Why is the first homology group of the torus is $\mathbb{Z}\oplus\mathbb{Z}$ instead of $\mathbb{Z}*\mathbb{Z}$ or $\mathbb{Z}\times\mathbb{Z}$?

Question

For the torus below:

I would like to compute $H_1^\Delta(T)$. Here is how I did:

We have $C_1=\Delta_1(T)=\mathbb{Z}$ and $C_2=\Delta_2(T)=\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$ am I right? $C_i=\Delta_i(T)$ is always a free abelian group isn't it?

$H_1^\Delta(T)=\frac{\ker\partial_1}{\text{im }\partial_2}=\frac{\mathbb{Z}*\mathbb{Z}*\mathbb{Z}}{\mathbb{Z}}=\mathbb{Z}*\mathbb{Z}$, where $*$ denotes the free product.

But the working solution given in Hatcher's page 106 is:

My question is:

1. Is my working correct? And why is the first simplicial homology group of the torus is $\mathbb{Z}\oplus\mathbb{Z}$ instead of $\mathbb{Z}*\mathbb{Z}$ or $\mathbb{Z}\times\mathbb{Z}$?

2. What is the difference between $\mathbb{Z}\oplus\mathbb{Z}$ and $\mathbb{Z}\times\mathbb{Z}$?

I am really confused. I appreciate any helps and explanation. Thanks!

Answer 1

The notation $*$ denotes the free product of groups, which notably is an operation on groups, not just abelian groups. So $\mathbb{Z}*\mathbb{Z}$ is the free (nonabelian) group on two generators, not the free abelian group on two generators.

There is no difference between $\mathbb{Z}\times\mathbb{Z}$ and $\mathbb{Z}\oplus\mathbb{Z}$: they mean the exact same thing (or in some contexts, their definitions may be different but you can prove they are canonically isomorphic).

(There is, however, a difference when talking about infinite products versus infinite direct sums: $\prod_{i\in I} A_i$ denotes the set of all tuples indexed by the set $I$ where the $i$th coordinate is in $A_i$ for all $i$, while $\bigoplus_{i\in I}A_i$ denotes the subgroup of $\prod_{i\in I}A_i$ consisting of elements that are $0$ on all but finitely many coordinates.)