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Embeddability of joinpowers, and minimal rank of partial matrices

A. Skopenkov, O. Styrt

TL;DR

The paper advances ${\mathbb Z}_2$-embeddability criteria for $k$-complexes into $2k$-manifolds, yielding Kuratowski-type obstructions for graphs and a higher-dimensional converse to the Dzhenzher-Skopenkov condition. It recasts the embeddability problem as a purely algebraic matrix problem involving a matrix $Y$ and the intersection form $\Omega_M$, linked to a detailed homology analysis of configuration spaces via the deleted product of the joinpower $[n]^{*k+1}$. The authors establish a complete sufficiency framework by connecting algebraic data with generators in the relevant homology groups, and prove the 'if' parts using a homological ${\mathbb Z}_2$-embedding criterion. These results simultaneously generalize classical obstruction theory to higher dimensions and provide a concrete bridge between topology and linear-algebraic matrix conditions, with implications for the mod $2$ Kühnel conjecture and related embedding questions.

Abstract

A general position map $f:K\to M$ of a $k$-dimensional simplicial complex to a $2k$-dimensional manifold (for $k=1$, of a graph to a surface) is a $\mathbb Z_2$-embedding if $|fσ\cap fτ|$ is even for any non-adjacent $k$-faces $σ,τ$. We present criteria for $\mathbb Z_2$-embeddability of certain $k$-dimensional complex (for $k=1$, of any graph) to $2k$-dimensional manifolds. These criteria are $\bullet$ a `Kuratowski-type' version of the Fulek-Kynčl-Bikeev criteria (for $k=1$), and $\bullet$ a converse to the Dzhenzher-Skopenkov necessary condition (for $k>1$). Our higher-dimensional criterion allows us to reduce the modulo 2 Kühnel problem on embeddings to a purely algebraic problem. Our proof is interplay between geometric topology, combinatorics and linear algebra. It is based on calculation of generators in the homology of certain configuration space (the deleted product) of certain complex (joinpower).

Embeddability of joinpowers, and minimal rank of partial matrices

TL;DR

The paper advances -embeddability criteria for -complexes into -manifolds, yielding Kuratowski-type obstructions for graphs and a higher-dimensional converse to the Dzhenzher-Skopenkov condition. It recasts the embeddability problem as a purely algebraic matrix problem involving a matrix and the intersection form , linked to a detailed homology analysis of configuration spaces via the deleted product of the joinpower . The authors establish a complete sufficiency framework by connecting algebraic data with generators in the relevant homology groups, and prove the 'if' parts using a homological -embedding criterion. These results simultaneously generalize classical obstruction theory to higher dimensions and provide a concrete bridge between topology and linear-algebraic matrix conditions, with implications for the mod Kühnel conjecture and related embedding questions.

Abstract

A general position map of a -dimensional simplicial complex to a -dimensional manifold (for , of a graph to a surface) is a -embedding if is even for any non-adjacent -faces . We present criteria for -embeddability of certain -dimensional complex (for , of any graph) to -dimensional manifolds. These criteria are a `Kuratowski-type' version of the Fulek-Kynčl-Bikeev criteria (for ), and a converse to the Dzhenzher-Skopenkov necessary condition (for ). Our higher-dimensional criterion allows us to reduce the modulo 2 Kühnel problem on embeddings to a purely algebraic problem. Our proof is interplay between geometric topology, combinatorics and linear algebra. It is based on calculation of generators in the homology of certain configuration space (the deleted product) of certain complex (joinpower).
Paper Structure (6 sections, 20 theorems, 34 equations, 1 figure)

This paper contains 6 sections, 20 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Generators in homology of the complex
  3. Generators in homology of the deleted product
  4. Proof of the 'if' part of Theorem \ref{['t:embntfk']}
  5. Appendix: a direct proof of Lemma \ref{['r:dedu']}
  6. Appendix: some alternative proofs

Key Result

Theorem 1.1

A ${\mathbb Z}_2$-embedding $K_n\to N$ exists if and only if there is a $\beta(N)\times{n\choose3}$-matrix $Y$ such that $Y^T\Omega_NY$ is an independent additive non-trivial matrix.

Figures (1)

  • Figure 1: A cycle of length 6 is the sum of three cycles of length 4

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 39 more