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

A. Skopenkov

TL;DR

This work proves that for k ≥ 3 odd K embeds into M if and only if there are a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), and a collection of orientations on k-faces of K such that for any nonadjacent k- faces σ, τ the element Aσ,τ equals to the algebraic intersection of fσ and fτ.

Abstract

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 $\bullet$ orientations on $k$-faces of $K$ such that for any nonadjacent $k$-faces $σ,τ$ of $K$ the element $A_{σ,τ}$ equals to the algebraic intersection of $fσ$ and $fτ$. 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 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 into $2k$-manifolds.

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

TL;DR

This work proves that for k ≥ 3 odd K embeds into M if and only if there are a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), and a collection of orientations on k-faces of K such that for any nonadjacent k- faces σ, τ the element Aσ,τ equals to the algebraic intersection of fσ and fτ.

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 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-Kyn\v cl criteria for the - and -embeddability of graphs to surfaces, and are related to the Harris-Krushkal-Johnson-Paták-Tancer criteria for the embeddability of -complexes into -manifolds.
Paper Structure (10 sections, 20 theorems, 19 equations, 1 figure)

This paper contains 10 sections, 20 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Almost embeddings, ${\mathbb Z}_2$- and ${\mathbb Z}$-embeddings
  4. Criteria in terms of realizability
  5. Proofs
  6. Linear algebraic lemmas
  7. Construction of ${\mathbb Z}_2$- or ${\mathbb Z}$-embeddings
  8. Deduction of realizability from ${\mathbb Z}_2$- or ${\mathbb Z}$-embeddability
  9. Proofs of Proposition \ref{['p:addi']} and Corollaries \ref{['c:rkcom']}.bc
  10. Appendix: unreliability of some papers on embeddings

Key Result

Theorem 1.1.1

Assume that $k\ge3$ is odd, and $M$ is a closed orientable $(k-1)$-connected $2k$-manifold. The complex $K$ embeds into $M$ if and only if $K$ is compatible to a skew-symmetric matrix whose rank does not exceed $\mathop{\fam0 rk}_{\mathbb Z} M$.

Figures (1)

  • Figure 1: Two curves intersecting at an even number of points the sum of whose signs is zero (left) or non-zero (right).

Theorems & Definitions (44)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Proposition 1.1.3: additivity
  • Corollary 1.1.4: additivity
  • Theorem 1.1.5: proved in §\ref{['ss:proof']}, §\ref{['ss:real']}
  • Definition 1.1.6
  • Remark 1.1.7: relation to known results
  • Theorem 1.2.1
  • proof : Comments on the proof of (a)
  • Remark 1.2.2
  • ...and 34 more