Suppose a group G acts on a chain complex K and induced action on H(K) is trivial. What "secondary operations" on H(K) can be defined in this situation?




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**Example.** If $G=\langle\sigma\rangle/\sigma^n$ acts trivially on H(K) then $x-\sigma x=dl(x)$ (for some function $l$) and a secondary operation $x\mapsto l(x)+\sigma l(x)+\dots+\sigma^{n-1}l(x)$ is well-defined mod n. And this operation is non-trivial (consider a complex $Z[G]\to Z[G]$, $x\mapsto (1-\sigma)x$).


So looks like these operations has something to do with group homology, but details elude me.




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**Update.** [Two](https://mathoverflow.net/questions/62016/secondary-operations-for-a-group-acting-on-a-chain-complex/62037#62037) nice [answers](https://mathoverflow.net/questions/62016/secondary-operations-for-a-group-acting-on-a-chain-complex/62036#62036) explain what is the meaning of the operation from the example above (and how it can be defined for an arbitrary group).


But does this construction give *all* operations? I.e. what structure on H(K) one needs to recover K (up to q/iso)? Like,


* associative multiplication on K $\Leftrightarrow$ $A\_\infty$-structure on H(K);
* G-action on K $\Leftrightarrow$ ??? on H(K).


(Perhaps, there is a very general answer: not just for k[G] but for an arbitrary algebra — or even arbitrary operad, maybe. Probably, [Tyler Lawson](https://mathoverflow.net/users/360/tyler-lawson)'s [comment](https://mathoverflow.net/questions/62016/secondary-operations-for-a-group-acting-on-a-chain-complex/62036#62036) is relevant — if somebody could elaborate on that...)



"Secondary operations" for a group acting on a chain complex
One family of secondary operations that arise come from trying to find the difference between "classes that look like they are acted on trivially" and "classes that are genuinely (or coherently) acted on trivially".


Let $\cdots \to F\_2 \to F\_1 \to F\_0 \to \mathbb{Z} \to 0$ be a free resolution of $\mathbb{Z}$. We get a sequence of chain complexes $0 = C\_{-1} \subset C\_0 \subset C\_1 \subset C\_2 \subset \cdots$ where 
$$C\_i = \cdots \to 0 \to F\_i \to F\_{i-1} \to \cdots \to F\_1 \to F\_0 \to 0$$
and we can look at mapping complexes of morphisms out:
$$\cdots \to Hom\_{\mathbb{Z}[G]}(C\_2, K) \to Hom\_{\mathbb{Z}[G]}(C\_1, K) \to Hom\_{\mathbb{Z}[G]}(C\_0, K) \to 0$$
This gives us a filtered chain complex and there's an associated spectral sequence.


The spectral sequence starts with
$$E\_1^{p,q} = Hom(F\_p, H^q(K))$$
and has next term
$$E\_2^{p,q} = H^p(G, H^q(K)).$$
In particular, $E\_2^{0,q}$ consists of the elements in $H\_q(K)$ which are fixed by the $G$-action.


From here, we get a "secondary" operation: the $d\_2$-differential $H^0(G, H^q(K)) \to H^2(G, H^{q-1}(K))$. For cyclic groups, you've described this on elements (modulo you need to be careful about how well-defined it is). The higher differentials give tertiary and higher operations. For a given element $x \in H^q(K)$, these measure a sequence of obstructions to finding a chain complex $L \sim K$ such that $x$ is represented in $L$ by a class honestly fixed by $G$.


(I cohomologically indexed $K$, but only because people tend to complain if I don't.)


