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Exact Sequences

  1. Feb 19, 2009 #1
    Hey, can anyone help me with this please. I am doing algebraic topology and am particularly stuck on exact sequences. I "understand" the idea of the definition for example:

    [tex] 0\rightarrow A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0 [/tex]

    in this short exact sequence, alpha has to be mono or injective and beta has to be epi or surjective. However, what i dont get is: given a sequence of some groups how does one test whether it is exact. In other words how does one construct function between groups which are either epi or mono? for example from Z2 to Z4 or others.

    (this may seem a strange question for someone doing "algebraic ..." but i have not done any algebra beyond an introduction few years ago)

    thank you
     
  2. jcsd
  3. Feb 19, 2009 #2
    In algebraic topology exact sequences are almost always given to you. You do not construct them. They arise naturally in comparing chain complexes.

    In the the case of Z2 and Z4 there is only one possible exact sequence,

    0 -> Z2 -> Z4 -> Z2 -> 0

    It is easy to construct.

    The generator of Z2 must be mapped to the unique element of order 2 in Z4. The generators of Z4 are mapped onto the generator of Z2.

    I would be happy to help you with the exact sequences used algebraic topology.
     
    Last edited: Feb 19, 2009
  4. Feb 19, 2009 #3
    I am looking at hatcher's book and one excercise asks to determine if there exist a short exact sequence: [tex] 0 \rightarrow \mathbb{Z}_{4}\rightarrow \mathbb{Z}_{8} \oplus\mathbb{Z}_{2} \rightarrow \mathbb{Z}_{4} \rightarrow 0 [/tex]

    Now when i look at that i have absolutely no idea where to start. So what i am hoping for is a general procedure to determine. How did you determine that there is a unique exact sequence for
    0->Z2->Z4->Z2->0

    Could you send my way, if you know any book or website that dwells in details into this.
     
  5. Feb 19, 2009 #4
    With Abelian groups the logic is simple. You do not need a general procedure in my opinion.
    This is how to do 0->Z2->Z4->Z2->0. If Z2 -> Z4 is injective then its generator must map to an element of order 2 in Z4. There is only one element of order 2 in Z4 so there is only one injective map. The quotient is clearly isomorphic to Z2 so you are done.

    For Z4 -> Z8 + Z2 the generator must map to an element of order 4. These are the elements (z^2,0) (z^2,1) (Z^6,0) and (z^6,1) where z is any generator of Z_8. Just check it out to see if any quotient is isomorphic to Z_4. Maybe you can generalize from this example.
     
  6. Feb 20, 2009 #5
    Try this problem. How many exact sequences are there of the form 0 -> Z2 -> Z2 + Z2 ->Z2 ->0? Construct all of them. How about 0 -> Z2 -> Z4 + Z4 -> G - > 0. What are the possible groups,G?

    Here is an exact sequence of groups that are not all Abelian. 0 -> L2 -> G -> Z2 -> 0
    G is not Abelian. L2 is the standard lattice in the plane i.e. all points in the plane whose co-ordinates are integers. It is a free Abelian group on two generators. G is the group generated by this lattice and one other transformation of the plane. This other transformation is multiply the y coordinate by -1 then translate by 1/2 in the x direction i.e.
    (x,y) -> (x+1/2,-y). G is the fundamental group of the Klein bottle. Show that g/L2 is Z2 and prove that this is an exact sequence.
     
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