Prove that singular homology groups $H_k(M_{diff}, \mathbb{Z})$ and $H_k(M, \mathbb{Z})$ are isomorphic

Question

I'm trying to complete the proof from David Mumford's Algebraic Geometry I, Complex Projective Varieties that the singular homology groups $H_k(M_{diff}, \mathbb{Z})$ and $H_k(M, \mathbb{Z})$ for a $C^{\infty}$-manifold $M$ are isomorphic at page 91. About notations:

The singular homology groups are defined as follows: Let $\Delta^k = $ convex hull in $\mathbb{R}^{k+1}$ of unit vectors $u_i = (0, ..., 1,..., 0) \}$

$\epsilon_i: \Delta^{k-1} \to \Delta^k = $ the linear map talking the vertices $u_0,..., u_{k-1} $ of $ \Delta^{k-1} $ to the vertices $u_0,...,\hat{u_i},..., u_{k-1} $ of $ \Delta^{k}$

$C_k(M) = $ group of linear combinations $\sum n_i f_i, n_i \in \mathbb{Z}, f_i$ continuous maps from $\Delta^k$ to $M$.

The boundary maps $\partial_k: C_k(M) \to C_{k-1}(M) =$ linear map defined on the generators $f$ of $C_k(M)$ by

$$ \partial_k f = \sum_{i=0}^k (-1)^i f \circ \epsilon_i $$

And finially $H_k(M, \mathbb{Z}) = Ker(\partial_k) / Im(\partial_{k+1}) $

The differentiable chains considered as subgroup $C_k(M_{diff}) \subset C_k(M)$ are $\mathbb{Z}$-generated by the $C^{\infty}$ (= analytic) maps $f: \Delta^k to M$. As before, we can form $Ker(\partial) / Im(\partial)$ on these smaller chain groups the obtain the map $H_k(M_{diff}, \mathbb{Z}) \to H_k(M, \mathbb{Z})$ which should be verified to be an isomorphism.

To prove it Mumford suggested to do $5$ steps:

1. introduce a covering $\{U_{\alpha} \}$ such that all intersections $U_S = U_{\alpha_1} \cap ... \cap U_{\alpha_k} $ are diffeomorphic to convex subsets of $\mathbb{R}^n $

2. Let $C^*(M_{diff}), C^*(M)$ be subcomplexes generated by functions/simplices $f$ such that $Im(f) \subset U_{\alpha}$ for some $\alpha$

3. show that $H_k(C^*(M_{diff}) \simeq H_k(C(M_{diff})$ and $H_k(C^*(M) \simeq H_k(C(M)$.

4. Show then that for all $(\alpha_1, ...,\alpha_k)$, $H_k(C(M_{S, diff}) \simeq H_k(C(M_S) =0$ if $k >0$ and $= \mathbb{Z}$ if $k=0$.

5. Show that the qotient complex $C^*(M)/C^*(M_{diff})$ with carriers $C^*(M_S)/C^*(M_{S, diff})$ is acyclic by the lemma of acyclic carriers

The lemma which I not want to quote here immediately gives to result, but I have a lot of problems to show some claims from 3) and 4):

Problems: Firstly for every $k$ we obviously have canonical inclusions $C_k^*(M) \subset C_k(M)$ (and the same for diffs). And they are compatoble with boundary respectively restricted boundary maps, therefore we obtain moreover an inclusion $C_{\bullet}^*(M) \subset C_{\bullet}(M)$ of complexes. But I don't see how to show that after passing to homology groups they become isomorphisms $H_k(C^*(M) \simeq H_k(C(M)$. It seems that this can be proved by a so called 'subdivision argument' which I wish to discuss here.

Secondly I not know how why we need the exotic assumption that all intersections $U_S = U_{\alpha_1} \cap ... \cap U_{\alpha_k} $ are diffeomorphic to convex subsets of $\mathbb{R}^n $? With viewpoint to Step 4) in order to assure that the $U_S$ should have homology groups of a point set: isn't to require that all intersections $U_S$ are just differomorphic contractable.

Thirdly: Why $C^*(M_S)/C^*(M_{S, diff})$ carries $C^*(M_S)/C^*(M_{S, diff})$?