Tautological classes and smooth bundles over BSU(2)

TL;DR

This paper investigates the distinction between smooth group actions and bundle constructions over BSU(2), revealing examples of bundles not arising from actions and establishing constraints on tautological classes.

Contribution

It provides the first examples of smooth bundles over BSU(2) not induced by actions and introduces a new constraint on tautological classes from group actions.

Findings

Existence of smooth bundles over BSU(2) not induced by actions

A new constraint on tautological classes in cohomology

Identification of differences between actions and bundles

Abstract

For a Lie group G and a smooth manifold W, we study the difference between smooth actions of G on W and bundles over the classifying space of G with fiber W and structure group Diff(W). In particular, we exhibit smooth manifold bundles over BSU(2) that are not induced by an action. The main tool for reaching this goal is a technical result that gives a constraint for the values of tautological classes pulled back to the cohomology of BSU(2) along a map induced by an action.