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

Author:

Akhmet'ev Petr Mikhailovich12,Muranov Yury Vladimirovich3

Affiliation:

1. Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow, Russia

2. Tikhonov Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow, Russia

3. University of Warmia and Mazury in Olsztyn, Olsztyn, Poland

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.

Funder

Russian Science Foundation

Publisher

Steklov Mathematical Institute

Subject

Algebra and Number Theory

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