Real spin bundles over $\mathbb{C}\mathrm{P}^3$ and a new Euclidean embedding of $\mathbb{R}\mathrm{P}^7$

TL;DR

This paper extends the concept of the alpha-invariant to classify real spin bundles over complex projective 3-space and demonstrates a new smooth embedding of real projective 7-space into real projective 11-space.

Contribution

It introduces a generalized alpha-invariant for real spin bundles and applies it to classify bundles over , leading to a novel Euclidean embedding of 7.

Findings

Classified real spin bundles over  using the generalized alpha-invariant.

Proved 7 can be smoothly embedded in 11.

Extended the applicability of the alpha-invariant to real bundles.

Abstract

We generalize the $\alpha$-invariant introduced by Atiyah and Rees to an invariant of real spin bundles and use it to classify real bundles over $\mathbb{C}\mathrm{P}^3$ admitting spin structure. We apply this result to show that $\mathbb{R}\mathrm{P}^7$ can be smoothly embedded in $\mathbb{R}\mathrm{P}^{11}$.