Fibre bundles and measure

Abstract

In the study of Banach algebra bundles [2], there arises quite naturally the question of integration of bundle sections if the fibre is a vector space and the base space supports a measure. The possibility of such integration follows from the theorem below. It is published separately because of its independent interest and because of its relevance to the following results in topology. 1. (DOYLE-HOCKING). Every topological closed n-manifold Mn is the union of an open n-cell and a closed (n-1)-dimensional subset disjoint from the cell [1]. 2. (BROWN-CASSLER). Every closed topological n-manifold Mn is the continuous image by a map 4 of the closed n-cell In _[O, l]X[0, l]X * X [O, 1] (n factors) so that (i) al 0n is a homeomorphism ((In = Inn(Rn\I)-, (ii) 4-1'0(aI1) =31n and (iii) dim q5(,91n) _ n -l [l]. 3. Any (coordinate) bundle 63 over the n-sphere Sn is strictly equivalent to a (coordinate) bundle 6W' in normal form. In 6W' the open covering of Sn consists of two sets Vi and V2 each of which is a zone of Sn and such that Vnl V2 is a narrow equatorial band. The width of the band can be made arbitrarily small [4]. Result 2 is regarded as complementary to 1. Results 1 and 3 show that fibre bundles over many manifolds are really fibre bundles over contractible sets united with sets of small dimension or small measure. Such bundles are trivial over the contractible parts of their base spaces. Consequently, each may be exemplified by a coordinate bundle for which the transition functions (coordinate transformations) are constant and equal to the group identity over of the base space. The theorem of this paper extends the results above. Broadly paraphrased, the theorem says that a fibre bundle over any topological space X endowed with a reasonable topological measure may be exemplified by a coordinate bundle whose transition functions are, on most of X, constant and equal to the identity of the group of the bundle. More precisely we have the