On the computation of the projective surgery obstruction groups
The article deals with the computation of the projective surgery obstruction groups \(L^ p_ n(\mathbb{Z}[G])\) for finite groups, in particular for 2-hyperelementary groups. These play an important role in the context of bounded and controlled surgery theory. They deviate from the classical surgery groups only by 2-torsion and the deviation is measured by the Rothenberg sequence. Let \(G\) be a 2-hyperelementary group. A group is called basic if all normal abelian subgroups are cyclic. If \(G\) is the semi-direct product of the cyclic group \(\mathbb{Z}/m\) of odd order and a 2-group \(\sigma\), then \(\sigma_ 1\) is the kernel of the obvious homomorphisms from \(\sigma\) to the automorphisms of \(\mathbb{Z}/m\). Put \(\overline{G} = G/[\sigma_ 1,\sigma_ 1]\). A generalized restriction is to be understood as restriction followed by a map induced by a surjection of groups. Then Theorem A: The sum of all generalized restriction maps \[ L^ p_ n(\mathbb{Z}[G]) \to L^ p_ n(\mathbb{Z}[\overline{G}]) \oplus (\oplus\{L^ p_ n(\mathbb{Z}[H/H])\mid H/N \text{ a basic subquotient of }G\}) \] is a natural split injection. This reduces the computation of the projective surgery group to the case of smaller groups where the calculation can be reduced to standard calculations in number theory. The next result is important for the problem to decide whether a surgery obstruction is zero. Theorem B: For any oriented degree 1 normal map its projective surgery obstruction is detected by the multi-signature, Arf invariants, semicharacteristic, cohomology finiteness obstruction and the \(\delta\)- invariant. The first three invariants in this theorem are classical invariants. The cohomology finiteness obstruction and the \(\delta\)-invariant are computable by another classical invariant, the Reidemeister torsion.
projective surgery obstruction groups for finite groups
scientific article; zbMATH DE number 549707
scientific article