Arf invariants of codimension one in a Wall group of the dihedral group

Abstract

An element $x$ is specified in the Wall group $L_3(D_3)$ of the dihedral group of order $8$ with trivial orientation character, such that $x$ is an element of the third type in the sense of Kharshiladze with respect to any system of one-sided submanifolds of codimension $1$ for which the splitting obstruction group along the first submanifold is isomorphic to $LN_1(\mathbb Z/2\oplus \mathbb Z/2\to D_3)$. The element $x$ is not realisable as an obstruction to surgery on a closed $\mathrm{PL}$-manifold. It is also proved that the unique nontrivial element of the group $LN_3(\mathbb Z/2\oplus \mathbb Z/2\to D_3^-)$ can be detected using the Hasse-Witt $Wh_2$-torsion. Bibliography: 25 titles.