 By Dale Husemoller (auth.)

ISBN-10: 1475722613

ISBN-13: 9781475722611

ISBN-10: 147572263X

ISBN-13: 9781475722635

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Extra resources for Fibre Bundles

Example text

We assume the theorem is true for all B with dim B < n. This is the case for n = 0 because B =A. We let B be of dimension n. By the inductive hypothesis we have a cross section s' of ~1Bn_ 1 with s'JA = s. We let C be an n-cell of B with attaching map uc: r--+ B. The bundle u~(O over r is locally trivial, and since In is compact, we can dissect In into equal cubes K of length 1/k such that u~(~)IK is trivial. 10) the cross sections' defines a cross section (J' of u~(~)l8r. Applying the inductive hypothesis to (J', we can assume that (J' is defined on the (n - i)-skeleton of r decomposed into cubes K oflength 1/k.

This principal Zz-space defines a principal Z 2 bundle with base space RP". Finally, we prove that a principal G-bundle is a bundle with fibre G. 3. 6 Proposition. with fibre G. Let~= (X,p,B) be a principal G-bundle. 43 Then~ is a bundle Proof For x E p- 1 (b) we define a bijective map u: G ~ p- 1 (b) by the relation u(s) = xs. The inverse function of u is x' ~ r(x, x'), which is continuous, and u is a homeomorphism. 3. 1 Definition. A morphism (u,f): (X, p, B)~ (X', p', B') between two principal G-spaces is a principal morphism provided u: X ~X' is a morphism between G-spaces.

And from this u is also continuous. 3 Corollary. There exists a Gauss map g: E(~) ~ pm (k ~ m ~ +oo) if and only if~ is B(O-isomorphic with f*(yk) for some map f: B(~) ~ Gk(Fm). Proof. 2). 5), we construct a Gauss map for each vector bundle over a paracompact space. First, we need a preliminary result concerning the open sets over which a vector bundle is trivial. 4 Proposition. Let ~ be a vector bundle over a paracompact space B such that ~IV;, i E I, is trivial, where {U;}, i E I, is an open covering.