Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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


share|cite|improve this question
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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.