Consider a positively weighted sphere system \(S\) in \(M_n = \#_n S^1 \times S^2\). By PoincarĂ© duality, we can consider \([S] \in H_2(M_n)\) as an element of \(H^1(M_n) = H^1(F_n)\) (where \(F_n\) is the free group of rank \(n\)). Fix some \(\alpha \in H^1(F_n)\). Then we can define the *cocycle complex* \(X_\alpha\) to be the collection of weighted sphere systems representing \(\alpha\) under this identification.

As for the cycle complex, the question now becomes: is the cocycle complex contractible?