$SU(2)$-bundles over highly connected $8$-manifolds

TL;DR

This paper classifies the homotopy types of total spaces of $SU(2)$-bundles over highly connected 8-manifolds and studies related 11-dimensional complexes formed from spheres.

Contribution

It provides a classification of $SU(2)$-bundles over 3-connected 8-manifolds and analyzes the homotopy types of certain 11-dimensional complexes.

Findings

Classification of homotopy types of total spaces of $SU(2)$-bundles

Description of 11-dimensional complexes formed from spheres

Insights into the topology of highly connected 8-manifolds

Abstract

In this paper, we analyze the possible homotopy types of the total space of a principal $SU(2)$-bundle over a $3$-connected $8$-dimensional Poincar\'{e} duality complex. Along the way, we also classify the $3$-connected $11$-dimensional complexes $E$ formed from a wedge of $S^4$ and $S^7$ by attaching a $11$-cell.