0P8Z Lemma 7.1.22. Given c∈T(X)c\in T(X), the map mcm_{c} induces a morphism of groups R(X)→𝐙R(X)\to{\mathbf{Z}}.
0P90 Proof. Consider γ\gamma and γ′\gamma^{\prime} two injective composable paths such that γ∘γ′\gamma\circ\gamma^{\prime} is injective. We have mc±([γγ′])=mc±([γ])+mc±([γ′])m_{c}^{\pm}([\gamma\gamma^{\prime}])=m_{c}^{\pm}([\gamma])+m_{c}^{\pm}([\gamma^{\prime}]). Consider now γ\gamma a minimal path. We have mc±([γ])=mc∓([γ−1])m_{c}^{\pm}([\gamma])=m_{c}^{\mp}([\gamma^{-1}]), hence mc([γ])+mc([γ−1])=0=mc([γ−1∘γ])m_{c}([\gamma])+m_{c}([\gamma^{-1}])=0=m_{c}([\gamma^{-1}\circ\gamma]). The lemma follows. ∎