Semi-linear homology $G$-spheres and their equivariant inertia groups

This paper introduces an abelian group $H\Theta _V^G$ for all semi-linear homology $G$-spheres, which corresponds to a known abelian group $\Theta _V^G$ for all semi-linear homotopy $G$-spheres, where $G$ is a compact Lie group and $V$ is a $G$-representation with $\dim V^G>0$. Then using equivariant surgery techniques, we study the relation between both $H\Theta _V^G$ and $\Theta _V^G$ when $G$ is finite. The main result is that under the conditions that $G$-action is semi-free and $\dim V-\dim V^G\geq 3$ with $\dim V^G >0$, the homomorphism $T: \Theta _V^G\longrightarrow H\Theta _V^G$ defined by $T([\Sigma ]_G)=\langle \Sigma \rangle _G$ is an isomorphism if $\dim V^G\not =3,4$, and a monomorphism if $\dim V^G=4$. This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology $G$-spheres.