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

(joint work with E. Kogan) We present criteria for embeddability of $k$-dimensional simplicial complexes in $2k$-dimensional manifolds. The criteria are formulated in terms of minimizing the rank of a certain partial symmetric matrix (a version of the Netflix problem). The partial matrix is associated to the intersection form of the manifold, and to certain cohomological (van Kampen) obstruction. The proof is based on the known equivalence (for $k>2$) of embeddability and of $\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kynčl-Schaefer-Stefankovič criteria for the $\mathbb Z_2$- and $\mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Paták-Tancer criteria for the embeddability of $k$-complexes in $2k$-manifolds.

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