today one of my friends ask me what is the homology groups for mobius band and cylinder?
the first thing that comes to my mind is that both of them deformation retract to a circle,the circle is path component so $H_0(X)=\mathbb{Z}$ when $X$ is circle.for $H_1(X)$ ,the fundamental group of the circle is $\mathbb{Z}$ and because it is abelian so $H_1(X)=\mathbb{Z}$ and for $n \geq 2$ $H_n(X)$ is zero because there is no 2-simplex in a circle.
now please tell my Idea was right or wrong,actually I have doubt because mobius band is a circle with $z \rightarrow z^2$ ,so help me with your knowledge,thanks a lot.
Best Answer
Let $M$ be a closed connected $n$ manifold then $H_i(M;\mathbb Z)=0$ for each $i>n$. If in addition $M$ is orientable then $H_n(M;\mathbb Z)\cong \mathbb Z$.
For a proof see Hatcher Theorem 3.26.
Apply this to $M=S^1$ which is a closed connected orientable $1$-manifold.