
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.
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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 chainaddition by definition of the cone operation...So what is left to do? can someone help me?
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Thanks in advance

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 $\displaystyle \partial{c}=\sum_{k=0}^{p+1}(1)^k [v_0,...,v_{k1},v_{k+1},...,v_{p+1}]$ 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

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!

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.

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

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

Re: Suspension homology
$\displaystyle w_0  w_1 $ ?

Re: Suspension homology
NVM, I got it !
THanks a lot!