- #1
Bacle
- 662
- 1
Hi, everyone:
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.
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.