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*(B π) → A, where A is a suitable abelian group. The 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) → H1JO(π) (KOπ-* -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.