[I thought that I had already posted this question, but I couldn't find it in a search, so I apologize if I'm posting twice.]
Let $G$ be a finite group. Then the rational oriented bordism ring $\Omega_{2k-1}^{STop}(BG)\otimes{\mathbb Q}$ is trivial. As a result, if $N^{2k-1}$is an oriented Top manifold with fundamental group $G$, then there is a Top manifold $(M^{2k}, \partial M)$ with fundamental group $G$ such that $\partial M$ is the disjoint union of a number (say $m$) of copies of $N$.
Atiyah-Singer defines the rho invariant of $N$ to be $$\rho(N)=\frac{1}{m}\cdot\mathrm{sig}_G(\widetilde{M}) \in {\mathbb Q} R^\pm(G)/I_G,$$ where $\pm=(-1)^k$ and $I_G$ is the ideal generated by the regular representation.
My question is about $I_G$. Certainly the manifold $M$ might not be uniquely chosen. How do we get a well-defined rho invariant independent of this choice if we mod out by $I_G$?