The surgery obstruction groups of the infinite dihedral group

Abstract

This paper computes the quadratic Witt groups (the Wall $L$–groups) of the polynomial ring $\mathbb{Z}\left[t\right]$ and the integral group ring of the infinite dihedral group, with various involutions. We show that some of these groups are infinite direct sums of cyclic groups of order 2 and 4. The techniques used are quadratic linking forms over $\mathbb{Z}\left[t\right]$ and Arf invariants.