$\mathbb Z$-formality of spheres A topological space $X$ is $\mathbb Z$-formal, if the singular cochain complex $C^*(X,\mathbb Z)$ is
quasi-isomorphic to $H^*(X, \mathbb Z)$ as an augmented differential graded ring.

It's quite simple to write down specific quasi-isomorphisms to show that the Spheres $S^n$ are $\mathbb Q$-formal spaces by fixing a volume form $v \in \Omega^n(S^n)$ and considering the maps $H^*(S^n)=\operatorname{span}(1,[v]) \to \Omega^*(S^n)$ sending $1$ to the $1$-form and $[v]$ to $v$
and the canonical map $C^*(S^n) \to \Omega^*$.
Is it also possible to show the $\mathbb Z$-formality of the Spheres $S^n$ by writing down specific quasi-isomorphisms, or is it easier to use another method for showing $\mathbb Z$-formality?
 A: Consider the simplicial set $\def\S{{\bf S}} \def\Sing{\mathop{\rm Sing}} \S^n=Δ^n/∂Δ^n$, which has exactly two nondegenerate simplices: a 0-simplex and an $n$-simplex.
Consider the map $\S^n→\Sing S^n$ that sends the only vertex of $\S^n$ to the given basepoint of $S^n$ and the only nondegenerate $n$-simplex of $\S^n$ to some singular $n$-simplex $Δ^n→S^n$ that sends the boundary to the basepoint and induces a degree 1 map once we mod out the boundary.
The map $\S^n→\Sing S^n$ is a simplicial weak equivalence.
Thus, the induced map on integral normalized simplicial cochains $\def\Z{{\bf Z}} \def\C{{\rm C}} \C(S^n,\Z)=\C(\Sing S^n,\Z)→\C(\S^n,\Z)$ is a quasi-isomorphism of augmented differential graded rings.
Observe that $\C(\S^n,\Z)$ is a differential graded ring
with exactly two generators: one in degree 0 and another one in degree $n$.  The differentials of both generators vanish.  Furthermore, the degree $n$ generator squares to 0 for dimensional reasons.
Thus, the normalized simplicial cochain ring $\C(\S^n,\Z)$ is precisely the graded cohomology ring $\def\H{{\rm H}} \H(S^n,\Z)$.
