Let $X$ be a simply connected space such that $H^k(X)$ is finitely generated for every $k$ and $H^k(X)=0$ for $k>n$. How can I find a finite $n$-(CW) complex $K$ homotopy equivalent to $X$?
The usual construction of CW approximation does not help here, since although the homotopy groups $\pi_k(X)$ are also finitely generated (by Serre mod C theory), we could have infinitely many nonzero homotopy groups. So the construction will only give a CW complex with finitely many cells of each dimension, not a finite CW complex.
One idea is to first construct a CW approximation as described above, and try to remove the higher dimensional cells to modify the homology groups, and finally use (relative) Hurewicz theorem to show homotopy equivalence. However I do not know how to kill off homology groups without adding new generators (in higher dimensions).
For your information, the reference given in the paper for this result is Seminare Cartan 1954-1955 Expose 22 Apendix, which is available here: http://www.numdam.org/issues/SHC_1954-1955__7_1. However I cannot see how the results there are related to this result I want to prove here.
Any help is appreciated!