On splitting topological \(R^ 4\)-bundles
Suppose \(\xi\) is an \(\mathbb{R}^ 4\)-bundle over a Poincaré complex of formal dimension at most 4, and \(\alpha\) is either a line or a plane bundle over \(X\). This paper considers conditions involving characteristic classes of \(\xi\) and \(\alpha\) which determine when there is a Whitney sum decomposition \(\xi\approx \alpha\oplus\beta\), for some bundle \(\beta\). As an example of an application, it is shown that the tangent bundle of a closed connected 4-manifold splits off a given line bundle if and only if both the Euler number and the Kirby-Siebenmann number of the manifold vanish.
\(\mathbb{R}^ 4\)-bundle over a Poincaré complex of formal dimension at most 4
tangent bundle of a closed connected 4- manifold
scientific article; zbMATH DE number 60705