Let \(G\) be a countable, finitely generated group and let \(\ell_2G\) denote the Hilbert space of square summable real functions on \(G\). If \(P\) is a finitely generated, projective \(\mathbb{Z} G\)-module, then one has various ranks defined: The naive rank is \(rkP= \dim_\mathbb{R} \mathbb{R} \otimes_GP\). The Kaplansky rank, \(\kappa P\), is the coefficient of 1 in the trace of an idempotent \(p:\mathbb{Z} G^n\to \mathbb{Z} G^n\) with image \(P\). One has \(\kappa P= r_p(1)\), where the Hattori-Stallings rank \(r_p: G\to\mathbb{Z}\) is a conjugacy invariant function derived from the full trace of the above \(p\). The strong (respectively, weak) Bass-conjecture for \(G\) says that \(r_p (x) =0\) for all \(x\neq 1\) (respectively, that \(\text{rk}P =\kappa P)\). Since it is known that \(\sum_x r_p(x) =r_kP\), the strong conjecture does imply the weak one. In this paper the groups \(G\) considered are assumed to satisfy the strong Bass conjecture, or to be residually finite. It is shown that for such groups the ordinary and the \(\ell_2\) Euler characteristics coincide. Moreover, the \(\ell_2\) Wall obstruction vanishes for any chain complex of type \(FP\) over \(\mathbb{Z} G\). Applications to groups of type \(FG\) and Poincaré duality spaces are given.