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.