Let $X$ be a space. The symmetric group $\Sigma_{n+1}$ acts on the function space $$ X^{\Delta^n} $$ of continuous maps from the standard $n$-simplex to $X$. The action is induced by permuting the vertices.

Let $\Sigma_{n+1}$ act on $\Bbb Z$ by means of the sign representation.

Then the singular $n$-cochains $$ S^n(X) := \text{map}(X^{\Delta^n},\Bbb Z) $$ inherits a $\Sigma_{n+1}$-action given by conjugating functions.

**Definition.**
The *skew $n$-cochains* on $X$ is given by the invariants
$$
S^n(X)^{\Sigma_{n+1}}
$$
that is, by equivariant functions $X^{\Delta^n} \to \Bbb Z$.

Then an elementary calculation shows that the usual singular coboundary operator $\delta$ maps $S^n(X)^{\Sigma_{n+1}}$ into $S^{n+1}(X)^{\Sigma_{n+2}}$.

So we get a cochain complex:
$$
S^0(X) \overset \delta \to \cdots \overset \delta \to S^n(X)^{\Sigma_{n+1}} \overset \delta \to S^{n+1}(X)^{\Sigma_{n+2}} \overset \delta \to \cdots
$$
Define the *skew cohomology* of $X$ to be the cohomology of this cochain complex.

**Questions:** *What is it? What properties does it have? Has it ever before been studied?*