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Math Help - Homology Groups of the 2D Torus

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    Super Member Bernhard's Avatar
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    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  H_1 (T) \simeq  \mathbb{Z} \oplus \mathbb{Z} and  H_2 (T) \simeq  \mathbb{Z} .

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

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

    -----------------------------------------------------------------------------
    To compute  H_2 (T) , note that by (2) any 2-cycle d of T must be of the form  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

     H_2 (T) \simeq  \mathbb{Z}

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

    ------------------------------------------------------------------------------------------------


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

    Can someone please explain exactly why this follows?

    Would appreciate some help.




    -----------------------------------------------------------------------------


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

    Homology Groups of the 2D Torus-theorem-6.2-part-1-page-35-munkres-elements-algebraic-topology.png
    Homology Groups of the 2D Torus-theorem-6.2-part-2-page-35-munkres-elements-algebraic-topology.png


    Some of my thoughts ... ...

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

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

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

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

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

    Homology Groups of the 2D Torus-definition-carried-homologous-munkres-page-31-elements-algebraic-topology.png

    Hope someone can help.

    Peter
    Last edited by Bernhard; April 19th 2014 at 12:57 AM.
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