The discussion centers on the properties of fiber bundles, specifically the homeomorphism between the preimage of a point in the base space under a continuous map and the fiber. Participants explore the implications of local trivializations and the conditions under which these homeomorphisms hold, raising questions about the nature of neighborhoods in the base space and their relationship to the total space.
Participants do not reach consensus on several points, including the nature of neighborhoods in the base space and the properties of charts in topological manifolds. Multiple competing views remain regarding the conditions under which homeomorphisms hold in fiber bundles.
Limitations include unresolved assumptions about the nature of neighborhoods in the base space and the implications of disconnected components on the homeomorphic properties of fiber bundles.
They're only guanteed to be locally , only at times globally trivial. IIRC, the latter is true only for product spaces. The term " Trivial Bundle" wouldn't have been coined.cianfa72 said:A general question on tangent and cotangent bundle. Are they always trivial bundle regardless of the base space topological manifold ?
Yes, I'm keeping study Tu's book.jbergman said:Have you systematically worked through a book like Tu's Introduction to Manifolds?
Isomorphisms of the bundles not ##M##.cianfa72 said:Yes, I'm keeping study Tu's book.
In particular at the end of section 12.3 he talks of isomorphism between vector bundles over the same manifold ##\mathbb M##. But the manifold ##\mathbb M## in general may not have a vector space structure, so why he talks of isomorphisms and non just diffeomorphism ? Thanks.
However, in general, bundles do not have a vector space structure.martinbn said:Isomorphisms of the bundles not ##M##.
Maybe the manifolds themselves only have a topological structure, not a differentiable one?cianfa72 said:Yes, I'm keeping study Tu's book.
In particular at the end of section 12.3 he talks of isomorphism between vector bundles over the same manifold ##\mathbb M##. But the manifold ##\mathbb M## in general may not have a vector space structure, so why he talks of isomorphisms and non just diffeomorphism ? Thanks.
Yes, in that case (topological manifolds alone) the isomorphism is actually an homeomorphism.WWGD said:Maybe the manifolds themselves only have a topological structure, not a differentiable one?
The cylinder and the mobius strip are two different vector bundles over the base manifold of the circle. The cylinder is a trivial bundle and the Mobius strip is not. They are not isomorphic. I would work through Lee's chapter on vector bundles because it goes more in depth.cianfa72 said:Yes, I'm keeping study Tu's book.
In particular at the end of section 12.3 he talks of isomorphism between vector bundles over the same manifold ##\mathbb M##. But the manifold ##\mathbb M## in general may not have a vector space structure, so why he talks of isomorphisms and non just diffeomorphism ? Thanks.
You may want to look up some more into Stieffel-Whitney classes.cianfa72 said:Yes, in that case (topological manifolds alone) the isomorphism is actually an homeomorphism.
Btw up to dimension 3, every topological manifold admits a differentiable structure and all these structures are equivalent.
Yes, the initial topology will contain whole fibers. The bundle topology also has open subsets of fibers.cianfa72 said:As far as I can tell, the topology assigned on the tangent bundle is not the same as the initial topology from the canonical projection map ##\pi## on the first factor (see Tu's book section 12.1).
Yes, open sets in initial topology contain whole fibers, however each fiber ##\pi^{-1} \{p \}## will be open in the initial topology iff the base space B has the discrete topology.martinbn said:Why!
No, i meant why do you make these statements/questions?cianfa72 said:Yes, open sets in initial topology contain whole fibers, however each fiber ##\pi^{-1} \{p \}## will be open in the initial topology iff the base space B has the discrete topology.
Just to double check my understanding (also sometimes I've problem with reading english).martinbn said:No, i meant why do you make these statements/questions?