Sets of degrees of maps between SU(2)-bundles over the 5-sphere

TL;DR

This paper determines the possible degrees of maps between certain SU(2)-bundles over the 5-sphere, showing Steenrod squares are the key obstruction and constructing explicit maps for each degree.

Contribution

It explicitly computes the sets of degrees of maps between SU(2)-bundles over S^5 and clarifies the role of Steenrod squares as obstructions.

Findings

Steenrod squares are the only obstruction to map degrees.

Explicit maps are constructed for all realizable degrees.

The sets of degrees between these manifolds are fully characterized.

Abstract

We compute the sets of degrees of maps between principal $SU(2)$-bundles over $S^5$, i.e. between any of the manifolds $SU(2)\times S^5$ and $SU(3)$. We show that the Steenrod squares provide the only obstruction to the existence of a mapping degree between these manifolds, and construct explicit maps realizing each integer that occurs as a mapping degree. Added Erratum. After this manuscript was accepted for publication by Transformation Groups, Xueqi Wang pointed out a mistake in our paper. At the end of this arXiv version we add an erratum, where we correct the statement and the proof of Theorem 1.1.