Fungible obstructions to embedding 2-complexes

TL;DR

This paper constructs specific 2-complexes that cannot be piecewise-linearly embedded in four-dimensional space, explores their embedding obstructions, and demonstrates the limitations of certain known invariants.

Contribution

It provides explicit examples of 2-complexes with non-embeddability in R^4 and analyzes their obstructions using higher-order Milnor invariants, showing some existing obstructions vanish.

Findings

Examples of non-PL embeddable 2-complexes in R^4

Family of PL immersions hiding higher-order Milnor invariants

Krushkal's embedding obstructions vanish for these examples

Abstract

We give examples of finite, simplicial $2$-complexes that do not PL embed in $\mathbb{R}^4$ and exhibit, for each such complex, a family of PL immersions into $\mathbb{R}^4$ that hide the obstruction to embedding in "higher and higher order Milnor invariants". In addition, we show that the embedding obstructions defined by Krushkal in [8] vanish for our examples. We also answer a question in a paper of Avramidi-Okun-Schreve.