Where can i find the results about $L^{\ast}(\mathbb{Z}\pi)$ for $\pi$ a finite cyclic group?



Symmetric L-groups of integral group ring of finite cyclic groups
The symmetric $L$-groups $L^\*(Z[\pi])$ for finite cyclic groups $\pi$ have never been computed, perhaps for lack of applications: do you have any? In principle, it is possible to extend the known calculations (mainly due to Wall himself) of the quadratic $L$-groups $L\_\*(Z[\pi])$ to the symmetric $L$-groups. The failure of 4-periodicity in the symmetric $L$-groups was studied by Gunnar Carlsson in his 1979 paper *Desuspension in the symmetric $L$-groups*. In his 1985 paper *Surgery and the generalized Kervaire invariant* Michael Weiss interpreted the relative $L$-groups $\widehat{L}^\*(Z[\pi])$ (which are 8-torsion) in the long exact sequence
$$\dots \to L\_\*(Z\pi) \to L^\*(Z[\pi]) \to \widehat{L}^\*(Z[\pi]) \to L\_{\*-1}(Z[\pi]) \to \dots$$
as the twisted $Q$-groups $Q\_\*(B,\beta)$ of the universal chain bundle $(B,\beta)$ of $Z[\pi]$. The twisted $Q$-groups are homological in nature and so much more computable then either the quadratic or the symmetric $L$-groups, which by definition are generalized Witt groups. See Chapters 2 and 9 of my 1992 book *Algebraic $L$-theory and topological manifolds* for the general theory. Example 9.17 computes $L^0(Z[Z\_2])$, which is a start!


