Equivariant Cutting and Pasting of $G$ Manifolds

Abstract

Let $G$ be a finite abelian group and let $SK_{*}^{G}(pt,pt)$ be a cutting and pasting group (an SK group) based on $G$ manifolds with boundary. In this paper, we first obtain a basis for a $\mathbf{Z}$ module $\mathcal{T}_{*}^{G}$ consisting of all homomorphisms ($G$-SK invariants) $T:SK_{*}^{G}(pt,pt)\rightarrow\mathbf{Z}$. Let $SK_{*}^{G}$ be the SK group based on closed $G$ manifolds. We next study a relation between the theories $SK_{*}^{G}$ and $SK_{*}^{G}(pt,pt)$ by performing equivariant cuttings and pastings of $G$ manifolds, and characterize a class of multiplicative invariants which are related to $\chi^G$.