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 chain-addition by definition of the cone operation...So what is left to do? can someone help me?

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Thanks in advance