I have just read about commutative cochain problem(CCP) here and I'm trying understand it.
It states that you cannot turn(in nontrivial way) simplicial set $S$ to differential graded commutative algebra(DGCA) $\Lambda^*(S)$ which has the usual cohomology $H^*(S)$.
If I modify above problem slightly, is it still impossible?
Start wit simplicial complex $(S,\partial)$. Can we find finite chain complex $(T, \partial)$, such that $(S,\partial)\subset (T,\partial)$ and that we can turn $T$ into DGCA $(T,d,\wedge)$ such that $(S,d)$ is cochain subcomplex of $(T,d)$ and cohomolgy groups of $(S,d)$ are the same as of $(S,\partial)$?
Motivation: Sometimes it is useful to consider dual graph $S^*$ of $S$. As CCP states, it is impossible to construct graded commutative multiplication on cochains of $S$, but what if the product can mix chochains of $S$ and $S^*$?
edit: My main motivation is that I'm interested in discrete exterior calculus and I stumbled upon article Theoretical Limitations of Discrete Exterior Calculus in the Context of Computational Electromagnetics where they say that CCP is a problem, so I would like to understand how big the problem is.