3 Attachment(s)

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 and .

After some work I now (just!) follow the proof that but I need some help to understand a point in the proof of .

Munkres' argument to show is as follows:

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To compute , note that by (2) any 2-cycle d of T must be of the form 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

and this group has as generator the 2-cycle .

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**I do not fully understand why any 2-cycle d of T must be of the form 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:

Attachment 30710

Attachment 30711

Some of my thoughts ... ...

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

If where then .

We have, of course that

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

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

Attachment 30712

Hope someone can help.

Peter