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 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 
$$
S^n(X)^{\Sigma_n}
$$
that is, by equivariant functions $X^{\Delta^n} \to \Bbb Z$.


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

So we get a cochain complex:
$$
S^0(X) \overset \delta \to \cdots \overset \delta \to S^n(X)^{\Sigma_n} \overset \delta \to S^{n+1}(X)^{\Sigma_{n+1}} \overset \delta \to \cdots
$$
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?*