# Group of surface homeomorphisms is locally path-connected

I think the following is true and I need a reference for the proof. (Given a closed surface $$S$$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $$S$$ with a distance generating its topology, and endow the set of self-homeomorphisms of $$S$$ with the distance max(uniform distance between two maps, uniform distance between their inverses)).

For any $$\varepsilon>0$$ there exists $$\eta>0$$ such that for any self-homeomorphism $$f$$ of $$S$$ whose distance to the identity is $$<\eta$$, there is an isotopy $$t\mapsto f_t$$ from $$f$$ to the identity that stays at distance $$<\varepsilon$$ to the identity.

I'd be interested in a reference and/or a hint of the proof.

Note: Unless I'm too tired and got it wrong, this result implies (and is implied by the fact) that the group of self-homeomorphisms of $$S$$ is locally path-connected (in the sense that every point has a path-connected neighbourhood, not necessarily open), whence the title of this question.

In the particular case of surfaces, I found the following reference which includes a proof that is not too complicated: Regular Mappings and the Space of Homeomorphisms on 2-Manifolds by Hamstrom and Dyer. They prove local contractibility, which is more than I asked. It works for surfaces with or without boundaries and includes a slight generalization with fixed points on the boundary. This is Theorem 1 in this article. The proof fits in 6 pages, is a bit heavy on notations but this remains managable. Unfortunately there is no figure. The proof uses conformal maps for a couple of lemmas, Alexander's trick, and a technique due to J.H. Roberts (Local arcwise connectivity in the space $$H^n$$ of homeomorphisms of $$S^n$$ onto itself, Summary of Lectures, Summer Institute on Set Theoretic Topology, Madison, Wisconsin, 1955, p. 100) but I cannot find the corresponding reference. They also cite a German article of Kneser (Die Deformationssätze der einfach zusammenhägenden Flächen, Mathematische Zeitschrift, Vol. 25(1926), pp. 362-372) as a source of inspiration, but my knowledge of German is very basic so reading it would represent quite an investment.