Embeddability of joinpowers, and minimal rank of partial matrices

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. It is based on calculation of generators in the homology of certain configuration space (the deleted product) of certain complex (joinpower).