Equivariant bundle

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In geometry and topology, given a group G (which may be a topological or Lie group), an equivariant bundle is a fiber bundle π : E → B {\displaystyle \pi \colon E\to B} {\displaystyle \pi \colon E\to B} such that the total space E {\displaystyle E} {\displaystyle E} and the base space B {\displaystyle B} {\displaystyle B} are both G-spaces (continuous or smooth, depending on the setting) and the projection map π {\displaystyle \pi } {\displaystyle \pi } between them is equivariant: π ∘ g = g ∘ π {\displaystyle \pi \circ g=g\circ \pi } {\displaystyle \pi \circ g=g\circ \pi } with some extra requirement depending on a typical fiber.

For example, an equivariant vector bundle is an equivariant bundle such that the action of G restricts to a linear isomorphism between fibres.

References

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