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

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Title: Embeddings of $k$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}k$-complexes in $2k$$2k$-manifolds and minimum rank of partial symmetric matrices

Authors: E. Kogan and A. Skopenkov

Abstract: Let $K$$K$ be a $k$$k$-dimensional simplicial complex having $n$$n$ faces of dimension $k$$k$ and $M$$M$ a closed $(k-1)$$(k-1)$-connected PL $2k$$2k$-dimensional manifold. We prove that for $k\ge3$$k\ge3$ odd $K$$K$ embeds into $M$$M$ if and only if there are

$\bullet$$\bullet$ a skew-symmetric $n\times n$$n\times n$-matrix $A$$A$ with $\mathbb Z$$\mathbb Z$-entries whose rank over $\mathbb Q$$\mathbb Q$ does not exceed $rk H_k(M;\mathbb Z)$$rk H_k(M;\mathbb Z)$,

$\bullet$$\bullet$ a general position PL map $f:K\to\mathbb R^{2k}$$f:K\to\mathbb R^{2k}$, and

$\bullet$$\bullet$ a collection of orientations on $k$$k$-faces of $K$$K$

such that for any nonadjacent $k$$k$-faces $\sigma,\tau$$\sigma,\tau$ of $K$$K$ the element $A_{\sigma,\tau}$$A_{\sigma,\tau}$ equals to the algebraic intersection of $f\sigma$$f\sigma$ and $f\tau$$f\tau$.

We prove some analogues of this result including those for $\mathbb Z_2$$\mathbb Z_2$- and $\mathbb Z$$\mathbb Z$-embeddability. Our results generalize the Bikeev-Fulek-Kyncl-Schaefer-Stefankovic criteria for the $\mathbb Z_2$$\mathbb Z_2$- and $\mathbb Z$$\mathbb Z$-embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Patak-Tancer criteria for the embeddability of $k$$k$-complexes into $2k$$2k$-manifolds.

See [Kogan&Skopenkov2021].

References

• [Kogan&Skopenkov2021] E. Kogan and A. Skopenkov, Embeddings of $k$$k$-complexes in $2k$$2k$-manifolds and minimum rank of partial symmetric matrices, arXiv:2112.06636.