The biquotient model of the total space of the projectivization quaternionic bundles over certain manifolds

TL;DR

This paper determines the rational homotopy type of quaternionic projective bundles over certain manifolds, revealing that in specific cases, the total space can be modeled as a biquotient of Lie groups, advancing understanding of their topological structure.

Contribution

It provides a biquotient model for the total space of quaternionic projective bundles over quaternionic manifolds, especially when the base is quaternionic projective space.

Findings

Rational homotopy type of quaternionic projective bundles computed.

Biquotient model identified for the total space over $\

Sp(1)\backslash Sp(3) / Sp(1) \times Sp(1)$ in the case of $\

Abstract

In this paper, we compute the rational homotopy type of the quaternionic projective bundle $P(\tau): \mathbb{H}P^{n-1} \rightarrow P(E) \rightarrow M$ obtain from the quaternionic tangent bundle $\tau: \mathbb{H}^{n} \rightarrow E \rightarrow M$ over quaternionic manifold $M$. If the base space $M$ is the quaternionic projective space $\mathbb{H}P^{2}$, then the rational homotopy type of the total space $P(E)$ of the quaternionic projective bundle $P(\tau)$ is given by a biquotient model $Sp(1)\backslash Sp(3) / Sp(1) \times Sp(1)$.