*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.*




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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?



$\mathbb Z$-formality of spheres
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 $\def\H{{\rm H}} \H(S^n,\Z)$.


