Equivariant controllable cutting-pasting and cobordism with vector fields
Let G be a finite group of odd order, \({\mathcal M}_ n(G)\) the set of n- dimensional closed G-manifolds (smooth), \(N_ n(G)\) the cobordism group of n-dimensional closed G-manifolds, \(N^ R_ n(G)\) the Grothendieck group of the semigroup constructed by the Reinhart cobordism classes in \(N_ n(G)\), and \(SKK_ n(G)\) the Grothendieck group of the semigroup \({\mathcal M}_ n(G)/\sim\) with \(\sim\) the SKK-equivalence. In this paper, the author obtains two short exact sequences which contain the groups \(N_ n(G)\), \(N^ R_ n(G)\) and \(SKK_ n(G)\). And then he shows that for \(M,N\in {\mathcal M}_ n(G)\) with \(M^{H,1}=\emptyset\), \(N^{H,1}=\emptyset\) for any subgroup H of G, \([M]=[N]\) in \(N^ R_ n(G)\) iff \([M]=[N]\) in \(SKK_ n(G)\), where \(M^{H,1}\) denotes the 1- dimensional components of the H-fixed point set of the G-manifold M.
cobordism group of n-dimensional closed G-manifolds
scientific article; zbMATH DE number 4126332