Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of
  partial symmetric matrices   Let $K$ be a $k$-dimensional simplicial complex having $n$ faces of dimension
$k$ and $M$ a closed $(k-1)$-connected PL $2k$-dimensional manifold. We prove
that for $k\ge3$ odd $K$ embeds into $M$ if and only if there are
  $\bullet$ a skew-symmetric $n\times n$-matrix $A$ with $\mathbb Z$-entries
whose rank over $\mathbb Q$ does not exceed $rk H_k(M;\mathbb Z)$,
  $\bullet$ a general position PL map $f:K\to\mathbb R^{2k}$, and
  $\bullet$ a collection of orientations on $k$-faces of $K$ such that for any
nonadjacent $k$-faces $\sigma,\tau$ of $K$ the element $A_{\sigma,\tau}$ equals
to the algebraic intersection of $f\sigma$ and $f\tau$.
  We prove some analogues of this result including those for $\mathbb Z_2$- and
$\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kyn\v
cl-Schaefer-Stefankovi\v c criteria for the $\mathbb Z_2$- and $\mathbb
Z$-embeddability of graphs to surfaces, and are related to the
Harris-Krushkal-Johnson-Pat\'ak-Tancer criteria for the embeddability of
$k$-complexes into $2k$-manifolds.

5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph