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