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May16-07, 01:23 PM
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mathwonk's Avatar
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well i looked in bott tu p. 58 for the vector bundle case, and it is knid of like what i said.

one shows the isomorphism class of a bundle on YxI, restricted to Yx{0} cannot change locally near 0.

i.e. the homotopy gives a pull back bundle E on YxI, and we look at its restriction to Yx{0} and we just cross that restriction with I to get a family F of bundles on YxI, which have isomorphic restrictions to every Yxt.

So we have an isomorphism of these bundles E,F over 0 and want to extend the isomorphism say to Yxe where e is a small interval containing 0.

To do this view the isomorphism as a section defined over Yx{0}, of the subbundle Iso(E,F) of Hom(E,F).

Then since this last bundle has Eucklidean space fibers, we can extend, again by a Urysohn type lemma.

then near t=0 the section lies in Iso(E,F), and using compactness of Y and I we get our theorem over YxI.

It can be extended to paracompact Y. This argumenT uses that the bundle of isomorphisms of two vector bundies is itself a fiber bundle contained in a euclidean space bundle, which seems a little special for all fiber bundles..

to go further i guess we need to know your definition of a fiber bundle. but i will probably leave it here.