Abstract. Let π be a finite group. There is a geometric description of *KO _{*}(B π)* which represents classes by compact spin manifolds. Taking the

*p*-invariants of Dirac operators associated with these spin manifolds gives a homomorphism

*KO*, where A is a suitable abelian group. The

_{*}(B π) → A*ρ*-invariant in this form has been used to prove special cases of the Gromov-Lawson-Rosenberg conjecture.

On the other hand, there is Greenlees' algebraic approach to *KO*-homology which describes *KO _{*}(B π)* in terms of local cohomology groups of modules over the representation ring

*RO(π)*.

In our talks we explain how to relate the geometric construction of the *ρ*-invariant to the algebraic picture by constructing a homomorphism

ρ ̂_{*} : Ω_{*} ^{Spin} (Bπ) → H^{1}_{JO(π)} (KO_{π}^{-* -1})

taking values in the _rst local cohomology group of Atiyah-Segal π –equivariant KO-theory. We show that the invariant ρ ̂ is equivalent to ρ.