The panted cobordism groups of cusped hyperbolic 3-manifolds

For any oriented cusped hyperbolic 3-manifold $M$, we study its $(R,\epsilon )$-panted cobordism group, which is the abelian group generated by $(R,\epsilon )$-good curves in $M$ modulo the oriented boundaries of $(R,\epsilon )$-good pants. In particular, we prove that for sufficiently small $\epsilon >0$ and sufficiently large $R>0$, some modified version of the $(R,\epsilon )$-panted cobordism group of $M$ is isomorphic to $H_1(\text{SO}\left(M\right);Z)$.