How do I show that homeomorphic spaces have isomorphic homology groups?
Hatcher says that it is evident from the definitions, which makes me think that I didn't understand something. How does a homeomorphism between two spaces $X$ and $Y$ induce an isomorphism between $\ker{∂_n}/\operatorname{Im}∂_{n+1}$ and $\ker{∂'_n}/\operatorname{Im}∂'_{n+1}$ ($∂$'s are the boundary maps here)? I would send the equivalence class $\operatorname{Im}∂_{n+1}$ to the class $\operatorname{Im}∂'_{n+1}$ but where do I send the classes of the other elements of $\ker{∂_n}$?
Best Answer
Hint: Let $f:X\rightarrow Y$ be a homeomorphism, and $g:Y\rightarrow X$ its inverse. Let $f_*,g_*$ be the maps induced in homology. Then you can easily show that $f_*$ and $g_*$ are inverse one of the other.