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How can I construct a CW complex A with H_0(A) = Z, H_2(A) = Z/4Z, H_4(A) = Z ⊕ Z and all other homology groups trivial? Any idea?

Thanks!

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could you give some motivation? is this a homework question? the only reason I would think it is HW is because of the way you have written it. –  Sean Tilson Dec 16 '10 at 14:01
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Let $f:S^2\to S^2$ be a map of degree $4$. Attach a $3$-cell $B^3$ to $Y=S^2$ using $f$ to glue the boundary $\partial B^3=S^2$ to $Y$. Then $Z=Y\cup_f B^3$ has homology given by $$H_q(Z)=\begin{cases} \mathbb Z, & \mbox{if }q=0; \\ \mathbb Z/\mathbb 4Z, & \mbox{if }q=2; \\ 0, & \mbox{otherwise.} \end{cases} $$ Now consider the connected sum $Z\vee S^4\vee S^4$.

NB: The interesting part is of course the construction of $Z$. If I recall correctly, you can find in Hatcher's book information regarding Moore spaces, of which $Z$ is a simple example.

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