Equivariant approximate fibrations

Abstract

The theory of approximate fibrations is extended to an equivariant setting. Equivariant approximate fibrations are characterized by considering the maps on the fixed point sets. The theory is applied to equivariant fibrations over the circle.

Dihedral manifold approximate fibrations over the circle

“…Hence ρ r : R n → R n is a G-εhomotopy equivalence. Therefore, by (b) =⇒ (c) in [26,Theorem 3.4], we conclude that ρ r is a G-approximate fibration.…”

“…Hence ρ pF is µ-close to F. In any case, we have shown ρ pF is µ-close to F. Thus ρ p : M → R n is an approximate G-fibration for the class of compact, metric Gspaces. Therefore, by a result of S. Prassidis [26,Prop. 2.18], we conclude that ρ p is an approximate G-fibration for the class of all G-spaces.…”

“…The notion of (ε, ν)-fibration is due to Hughes [19]. See Prassidis [26] for a more complete treatment.…”

“…2.1], both the finite-dimensional spaces W /J 1 and S 1 are C 2 -ENRs, hence they are C 2 -ANRs for the class of separable metric spaces. Therefore, by Prassidis's recognition principle [26,Thm. 3.1], we conclude that p/J 1 is a C 2 -approximate fibration.…”

“…26 The Finiteness condition above says in the first place that the set {X b | b ∈ B} of non-isomorphic polyhedra that are links of barycenters is finite. In the second place, it says that for any given polyhedron X occurring as a link and any 0 ≤ k ≤ n, even though there might be infinitely many different open embeddings given of• c(X) × R k into B, there are only finitely many different induced metrics on • c(X) × R k .…”