Hi, everyone:(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Classifying Bundles; Fixed Base and Fiber

**Physics Forums | Science Articles, Homework Help, Discussion**