Smooth integral chains compute singular homology

The proof of [F2] performs replacements on finitely many simplices and their faces, and its homotopy identity is a signed finite sum of faces and prisms. Every coefficient in that construction is $0$, $1$, or $-1$; no division or real scalar operation is used. The same replacement and prism formulas therefore define integral chains, commute with the integral boundary, and satisfy the identical chain-homotopy identity generator by generator.

Let $z$ be an ordinary integral cycle. Apply the finite smoothing construction of [F2] to its support and all faces. It yields a smooth integral cycle $Qz$ and a finite integral chain $Pz$ with $Qz-z=\mathrm{\partial }(Pz)$. Thus every ordinary integral homology class has a smooth integral representative.

Now let a smooth integral cycle $z$ bound an ordinary integral chain $b$. Apply the construction to the finite supports of $z$ and $b$, using the fixed-face clause on every already-smooth simplex of $z$. Then $Qz=z$ and $\mathrm{\partial }(Qb)=Q(\mathrm{\partial }b)=Qz=z$. Hence $z$ already bounds a smooth integral chain.

The inclusion is a chain map by the subcomplex statement [F1], so it induces a homomorphism on homology. Step 2.1 proves surjectivity and step 2.2 proves injectivity. Postcomposition by a smooth map commutes with inclusion on each simplex, so the induced isomorphism is natural. The empty manifold and negative degrees have zero chain groups on both sides.