Thread: Homology Groups of the 2D Torus

1. Homology Groups of the 2D Torus

I am reading James Munkres' book, Elements of Algebraic Topology.

Theorem 6.2 on page 35 concerns the homology groups of the 2-dimensional torus.

Munkres shows that $\displaystyle H_1 (T) \simeq \mathbb{Z} \oplus \mathbb{Z}$ and $\displaystyle H_2 (T) \simeq \mathbb{Z}$.

After some work I now (just!) follow the proof that $\displaystyle H_1 (T) \simeq \mathbb{Z} \oplus \mathbb{Z}$ but I need some help to understand a point in the proof of $\displaystyle H_2 (T) \simeq \mathbb{Z}$.

Munkres' argument to show $\displaystyle H_2 (T) \simeq \mathbb{Z}$ is as follows:

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To compute $\displaystyle H_2 (T)$, note that by (2) any 2-cycle d of T must be of the form $\displaystyle p \gamma$ for some p. Each such 2-chain is in fact a cycle,by (4) , and there are no 3-chains for it to bound. We conclude that

$\displaystyle H_2 (T) \simeq \mathbb{Z}$

and this group has as generator the 2-cycle $\displaystyle \gamma$.

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I do not fully understand why any 2-cycle d of T must be of the form $\displaystyle p \gamma$ for some p.

Can someone please explain exactly why this follows?

Would appreciate some help.

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To give members of MHF the context of the post above, the text of Theorem 6.2 and its proof follow:

Some of my thoughts ... ...

Basically, to show that any 2-cyclce of L (i.e. T) os of the form $\displaystyle p \gamma$, we have to show the following:

If $\displaystyle d = \sum_i n_i \sigma_i$ where $\displaystyle \partial d = 0$ then $\displaystyle d = p \gamma$.

We have, of course that $\displaystyle \gamma = \sum_i \sigma_i$

Note that we have that if d is a 2-chain of L and if $\displaystyle \partial d$ is carried by A then d is a multiple of $\displaystyle \gamma$.

Munkres defines 'carried by' in the following text taken from page 31:

Hope someone can help.

Peter