Equivariant Witt groups of finite groups of odd order

Abstract

The orthogonal representations of a finite group over a Dedekind domain are studied. First, we study the equivariant Witt group W 0( D, DG) of a finite nilpotent group G over a Dedekind domain D. Introducing a Morita correspondence on the set of orthogonal representations, we determine the structure of W 0( D, DG) for a finite nilpotent group G of odd order. We next treat the exact sequence 0→ W 0( Z, Z G) → W 0( Q, Q G) → ∂ W 0( Z, Z G), which was introduced by A. Dress (1975, Ann. of Math. (Z) 102, 291–325). We show that the boundary homomorphism δ is surjective when G is a finite group of odd order. Our last aim is to show that W 0( Z, Z G) is sufficiently large to investigate the Witt group W 0( Z G) in L-theory when G is a finite group of odd prime power order.