Manifold Theory

Abstract

In this section we take up the discussion of fibrewise differential topology begun in Part I, Section 22. As we observed there, fibrewise manifolds appeared in the work of Atiyah and Singer [5] on the Index Theorem for families of elliptic pseudodifferential operators and were there called manifolds over a base. The definition that we give here is rather more general than that given by Atiyah and Singer, although their definition covers most of the important examples. We broaden the definition in such a way that open subspaces of fibrewise manifolds are also regarded as fibrewise manifolds.