Mackey and Frobenius Structures on Odd Dimensional Surgery Obstruction Groups

C. T. C. Wall formulated surgery-obstruction groups Ln(Z(G)) in terms of quadratic modules and automorphisms. C. B. Thomas showed that the Wall-group functors Ln(Z(−) ,w |−) are modules over the Hermitian-representation-ring functor G1(Z, −) if the orientation homomorph- ism w is trivial. A. Bak generalized the notion of quadratic module by introducing quadratic-form parameters, and obtained various K-groups related to quadratic modules and automorphisms. One of the authors established that some Bak groups Wn(Z(G) ,� ; w) are equivariant-surgery-obstruction groups and showed in the case of even dimension n that the Bak-group functor Wn(Z(−) ,� −; w|−) is a w-Mackey functor as well as a module over the Grothendieck-Witt-ring functor GW0(Z, −), where w is possibly nontrivial. In this paper, we prove the same facts in the case of odd dimension n. Mathematics Subject Classifications (2000): Primary 19G12, 19G24, 19J25; Secondary 57R67.