What are the cohomology groups of Moore space

algebraic-topologyhomological-algebrahomology-cohomology

I am using the following definition of Moore space:

Definition:

Let $G$ be an abelian group and $n$ an integer $\geq 2.$ A Moore space of type $(G,n)$ is $1-$connected, $CW-$complex $X$ such that:

$$ \widetilde{H}_i(X) = \begin{cases}
0 & if \, i \neq n, \\
G & if i = n.
\end{cases} $$

And I want to find:

$1- H_*(M(\mathbb Z_{p^a}, 2); \mathbb Z).$

$2- H^*(M(\mathbb Z_{p^a}, 2); \mathbb Z).$

$3- H_*(M(\mathbb Z_{p^a}, 2) \mathbb Z_{p^t}).$

$4- H^*(M(\mathbb Z_{p^a}, 2); \mathbb Z_{p^t}).$

My answers are:

1-Using the given definition of Moore space, I can tell that $H_0 = \mathbb{Z} \oplus \widetilde{H_0}$ and $H_i = \widetilde{H_i} \text{ for } i \geq 1.$Is that correct?

2- I do not know what are the cohomology groups of Moore space, could anyone tell me a source where I can find this information calculated? I know that it can be calculated using chain complex, but is not there any easier way of calculating it? I want to know what the final groups should be.

3- I know that, knowing $(1,)$ we can use the UCT for homology to calculate $H_*(M(\mathbb Z_{p^a}, 2); \mathbb Z_{p^t}).$ correct?

4- I know that, knowing $(2,)$ we can use the UCT for cohomology to calculate $H^*(M(\mathbb Z_{p^a}, 2); \mathbb Z_{p^t}).$ correct?

Could anyone help me in answering those questions, specifically question 2?

Best Answer

  1. Yes, it is correct.

2-4. Yes, the UCT makes it a purely algebraic exercise. But I think you misunderstand the UCT because in fact all 2-4 follow from (1) by it, in particular both (2) and (4). For (3) there will be $H_* \otimes -$ and $Tor(H_*, -)$ and for (2) and (4) there will be $Hom(H_*, -)$ and $Ext(H_*, -)$. Note that both times only homology groups are used.

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