what is the homology groups of a contracting circle in a torus to a point?

Question

I need to compute the homology groups of the space $X$ obtained by contracting a circle in a torus $T$ to a point.

I think $H_0(X)= \mathbb{Z}$ since $X$ is a connected. For $H_1(X)$ I think we need to calculate the fundamental group which is $\mathbb{Z}\times \mathbb{Z}$, and it is $H_1(X)=\mathbb{Z}\times \mathbb{Z}$. For $H_2(X)$, I believe it is $\mathbb{Z}$.

I really need help here, any help or comment would be appreciated.

Answer 1

Let $C$ be some essential embedding of $S^1$ in the torus. Write it as $(m,q)$ for the image of $1 \in \mathbb Z$ under the induced map $H_1(S^1) \to H_1(T)$ where $m$ is coprime to $q$. Here are some fun ways to see the result.

1. You can use the idenfitication $\tilde{H_n}(X)=H_n(T,C)$ and the short exact sequence to see that we have

$0 \to H_2(T) \to \tilde{H}_2(X) \to H_1(S^1) \to H_1(T) \to \tilde{H}_1(X) \to H_0(S^1) \to H_0(T) \to 0$

Where the map $H_1(S^1)= \mathbb Z \to \mathbb Z \oplus \mathbb Z= H_1(T)$ is given by $n \mapsto (nm,nq)$.

2. You can use the classification of surfaces with boundary to deduce that $T \setminus C \cong S^1 \times I$, and from Alexander-Lefschetz duality we know that $\tilde{H}_{n-k}(X)=H_{n-k}(T,C) \cong H^{k}(S^1)$.

3. There is a homeomorphism of the torus carrying $C$ to $(1,0)$ in the torus (basically because $T \setminus X$ is homeomorphic to $S^1 \times I$ you can use the homeomorphism there, and glue back together.) You can then argue geometrically that $X \simeq S^2 \vee S^1$ and use the Mayer-Vietoris sequence to compute its homology. A hint for this homotopy equivalence is that $X$ is homotopy equivalent to a sphere with two points identified.

---

If the curve $C$ is not essential, i.e. bounds a disk, we can see that $X=T \vee S^2$ and compute the homology this way, or use the long exact seqeuence from $1$, except that $H_1(S^1) \to H_1(T)$ will be the zero map