Equivariant bundle

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a> and <a href="/facts/Topology/2EqW47cU">topology</a>, given a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G (which may be a topological or Lie group), an equivariant bundle is a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a> 
 
 
 
 π
 :
 E
 →
 B
 
 
 {\displaystyle \pi \colon E\to B}
 
 such that the total space 
 
 
 
 E
 
 
 {\displaystyle E}
 
 and the base space 
 
 
 
 B
 
 
 {\displaystyle B}
 
 are both <a href="/facts/Lie_group_action/IIlu4h7M">G-spaces</a> (continuous or smooth, depending on the setting) and the projection map 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 between them is equivariant: 
 
 
 
 π
 ∘
 g
 =
 g
 ∘
 π
 
 
 {\displaystyle \pi \circ g=g\circ \pi }
 
 with some extra requirement depending on a typical fiber.
For example, an <a href="/facts/Equivariant_vector_bundle/sAItdnmQ">equivariant vector bundle</a> is an equivariant bundle such that the action of G restricts to a linear isomorphism between fibres.

<ul><li>Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag</li></ul>

Equivariant bundle open-in-new

Equivariant bundle