On \(G\)-fiberings over the circle with a cobordism class (Q1884364)

Let \({\mathfrak N}^G_*\) be the cobordism group of unoriented closed \(G\)-manifolds, where \(G\) is a finite group. When \(G=\{1\}\), Conner and Floyd characterized those cobordism classes of \({\mathfrak N}^G_*\) admitting a representative which fibers over the circle \(S^1\), and proved that the Euler characteristic \(\chi(\alpha)\equiv 0\mod 2\) if and only if \(\alpha\) is represented by a closed manifold that fibers over \(S^1\), where \(\alpha\in {\mathfrak N}^G_*\). In a previous paper, the author of the paper under review studied the equivariant case of Conner and Floyd's work when \(G\) is chosen as being \({\mathbb Z}_{2^r}\). In the present paper, the author considers the case in which \(G\) is a finite abelian group of odd order. \vskip .2cm The author first develops the theory of \(G\)-SK invariants establised by himself. He introduces the groups \(\overline{\text{SK}}{}^G_*\) and \(G\)-\(\overline{\text{SK}}\) invariants \(\theta_\sigma\) taking values in \({\mathbb Z}_2\). Then he shows that an element \(\alpha\) in \({\mathfrak N}^G_*\) has a representative \(M^{2n}\) that fibers over \(S^1\) if and only if \(\theta_\sigma(M)\equiv 0\mod 2\) for all slice types \(\sigma\). Note that the \(G\)-\(SK\) groups are trivial in odd dimensions. The author also shows that the kernel of the natural map \({\mathfrak N}^G_*\to{\overline{SK}}^G_*\) is represented by closed manifolds which fiber over \(S^1\).