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I am trying to find a result for the number of bundles (up to bundle iso.) over a fixed

base and fixed fiber. For example, for B=S<sup>1</sup> , and fiber I=[0,1]

I think that there are two; the cylinder and the Mobius strip.

I think that the reason there are two (classes of) bundles is that there are

only two isotopy classes in Hom(I,I) , where Hom is a homeomorphism;

the class of the identity, and the class of the map f(t)=1-t, so that one copy of I .

Also: how can we count the number of bundles over S<sup>1</sup> x

S<sup>1</sup>.? The obvious bundles are the ones defined as product

bundles of the bundles above, i.e., cylinderxcylinder, etc. Is there a way

of knowing if the product bundles are the only bundles over S<sup>1</sup> x

S<sup>1</sup>.?

I guess if the above ideas is correct, we should consider the isotopy classes

of Hom(IxI,IxI ).

Thanks.

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# Classifying Bundles; Fixed Base and Fiber

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