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Let $F \to E \to B$ be a topological fibre bundle with fibre $F$ and base $B$. It can be characterized by a map $B \to BAut(F)$. If it can also be characterized as a map $B \to BG$ (or say $G$ is a structure group of the bundle), then we say that $G$ is a symmetry of this bundle.

Most often, such $G$ is not unique. For example, if $F$ is a vector space, then such $G$ could be $GL(V)$. And if $B$ happens to be a Riemannian manifold, then $O(V)$ is also a symmetry. In an extreme case when the bundle is trivial, the trivial group $\{e\}$ is also a symmetry.

Question

  1. Given a bundle, what's the limit group of all symmetries, if exists? How about the limit of all symmetries that are subgroups of $Aut(F)$?

  2. Call the limit group, the "minimal symmetry" of the bundle. Then, fix a group $G$, how to characterize all bundles with fibre $F$ and base $B$ that has the minimal symmetry $G$?

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    $\begingroup$ If $B$ is connected then simplicial methods give a topological group $G$ of homotopy type $\Omega B$, and $B\simeq BG$. If you make your question precise in a homotopically meaningful way then you will surely end up with $\Omega B$ as the minimal symmetry group. $\endgroup$
    – Neil Strickland
    Commented Apr 14, 2021 at 9:41
  • $\begingroup$ @NeilStrickland but the minimal symmetry group depends not only on the base $B$. $\Omega B$ can't be the answer. $\endgroup$
    – Student
    Commented Apr 14, 2021 at 12:32
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    $\begingroup$ If you don't think that's right, you should spell out in more detail what you mean by "can be characterized as a map $B\to BG$". $\endgroup$
    – Neil Strickland
    Commented Apr 14, 2021 at 12:55
  • $\begingroup$ Thanks for your comment! I meant that $G$ is a structure group of the bundle. Edited. $\endgroup$
    – Student
    Commented Apr 14, 2021 at 13:05
  • $\begingroup$ That is not a remotely sufficient definition to support the rest of your question. For example, you want to take the limit of all symmetry groups, so you need to give a definition which gives rise to well-defined transition maps between different symmetry groups so that you can form a diagram and take the limit. And you should note that a diagram in the homotopy category is insufficient, you need a strictly commuting diagram of topological groups, or corresponding $\infty$-categorical data. $\endgroup$
    – Neil Strickland
    Commented Apr 14, 2021 at 13:22

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