# Definition of reduced relative homology group

In Hatcher's book on P140, he said that when $$n=1$$, the boundary map $$d_1:H_1(X^1,X^0)\rightarrow H_0(X^0)$$ is the same as the simplicial boundary map $$\Delta_1(X)\rightarrow\Delta_0(X)$$. I know the definition of $$d_n$$ is $$j_{n-1}\partial_n: H_n(X^n,X^{n-1})\rightarrow H_{n-1}(X^{n-1})\rightarrow H_{n-1}(X^{n-1},X^{n-2})$$. So I think when $$n=1$$, $$j$$ is just identity map. But I don't understand what does author mean to identify the $$d_1$$ with simplicial boundary map???

Since $$H_1(X^1,X^0)=H_1^{\Delta}(X^1,X^0)$$, $$H_0(X^0)=H_0^{\Delta}(X^0)$$ by homomorphisms induced by the canonical inclusion maps, we see that $$d_1=\partial_1:H_1(X^1,X^0)\to H_0(X^0)$$ is the same as the map $$\partial_1^\Delta:H_1^{\Delta}(X^1,X^0)\to H_0^{\Delta}(X^0)$$ in the corresponding long exact sequence of the short exact sequence of singular complexes $$\Delta_n(X^0)\to\Delta_n(X^1)\to\Delta_n(X^1,X^0)$$, hence we need only observe that $$\partial_1^\Delta$$ is the same as the simplicial boundary map.
Note that by definition $$H_1^{\Delta}(X^1,X^0)$$ is free abelian generated by $$1$$-cells of $$X$$ since any $$1$$-cell of $$X$$ is a relative cycle and $$\Delta_2(X^1,X^0)$$ is trivial. Also, $$H_0^{\Delta}(X^0)$$ is generated by $$0$$-cells of $$X$$, concluding that $$H_1^{\Delta}(X^1,X^0)=\Delta_1(X^1)$$ and $$H_0^\Delta(X^0)=\Delta_0(X^0)$$. It suffices to observe that $$\partial_1^\Delta$$ sends $$1$$-cells of $$X$$ to what the simplicial boundary map $$\partial:\Delta_1(X^1)\to\Delta_0(X^1)=\Delta_0(X^0)$$ sends to, which is rather straightforward by definition of $$\partial_1^\Delta$$: The quotient $$j: \Delta_1(X^1)\to\Delta_1(X^1,X^0)$$ is an isomorphism sending $$1$$-cells to $$1$$-cells and $$\Delta_0(X^1)=\Delta_0(X^0)$$ via the inclusion of $$X^0$$ into $$X^1$$.