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Tate cohomologies and Browder-Livesay groups of dihedral groups. (English. Russian original) Zbl 0840.57021

Math. Notes 54, No. 2, 798-805 (1993); translation from Mat. Zametki 54, No. 2, 44-55 (1993).
The paper deals with the computation of the Browder-Livesay groups \(LN_n (\pi \to D^\pm_{r + 1})\) where \(D_{r + 1}\) is the dihedral group of order \(2^{r + 1}\) \[ D_{r + 1} = \bigl\{ \tau,t \mid \tau^{2^r} = t^2 = 1,\;t^{-1} \tau t = \tau^{-1} \bigr\}, \] \(\pi\) is a cyclic subgroup of index 2 and \(\pm\) refers to orientation character \(w : D_{r + 1} \to \{\pm 1\}\) with \(w(t) = \pm 1\). In the orientable case \(w (\tau) = 1\) and in the non-orientable case \(w (\tau) = - 1\). The Browder-Livesay groups are the obstruction groups for splitting along a one-sided submanifold of codimension 1 and are related to the Wall groups \(L^Y_n (\mathbb{Z} \pi)\). The computation is based on Rothenberg sequences and computations of the corresponding Tate cohomology groups. Among other results the following theorem is proven:
1. In the orientable case the group \(LN_n (\pi \to D^-_{r + 1})\) is isomorphic to \(\mathbb{Z}/2\) for \(n = 2\) and for the other dimensions the following exact sequences hold \[ 0 \to (\mathbb{Z}/2)^{r - 1} \to LN_3 (\pi \to D^-_{r + 1}) \to (\mathbb{Z}/2)^{r^r - 1} \to 0 \] and \[ 0 \to LN_1 (\pi \to D^-_{r + 1}) \to (\mathbb{Z} / 2)^{2^r} \to (\mathbb{Z}/2)^{2^{r - 1} + r} \oplus (8 \mathbb{Z})^2 \to LN_0 (\pi \to D^-_{r + 1}) \to 0; \] 2. In the non-orientable case the Browder-Livesay groups are 2-periodic and \[ LN_{2n} (\pi \to D^\pm_{r + 1})\cong (4\mathbb{Z}) \oplus (\mathbb{Z}/2)^{2^{r - 2}}\qquad LN_{2n + 1} (\pi \to D^\pm_{r + 1})\cong (\mathbb{Z}/2)^{2^{r - 2} - 1}. \]
Reviewer: W.Lück (Mainz)

MSC:

57R67 Surgery obstructions, Wall groups
19G24 \(L\)-theory of group rings
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References:

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