On 4-manifolds and span-related numbers for CAT manifolds (Q1173606)

The author uses work of Donaldson and Quinn to extend the Hirzebruch-Hopf classification of O(4)-bundles on a smooth four-manifold to TOP(4) bundles on a topological four-manifold. As a sample, he shows that an orientable TOP(4) bundle over a four-dimensional complex has a sub- \(\mathbb{R}^ 1\)-bundle if and only if the Euler class vanishes; it has a sub-disk-bundle if and only if the Kirby-Siebenmann obstruction vanishes and it splits off a line bundle if and only if both the Euler class and the Kirby-Siebenmann obstruction vanish.