I am trying to do as many practise questions as possible for an A.T exam in June, which I am currently studying independently for.
I am stuck on the following;
I am wanting to construct a space X where the ith homology group is given by G_i for i = 1,...,n , where the G_i are some finitely generated abelian groups.
I have made little progress, aside from writing the G_i as direct sums using the fundamental theorem of finitely generated abelian groups, so any help at all would be greatly appreciated.
Thank you
Let's do this for $H_n = G$ and zero in all other dimensions - this is known as the Moore space and is denoted by $M(G, n)$. Since $G$ is finitely generated abelian, write it as the direct sum $\Bbb Z \bigoplus_{p} \Bbb Z/p^k\Bbb Z$.
$M(\Bbb Z, n)$ is $S^n$, and $M(\Bbb Z/m\Bbb Z, n)$ can be constructed by gluing an $(n+1)$-cell to $S^n$ by the degree $m$ map $S^n \to S^n$ (this indeed does have $H_{n+1} = 0$ - write down the long exact sequence of $(X, S^n)$ and note that the snake map coming out of the $n+1$-th term is injective)
By Dan Rust's hint write $M(\Bbb Z, n) \bigvee_p M(\Bbb Z/p^k \Bbb Z, n)$. That's the desired $M(G, n)$.
For given $G_1, \cdots, G_n$, look at $M(G_1, 1) \vee M(G_2, 2) \vee \cdots \vee M(G_n, n)$. This space has $H_i \cong G_i$ for each $i = 1, \cdots, n$ as desired.