# Adapting families of diffeomorphisms to an open cover

Has anyone seen the following result in the literature? I've asked a few experts but so far I've come up with nothing.

Given a manifold M and an open cover {U_i} of M, we want to see how families of diffeomorphisms of M can be adapted to {U_i}. We will think of families of diffeomorphisms as generators of C_*(Diff(M)), where C_*() denotes singular chains.

Def: A k-parameter family of diffeomorphisms f: P^k \times M -> M is supported on V \subset M if, for all y not in V, we have f(p, y) = f(q, y) for all p, q \in P. In other words, f is independent of the parameters P outside of V.

Define A_k \subset C_k(Diff(M)) to be the subcomplex generated by all k-parameter families (k-chains) of diffeomorphisms f such that f is supported on a union of at most k of the U_i's, and such that (inductively) the boundary of f is in A_{k-1}.

Claim: A_* is homotopy equivalent to C_*(Diff(M)).

There is a similar result if we replace Diff(M) with Maps(M -> T), where T is some topological space. It is used in the proof of the claim in this question.

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Maybe it's worth saying that we know the result is true, because we've written down a long and complicated proof, but that we'd be much happier if someone else has already done this or if there's a better explanation. –  Scott Morrison Oct 23 '09 at 16:13