Approximating topological metrics by Riemannian metrics

We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. More generally, if f : M → X f:M \to X is a continuous surjection from a smooth manifold to a compact metric space with f − 1 ( x ) {f^{ - 1}}(x) connected for every x ∈ X x \in X , then there is a metric d on X and a sequence of Riemannian metrics { ψ i } \{ {\psi _i}\} on M so that ( M , ψ i ) (M,{\psi _i}) converges to (X, d) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function ρ \rho and a sequence of Riemannian manifolds with ρ \rho as contractibility function so that lim ( M , ψ i ) \lim (M,{\psi _i}) is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds M and N and a contractibility function ρ \rho so that for every ε > 0 \varepsilon > 0 there are Riemannian metrics ϕ ε {\phi _\varepsilon } and ψ ε {\psi _\varepsilon } on M and N so that ( M , ϕ ε ) (M,{\phi _\varepsilon }) and ( N , ψ ε ) (N,{\psi _\varepsilon }) have contractibility function ρ \rho and d G H ( ( M , ϕ ε ) , ( N , ψ ε ) ) > ε {d_{GH}}((M,{\phi _\varepsilon }),(N,{\psi _\varepsilon })) > \varepsilon .