Spheres with more than 7 vector fields: all the fault of Spin(9)

TL;DR

This paper explores the maximum number of linearly independent vector fields on spheres by analyzing the role of the Spin(9) group, providing new insights into the algebraic structures influencing vector field existence.

Contribution

It offers a novel interpretation of the maximal vector fields on spheres through the Spin(9) representation, connecting algebraic structures to geometric properties.

Findings

Spin(9) explains the maximal vector fields on spheres beyond classical cases

Provides a new perspective on the role of algebraic groups in topology

Links Spin(9) to the existence of vector fields on spheres

Abstract

We give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R^16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionic structures on R^2, R^4 and R^8, respectively.