more generally, the assignment formula_24 actually does not map to formula_25 generally. since formula_5 is equivariant, if formula_27(the isotropy subgroup), then by equivariance, we have that formula_28, so in fact formula_5 will map to the collection of formula_30. in this case, one can replace the bundle by a homotopy quotient where formula_2 acts freely and is bundle homotopic to the induced bundle on formula_3 by formula_5.