A triple-point Whitney trick

TL;DR

This paper introduces a triple-point Whitney trick to classify certain manifold ornaments in high-dimensional Euclidean spaces using the μ-invariant, focusing on a minimal case of the construction.

Contribution

It presents a specific application of the triple-point Whitney trick to classify ornaments of orientable manifolds, highlighting a minimal case of the construction.

Findings

Ornaments of certain high-dimensional manifolds are classified by the μ-invariant.

A minimal case of the triple-point Whitney trick construction is demonstrated.

The approach is related to, but distinct from, similar work by Mabillard and Wagner.

Abstract

We use a triple-point version of the Whitney trick to show that ornaments of three orientable $(2k-1)$-manifolds in $\mathbb R^{3k-1}$, $k>2$, are classified by the $\mu$-invariant. A very similar (but not identical) construction was found independently by I. Mabillard and U. Wagner, who also made it work in a much more general situation and obtained impressive applications. The present note is, by contrast, focused on a minimal working case of the construction.