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clarify "can be characterized as a map"
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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 $G$?

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 $B \to BG$, 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 $G$?

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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Student
Student
  • 5.2k
  • 11
  • 33

Minimal symmetry of a fibre bundle

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 $B \to BG$, 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 $G$?