The min-max theory for minimal surface is developed for the area functional on the space of cycle $Z_n(M)$, producing an unstable minimal surface with area equal to the width. Of course, this is powerful when we are in a manifold with trivial homology (most of the papers on the subject considered this case). On a closed $n+1$-dimensional manifold $M$ with non-trivial $H_n$, we can minimize the area inside a non-trivial homology class to obtain an area-minimizing hypersurface.

My question is, do people do min-max on manifold with non-trivial homology too? Like, if we only consider sweep-outs into a fixed homology class $[\alpha]\neq 0$: $\phi: X\rightarrow [\alpha] \subset Z_n(M)$ and then define the width in the similar manner, can we find an unstable minimal hypersurface in $[\alpha]$, providing that the width is greater than the area of the minimizing one in $[\alpha]$?

If this is known, is there a case where we can find the exact width of a homology class $\alpha$? Appreciate any lead or literature on this.