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

Abstract

This paper introduces an abelian group H Θ V G H\Theta _V^G for all semi-linear homology G G -spheres, which corresponds to a known abelian group Θ V G \Theta _V^G for all semi-linear homotopy G G -spheres, where G G is a compact Lie group and V V is a G G -representation with dim ⁡ V G > 0 \dim V^G>0 . Then using equivariant surgery techniques, we study the relation between both H Θ V G H\Theta _V^G and Θ V G \Theta _V^G when G G is finite. The main result is that under the conditions that G G -action is semi-free and dim ⁡ V − dim ⁡ V G ≥ 3 \dim V-\dim V^G\geq 3 with dim ⁡ V G > 0 \dim V^G >0 , the homomorphism T : Θ V G ⟶ H Θ V G T: \Theta _V^G\longrightarrow H\Theta _V^G defined by T ( [ Σ ] G ) = ⟨ Σ ⟩ G T([\Sigma ]_G)=\langle \Sigma \rangle _G is an isomorphism if dim ⁡ V G ≠ 3 , 4 \dim V^G\not =3,4 , and a monomorphism if dim ⁡ V G = 4 \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 G -spheres.