Kairvaire Problems in Stable Homotopy Theory

TL;DR

This paper reviews the construction of a specific 30-dimensional manifold with a unique invariant in stable homotopy theory and explores whether similar manifolds exist in higher dimensions like 126.

Contribution

It revisits the construction of the 30-dimensional manifold with Arf-Kervaire invariant 1 and investigates the possibility of analogous manifolds in higher dimensions.

Findings

Constructed the 30-dimensional manifold with Arf-Kervaire invariant 1.

Analyzed properties of the 30-dimensional manifold.

Discussed the open problem of existence in dimension 126.

Abstract

The Kervaire Problem is an unsolved problem in Stable Homotopy Theory. The first interesting example is in dimension $30$. There exists a closed stably-parallelizable manifold $\tilde{M}^{30}$ with Arf-Kervaire invariant 1. It is unknown, if such a manifold exists in dimension $126$? The goal of the paper is to recall the construction of the manifold $M^{30}$ and investigate its properties.