Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules.
In studying objects related to Wall’s D2 problem on CW-complexes I have encountered a situation in which a particular set of stably-free $\mathbb{Z}G$-modules, those that are “geometrically realisable”, happens to also form a submonoid of $M_G$, let it be called $N_G$.
From this result I would like to in some way understand the extent to which $M_G=N_G$ fails, noting that without the additional monoidal structure of $N_G$ I expect it is far harder to make this a well defined notion.
At first I naively hoped that the quotient of monoids $M_G/N_G$ could be a useful “measure” for the failure of this property and would contain some interesting information. However, soon after I noticed that my particular submonoid $N_G$ contains the free module $\mathbb{Z}G$, and hence the above quotient is trivial as $M_G$ consists of only stably-free modules.
Are there any methods I could employ to “measure” the extent to which $M_G=N_G$ fails, hopefully in the form of some algebraic object constructed from these two monoids as a sophisticated alternative to the quotient construction. In particular an invariant of $G$ that vanishes if and only if $M_G=N_G$ would be of interest.
