9. Soft Sheaves

In this chapter we study soft sheaves. The class of soft sheaves is on one hand large enough to include all kinds of interesting sheaves on real \(C^{\alpha}\)-manifolds if \(\alpha\leq\infty\), for instance the structure sheaf. On the other hand the cohomology of soft sheaves vanishes, and this yields immediately several local-global principles. The chapter starts with the definition and general properties of soft sheaves. In particular we will see that every global section has an arbitrary fine partition (Proposition 9.9). In Sect. 9.2 we will show that the structure sheaf of a real \(C^{\alpha}\)-manifold for \(\alpha\leq\infty\) is a soft sheaf. In the last section we show that the first cohomology vanishes for any soft sheaf of groups on a paracompact Hausdorff space. We obtain several immediate corollaries for real \(C^{\alpha}\)-manifolds with \(\alpha\leq\infty\): The fact that sections of vector bundles over closed sets can always be extended to a global section (and in particular a smooth version of Tietze’s extension theorem), a description of the Picard group, or the fact that every subbundle of a vector bundle is split.