also closely related to the theory of face numbers of triangulated manifolds.
  In the spirit of K\"uhnel's conjecture, we prove that if the $k$-skeleton of
the $n$-simplex embeds in a $2k$-manifold with $k$th $\mathbb Z_2$-Betti number
$b_k$, then $n \le 2b_k\binom{2k+2}{k} + 2k + 4$. This bound is weaker than the
generalized Heawood inequality, but does not require the assumption that $M$ is
$(k-1)$-connected. Our results generalize to maps without $q$-covered points,
in the spirit of Tverberg's theorem, for $q$ a prime power. Our proof uses a
result of Volovikov about maps that satisfy a certain homological triviality
condition.
