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The surgery L-groups of poly-(finite or cyclic) groups. (English) Zbl 0657.57015
Let \(\Gamma\) be a torsionfree poly-(finite or cyclic) group. The main result of the paper yields the following calculation of the L-theory of \(\Gamma\) : Let \(\omega\) : \(\Gamma\) \(\to {\mathbb{Z}}/2{\mathbb{Z}}\) be a homomorphism and M a closed aspherical manifold with \(\pi_ 1(M)=\Gamma\). Let E be the total space of an I-bundle \((I=[-1,1])\) with base space M and first Stiefel-Whitney class \(\omega\). Then \(L^ s_ i(\Gamma,\omega)=L^ h_ i(\Gamma,\omega)\) is isomorphic to the group of homotopy classes of maps on \(E\times I^ j\) rel boundary, where i and \(j>0\) are related via \(j\equiv i-(cd \Gamma +1)mod 4\). This extends results of F. T. Farrell and W. C. Hsiang [Am. J. Math. 105, 641-672 (1983; Zbl 0521.57018)].
Reviewer: M.Kolster

MSC:
57R67 Surgery obstructions, Wall groups
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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