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The rho-invariant and the local cohomology approach to KO-homology

The rho-invariant and the local cohomology approach to KO-homology
Datum:

23.05.2013 

Ort:

Raum 1C-04
im Allianzgebäude (05.20)  

Referent:

Dr. Malte Röer 

Zeit:

14:00 Uhr 

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.