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Thread: Suspension homology

  1. #1
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    Suspension homology

    Let $\displaystyle K$ be a complex, and let $\displaystyle w_0 *K $ , $\displaystyle w_1 * K $ be two cones whose polytopes intersect only in $\displaystyle |K|$.
    The complex $\displaystyle S(K)=(w_0 * K ) \cup (w_1 * K ) $ is called "SUSPENSION" of K.
    Define: $\displaystyle \phi : C_p(K) \to C_{p+1} (S(K)) $ by the equation -
    $\displaystyle \phi (c_p) = [w_o , c_p] - [w_1 , c_p ] $ .
    Show that $\phi$ induces a homomorphism:
    $\displaystyle \phi _{*} : H_p(K) \to H_{p+1} (S(K)) $.

    By the way, the homology groups here are the reduced ones... Which has no importance for p>0.
    [
    What I've tried:
    I thought I should define:
    $\displaystyle \phi _{*} (c_p + B_p(K) ) = \phi (c_p) +B_{p+1} (S(K)) $
    but what should I prove in order for it to be a homomorphism? it presereves the operation of chain-addition by definition of the cone operation...So what is left to do? can someone help me?
    ]
    Thanks in advance
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  2. #2
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    Re: Suspension homology

    $\displaystyle \phi$ is obviously a homomorphism. To verify it induces a homomorphism between the homology groups, we need only to show $\displaystyle \phi(\partial{c})$ ~ 0, that is $\displaystyle \phi$ maps boundary chains to boundary chains. This is only a straightforward computation:
    It is easy to verify that $\displaystyle \partial[w_0, c]=c-[w_0,\partial{c}]$, just write down $\displaystyle c=[v_0,...,v_{p+1}]$ and $\displaystyle \partial{c}=\sum_{k=0}^{p+1}(-1)^k [v_0,...,v_{k-1},v_{k+1},...,v_{p+1}]$ you can verify that.
    So we have $\displaystyle \phi(\partial{c})=[w_0,\partial{c}]-[w_1,\partial{c}]$
    $\displaystyle =(c-\partial[w_0,c])-(c-\partial[w_1,c])$
    $\displaystyle =\partial([w_1,c]-[w_0,c])$
    $\displaystyle =\partial(-\phi(c))$ ~ 0, we're done
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  3. #3
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    Re: Suspension homology

    Hi xxx9,

    Thanks for your answer. I've several questions regarding what you wrote:

    1) What we've actually proved is that $\displaystyle \phi (Im \partial _p ) \subseteq Im(\partial _{p+1} S(X)) ) $ for all p>0. But what about the case of p=0? We don't need it because the reduced homology group of order 0 contains only the expression of $\displaystyle Im \partial _ 1 $ ?

    2) After we proved boundaries are copied into boundaries, shouldn't we prove it for cycles as well ? And then my $\displaystyle \phi _{\ast} $ will be obviously a homomorphism? If not, can you please explain me this last thing?


    Hope you'll be able to help me with this

    Thanks a lot again, you've been very helpful so far!
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  4. #4
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    Re: Suspension homology

    Actually what we have shown is $\displaystyle \phi(\partial)=-\partial(\phi)$ for any $\displaystyle p \ge 0$, using this you can easily verify your 1) and 2), by just unwinding the definitions.
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  5. #5
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    Re: Suspension homology

    I still can't get it... The definition of the cone boundary operator for p=0 isn't the same as in the case of p>0 ... So we've actually shown it for p>0 .
    From the fact that $\displaystyle \phi (\partial ) = - \partial ( \phi ) $ we can deduce that $\displaystyle \phi (B_p(X)) \subseteq B_{p+1} (S(X)) $ . Can you please explain why this is what we needed in order to prove $\displaystyle \phi _{\ast} $ ?

    I really need your help in this

    Thanks in advance
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  6. #6
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    Re: Suspension homology

    How do you define $\displaystyle \phi(0)$?
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  7. #7
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    Re: Suspension homology

    $\displaystyle w_0 - w_1 $ ?
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  8. #8
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    Re: Suspension homology

    NVM, I got it !

    THanks a lot!
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