# A new problem

Consider a positively weighted sphere system $$S$$ in $$M_n = \#_n S^1 \times S^2$$. By Poincaré 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?