Homotopy equivalences and mapping torus projections

Equivariant approximate fibrations

“…Then for each subgroup H of G the map / H : X H -> X H is a homotopy equivalence and therefore it induces a bijection between path components and for each path component X% the restriction of /, f^ is a homotopy equivalence. By [12], Theorem A, the map T(f") -> S 1 is an approximate fibration. Therefore, for each subgroup H of G, p H : T(f H ) -> S 1 is an approximate fibration.…”

“…A G-map p : E -> B can be stably approximated by a locally trivial G-bundle if the map E x Q G -^ E ~-^ Β can be approximated by a locally trivial G-bundle. The following is the equivariant analogue of [12], Theorem B. Proof.…”

“…Let p be a Serre G-fibration. Then / is necessarily onto and by Proposition 4.1 in [12], p is an open map. By Proposition 4.8 above / is a G-£/F°°-mapping.…”

“…: T(f) -»S 1 where (/) is the mapping torus of /. We extend the results of [12] to show that p is an approximate G-fibration if and only if / is a G-homotopy equivalence and/? is a Hurewicz G-fibration if and only if / is a G-CE map.…”

“…First of allnotice that/isasurjectivemap by [12], Proposition 4.1. Let^bean element of A'with isotropy group G x and U an open G x -neighborhood of/" 1 .*.…”