The rhoinvariant and the local cohomology approach to KOhomology

Date:
23.05.2013

Speaker:
Dr. Malte Röer

Time:
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 pinvariants of Dirac operators associated with these spin manifolds gives a homomorphism KO_{*}(B π) → A, where A is a suitable abelian group. The ρinvariant in this form has been used to prove special cases of the GromovLawsonRosenberg conjecture.
On the other hand, there is Greenlees' algebraic approach to KOhomology 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 ρinvariant 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 AtiyahSegal π –equivariant KOtheory. We show that the invariant ρ ̂ is equivalent to ρ.

Place:
Raum 1C04
im Allianzgebäude (05.20)