Example 37.1

Let us calculate the relative homology groups $H_n(X, A)$ where $X$ is the Möbius band and $A$ is its boundary. Since the central circle is a deformation retract of $X$, we see that $H_n(X) = H_n(A) = 0$ when $n\geq 2$ and we infer from (37.3) that $H_n(X, A) = 0$ when $n \geq 3$. We now recall that the map $i_*:\pi_1(A)\longrightarrow \pi_1(X)$ induced by inclusion is the group homomorphism of $\mathbb{Z}$ into itself given by $x \mapsto 2x$. Since the fundamental groups are abelian the map $H_1(i) = i_*$ and so the kernel of $H_1(i)$ is trivial. The portion of the exact sequence (37.3) with $n = 2$ gives $H_2(M, A) = 0$. Finally since $H_0(i):H_0(A)\longrightarrow H_0(M)$ is an isomorphism (why?), we conclude from (37.3) (with $n = 0$) that the map $H_1(X)\longrightarrow H_1(X, A)$ is surjective with kernel $2\mathbb{Z}$. Hence $H_1(X, A) = \mathbb{Z}_2$.