If $G$ is a finite group, then the rational oriented cobordism group $\Omega_{2k-1}^{Stop}(BG)\otimes{\mathbb Q}=0$, so if $N^{2k-1}$ is an orientable odd-dimensional Top manifold with fundamental group $G$, then there is an oriented Top manifold $(M^{2k}, \partial M)$ such that $\pi_1(M)=G$ and $\partial M$ is the disjoint union of $m$ copies of $N$. Atiyah and Singer define the rho invariant $$\rho(N)=\frac{1}{m}\cdot\mathrm{sig}_G(\widetilde{M})\in {\mathbb Q} R^{\pm}(G)/I_G,$$ where $\pm=(-1)^k$ and $I_G$ is the ideal generated by the regular representation.
My question is about $I_G$. Of course the chosen manifold $M$ may not be unique, but I do not quite understand how the rho invariant is independent of this choice once you mod out by $I_G$.