I was trying to construct a space that has first $n$ homology groups any given abelian groups $G_1, ..., G_n$. To show this I would like to be able to do the following: Given any space $X$, I can form some $X'$ such that $H_j(X') = 0$ some fixed $j$ and $H_i (X) = H_i (X')$ for all $i \not = j$, i.e. a process of 'filling in $j$-dimensional holes'.
I do not see a way to proceed. It seems plausible but perhaps as I can only draw 'nice' spaces in my head.
Does any such process exist?
You can do this by taking a wedge product of Moore spaces. Your question can then be answered by reading how Moore spaces are constructed. See Hatcher's book, Example 2.40 and Example 2.41 (2015 print).