Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices
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Authors: E. Kogan and A. Skopenkov
Abstract: Let be a -dimensional simplicial complex having faces of dimension and a closed -connected PL -dimensional manifold. We prove that for odd embeds into if and only if there are
a skew-symmetric -matrix with -entries whose rank over does not exceed ,
a general position PL map , and
a collection of orientations on -faces of
such that for any nonadjacent -faces of the element equals to the algebraic intersection of and .
We prove some analogues of this result including those for - and -embeddability. Our results generalize the Bikeev-Fulek-Kyncl-Schaefer-Stefankovic criteria for the - and -embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Patak-Tancer criteria for the embeddability of -complexes into -manifolds.