A homomorphism between an equivariant SK ring and the Burnside ring for \(Z_4\)
One can define an equivariant \(SK\)-equivalence relation on smooth \(G\)-manifolds, \(G\) a compact Lie group, by cutting and pasting (\textit{S}chneiden und \textit{K}leben) of manifolds, obtaining the ring \(SK_*^G\) which is an \(SK_*\)-algebra, where \(SK_*\) is the ring \(SK_*^{\{1\}}\), \(\{1\}\) the trivial group. In the second author's previous paper [Hokkaido Math. J. 14, 169-174 (1985; Zbl 0591.57015)], \(SK_*^{Z_2}\) was investigated, where \(Z_2\) is the cyclic group of order \(2\). The paper under review investigates \(SK_*^{Z_4}\), where \(Z_4\) is the cyclic group of order \(4\). The authors obtain a minimal set of generators of \(SK_*^{Z_4}\) as \(SK_*\)-algebra, and determine the ring structure of it. The module structure of it is already determined in [\textit{C. Kosniowski}, Actions of finite abelian groups (1978; Zbl 0373.57018)]. There is a natural homomorphism \(\phi : SK_*^{Z_4} \to A(Z_4)\), where \(A(Z_4)\) is the Burnside ring of \(Z_4\). This is obiously surjective. The authors describe the kernel of \(\phi\). Finally they consider the restriction homomorphism \(SK_*^{Z_4} \to SK_*^{Z_2}\) and the extension homomorphism \(SK_*^{Z_2} \to SK_*^{Z_4}\).
scientific article; zbMATH DE number 1396093