The Arf-Kervaire invariant of framed manifolds as an obstruction to embeddability

TL;DR

This paper establishes that certain highly connected, stably parallelizable manifolds with Arf-Kervaire invariant 1 cannot be smoothly embedded into Euclidean spaces of specific dimensions, revealing obstructions related to the invariant.

Contribution

It proves new non-embeddability results for high-dimensional manifolds based on the Arf-Kervaire invariant, linking algebraic invariants to geometric embedding properties.

Findings

No 14-connected, stably parallelizable 30-manifold with Arf-Kervaire invariant 1 embeds in R^{36}.

No 30-connected, stably parallelizable 62-manifold with Arf-Kervaire invariant 1 embeds in R^{83}.

Establishes the Arf-Kervaire invariant as an obstruction to certain smooth embeddings.

Abstract

We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\mathbb{R}^{36}$ (resp. $\mathbb{R}^{83}$).