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 $$ S^n(X)^{\Sigma_n} $$ 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?

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?

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?

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 $$ S^n(X)^{\Sigma_n} $$ 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?