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:
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:
Hope someone can help.
Peter