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LSpice
LSpice
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Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map $$ c(a_1,a_2) \longmapsto \frac{c(a_1,a_2)}{c(a_2,a_1)}$$ induces a surjective homomorphism $H^2(A,\mathbb{C}^\times) \to \bigwedge^2(A,\mathbb{C}^\times)$, where $\bigwedge^2(A,\mathbb{C}^\times)$ is the group of alternating bihomomorphisms on $G$. If I remember correctly, this is even an isomorphism, but I couldn't find the statement anymore.

Now, let $d \in Z^3(A,\mathbb{C}^\times)$ be a 3-cocycle on $A$. We can now define a higher skew-map $$d(a_1,a_2,a_3) \longmapsto \prod_{\sigma \in S_3} d(a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)})^{sgn(\sigma)}.$$$$d(a_1,a_2,a_3) \longmapsto \prod_{\sigma \in S_3} d(a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)})^{\operatorname{sgn}(\sigma)}.$$ Again, this induces a homomorphism $H^3(A,\mathbb{C}^\times) \to \bigwedge^3(A,\mathbb{C}^\times)$, where $\bigwedge^3(A,\mathbb{C}^\times)$ is now the group of alternating trihomomorphisms. By alternating trihomomorphisms I mean homomorphisms $t:A \otimes A \otimes A \to \mathbb{C}^\times$, s.t. $t(a,a,b)=1=t(a,b,b)$ for all $a,b \in A$. I didn't check, but I believe this is possible for higher $n$-cocycles as well with the obvious definition of a higher skew map.

I believe this is all known, but I couldn't find it anywhere. In particular, I would like to know if the induced homomorphisms are again surjective (or even bijective).

Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map $$ c(a_1,a_2) \longmapsto \frac{c(a_1,a_2)}{c(a_2,a_1)}$$ induces a surjective homomorphism $H^2(A,\mathbb{C}^\times) \to \bigwedge^2(A,\mathbb{C}^\times)$, where $\bigwedge^2(A,\mathbb{C}^\times)$ is the group of alternating bihomomorphisms on $G$. If I remember correctly, this is even an isomorphism, but I couldn't find the statement anymore.

Now, let $d \in Z^3(A,\mathbb{C}^\times)$ be a 3-cocycle on $A$. We can now define a higher skew-map $$d(a_1,a_2,a_3) \longmapsto \prod_{\sigma \in S_3} d(a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)})^{sgn(\sigma)}.$$ Again, this induces a homomorphism $H^3(A,\mathbb{C}^\times) \to \bigwedge^3(A,\mathbb{C}^\times)$, where $\bigwedge^3(A,\mathbb{C}^\times)$ is now the group of alternating trihomomorphisms. By alternating trihomomorphisms I mean homomorphisms $t:A \otimes A \otimes A \to \mathbb{C}^\times$, s.t. $t(a,a,b)=1=t(a,b,b)$ for all $a,b \in A$. I didn't check, but I believe this is possible for higher $n$-cocycles as well with the obvious definition of a higher skew map.

I believe this is all known, but I couldn't find it anywhere. In particular, I would like to know if the induced homomorphisms are again surjective (or even bijective).

Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map $$ c(a_1,a_2) \longmapsto \frac{c(a_1,a_2)}{c(a_2,a_1)}$$ induces a surjective homomorphism $H^2(A,\mathbb{C}^\times) \to \bigwedge^2(A,\mathbb{C}^\times)$, where $\bigwedge^2(A,\mathbb{C}^\times)$ is the group of alternating bihomomorphisms on $G$. If I remember correctly, this is even an isomorphism, but I couldn't find the statement anymore.

Now, let $d \in Z^3(A,\mathbb{C}^\times)$ be a 3-cocycle on $A$. We can now define a higher skew-map $$d(a_1,a_2,a_3) \longmapsto \prod_{\sigma \in S_3} d(a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)})^{\operatorname{sgn}(\sigma)}.$$ Again, this induces a homomorphism $H^3(A,\mathbb{C}^\times) \to \bigwedge^3(A,\mathbb{C}^\times)$, where $\bigwedge^3(A,\mathbb{C}^\times)$ is now the group of alternating trihomomorphisms. By alternating trihomomorphisms I mean homomorphisms $t:A \otimes A \otimes A \to \mathbb{C}^\times$, s.t. $t(a,a,b)=1=t(a,b,b)$ for all $a,b \in A$. I didn't check, but I believe this is possible for higher $n$-cocycles as well with the obvious definition of a higher skew map.

I believe this is all known, but I couldn't find it anywhere. In particular, I would like to know if the induced homomorphisms are again surjective (or even bijective).

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Bipolar Minds
Bipolar Minds
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Alternating $n$-homomorphism on abelian group is skew of $n$-cocycle

Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map $$ c(a_1,a_2) \longmapsto \frac{c(a_1,a_2)}{c(a_2,a_1)}$$ induces a surjective homomorphism $H^2(A,\mathbb{C}^\times) \to \bigwedge^2(A,\mathbb{C}^\times)$, where $\bigwedge^2(A,\mathbb{C}^\times)$ is the group of alternating bihomomorphisms on $G$. If I remember correctly, this is even an isomorphism, but I couldn't find the statement anymore.

Now, let $d \in Z^3(A,\mathbb{C}^\times)$ be a 3-cocycle on $A$. We can now define a higher skew-map $$d(a_1,a_2,a_3) \longmapsto \prod_{\sigma \in S_3} d(a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)})^{sgn(\sigma)}.$$ Again, this induces a homomorphism $H^3(A,\mathbb{C}^\times) \to \bigwedge^3(A,\mathbb{C}^\times)$, where $\bigwedge^3(A,\mathbb{C}^\times)$ is now the group of alternating trihomomorphisms. By alternating trihomomorphisms I mean homomorphisms $t:A \otimes A \otimes A \to \mathbb{C}^\times$, s.t. $t(a,a,b)=1=t(a,b,b)$ for all $a,b \in A$. I didn't check, but I believe this is possible for higher $n$-cocycles as well with the obvious definition of a higher skew map.

I believe this is all known, but I couldn't find it anywhere. In particular, I would like to know if the induced homomorphisms are again surjective (or even bijective).