Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of integers on $X$.
In proving this, I have seen several sources proceed more or less the same way:
- Let $\tilde{\mathcal{S}}^k$ be the sheafification of the presheaf $\mathcal{S}^k$ of singular cochains on $X$. Then $\underline{\mathbb{Z}} \to \tilde{\mathcal{S}}^0 \to \tilde{\mathcal{S}}^1 \to \dotsb$ is a complex of sheaves. It is exact because $X$ is locally contractible. The sheaves $\tilde{\mathcal{S}}^k$ can be shown to be flabby, and so acyclic. Therefore the homology of the complex $\tilde{\mathcal{S}}^\bullet(X)$ is just $H^\bullet (X, \underline{\mathbb{Z}})$.
- For each $k$, $\tilde{\mathcal{S}}^k(X) \cong \mathcal{S}^k(X)/\mathcal{S}^k(X)_0$, where $\mathcal{S}^k(X)_0 \subset \mathcal{S}^k(X)$ is those cochains $\sigma$ for which there exists an open cover $\mathcal{U}$ of $X$ so that $\sigma(s) = 0$ whenever $s$ is a singular simplex that lies completely in one of the sets of $\mathcal{U}$.
- $\mathcal{S}^\bullet(X) \to \mathcal{S}^\bullet(X)/\mathcal{S}^\bullet(X)_0$ is a homology equivalence.
It is the second step where I am finding trouble**. Consider the sheafification map $\mathcal{S}^k \xrightarrow{\mathrm{shf}} \tilde{\mathcal{S}}^k$. I am comfortable with the fact that the kernel of $\mathcal{S}^k(X) \xrightarrow{\mathrm{shf}_X} \tilde{\mathcal{S}}^k(X)$ is $\mathcal{S}^k(X)_0$. But why should the map be surjective?
Possible Reasons:
- The sheafification map is surjective on global sections if $X$ is paracompact and the presheaf satisfies the gluing axiom (proof here). Certainly $\mathcal{S}^k$ satisfies the gluing axiom. But $X$ is merely locally contractible, which doesn't imply paracompact. (A counterexample is a topological sum of uncountably many copies of $\mathbb{R}$.) (Edit: Eric Wofsey points out that this is not the right counterexample to choose. The long line works.)
- Consider the proof offered below (original source):
I don't see how they conclude that $\tilde{\beta}_i(\alpha) = \tilde{\beta}_j(\alpha)$. We know that $\tilde{\beta}_i$ and $\tilde{\beta}_j$ have the same germs on $V_i \cap V_j$, but that doesn't imply they're the same cochain (because $\mathcal{S}^k$ isn't a monopresheaf).
So far I haven't been able to figure it out. Anyone have any ideas?
**Although in showing that $\tilde{\mathcal{S}}^k$ are flabby, it is assumed that the sheafification map is surjective, and that's what I have a problem with in part 2.