Reidemeister-Franz torsion as a homomorphism

Let us denote the Diff-isomorphism classes of n-dimensional non-empty, closed, connected, oriented differentiable manifolds by Mn Diff. An n-manifold Mn is called highly connected if π1(Mn)=0 for i=0,…, ⌊n/2⌋-1. So the Diff-isomorphism classes of n-dimensional highly connected differentiable manifolds is given as MnDiff,hc= {Mn𝜖 MDiff | Mn is highly con-nected}. Hence by [1], MnDiff and MnDiff,hc are abelian monoids under the connected sum. By [1]-[3], the monoid M2nDiff,hc is a unique factorisation monoid provided that n ≡ 3,5,7 mod 8 except for n=15 or n=31. Suppose that W2n 𝜖 M2nDiff,hc .Then W2n admits a unique connected sum decomposition into 2n-manifolds that can not be decomposed any further, W2n=M1#M2#⋯#Mj. By using such a connected sum decomposition, we prove that Rei- demeister-Franz torsion can be seen as a monoid homomorphism |T RF − |: M2nDiff,hc → R+given by | T RF (W2n)| = | T RF (M1)| x |T RF (M2)| x ⋯ x | T RF (Mj)|.