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There is the general construct of a fibre bundle induced by a topological group action. Yet, one of the distinctive differences between this notion and the notion of a vector bundle is that the base space of the former need not be locally Euclidean.

It seems preferable to have a generalisation that would allow for something like a $C^{-1}$-manifold, ie. a manifold that is not locally Euclidean at all, in order to have statements that encompass both the case of vector bundles and the case of general $(F,G)$-bundles (where $F$ is the local space, ie. the analogue of the vector space of the vector bundle, an $G$ is the group).

Does such a concept exist? If so, what is its name?

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  • $\begingroup$ An example would be any statement such as the one associated to the so-called "clutching construction", where one takes a clutching map and makes a vector bundle out of it. Here, if one works with differentiable manifolds, one has to keep track of differentiability; otherwise, this issue is not important. If one had a concept of different degrees of differentiability including continuity and total absence of Euclidean structure, then one could unify the construction. $\endgroup$
    – Cloudscape
    Feb 20, 2019 at 20:49
  • $\begingroup$ I don't think I understand the question. A topological vector bundle can be defined over any topological space. If it is defined over a manifold, it can be smooth (but does not have to be). Also, what is a manifold if not locally Euclidean? $\endgroup$
    – Sebastian Goette
    Feb 21, 2019 at 21:41
  • $\begingroup$ @SebastianGoette "what is a manifold if not locally Euclidean?" - exactly. All the computations that need the object to be locally Euclidean are as yet specific to the case where the base space is at least a manifold; often $G$ is a Lie group and $F$ is a manifold, such as a vector space (in the case of vector bundles). I'm trying to find a concept that unifies all these cases, thus making it possible to treat them uniformly. (This may necessitate the use of the concept of vacuous truth at some places.) I could come up with such a concept, but I'd prefer to use an existent one. $\endgroup$
    – Cloudscape
    Feb 22, 2019 at 14:45

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