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.
 A: This is not a full answer, but there is a once-well-known technique in geometric topology that could help clarify this result.  Kirby and Siebenmann, in their old work on triangulability of manifolds, have a technique called handle straightening.  It is a very nice way to push around pieces of a diffeomorphism without disturbing other pieces.
To use this technique, give the manifold M a handle decomposition (which in your case should refine your open cover) and collar all of the handles.  Then you can isotop the 0-handles with without disturbing the 1-handles.  How?  You should shrink back the higher handles along the collars of the 0-handles, then isotop the 0-handles in a region that is that contains the original 0-handles but is disjoint from the trimmed higher handles.  Then you can extend the higher handles back again so that they attach to the 0-handles.  You can repeat this for the k-handles in turn, provided that the requested isotopy of the k-handles is the identity in a neighborhood of the (k-1)-skeleton.
I don't know if your whole result is standard, but this part, if you can use it in your construction, is standard.
