$H_p(X,X)=0$ for all $p\in\mathbb{Z}$

Question

Givena a Homology Theory (are satisfied Eilenberg-Steenrod Axioms), how I can prove that $H_p(X,X)=(0)$ for all $p\in \mathbb{Z}$? Some book says that this claim follows from the long exact sequence in homology, but I'm not able to check this. Any help?

Answer 1

Consider the long exact sequence for the pair $(X,X)$: $$ \dots\to H_p(X)\xrightarrow{i_*} H_p(X)\xrightarrow{j_*} H_p(X,X)\xrightarrow{\partial} H_{p-1}(X)\xrightarrow{i_*} H_{p-1}(X)\to \dots $$ Note that $i_*$ is the identity on $H_{p-1}(X)$, so $\partial$ is the zero map by exactness. Also since $i_*$ is the identity on $H_p(X)$, it follows by exactness that $j_*$ is the zero map. Do you see how it follows from $\text{im}(j_*)=\ker(\partial)$ that $H_p(X,X)$ is trivial?