Geodesic
Arf invariants of codimension one in a~Wall group of the dihedral group
Sbornik. Mathematics, Tome 214 (2023) no. 5, pp. 613-626

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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.
Keywords: Browder-Livesay groups, Wall groups, one-sided submanifolds, splitting obstructions
Mots-clés : codimension-one Arf invariant, Hasse-Witt torsion. 
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     author = {P. M. Akhmet'ev and Yu. V. Muranov},
     title = {Arf invariants of codimension one in {a~Wall} group of the dihedral group},
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     url = {http://geodesic.mathdoc.fr/item/SM_2023_214_5_a0/}
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P. M. Akhmet'ev; Yu. V. Muranov. Arf invariants of codimension one in a~Wall group of the dihedral group. Sbornik. Mathematics, Tome 214 (2023) no. 5, pp. 613-626. http://geodesic.mathdoc.fr/item/SM_2023_214_5_a0/