My question is
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dimensional CW complex?
(My thoughts on this which might not be useful at all.) Since an infinite dimensional CW complex could have only finitely many nontrivial homology groups($S^\infty$ for example), it seems to me that the relation between the dimension and the zeroness of the CW complexes is not very strong. On the other hand, we know that Moore spaces are unique up to homotopy equivalence and any CW complex with prescribed homology groups can be construct by taking the wedge sum of Moore spaces. If this statement above is true, then it means every CW complex with only finitely many nonzero homology groups is essentially built up in this way...
Please note that I meant finite dimensional CW complex instead of finite CW complex. Otherwise the infinite dimensional discrete space will serve the purpose. Also, I really appreciate it if someone can point it out whether the statement can become true buy adding some small conditions(One of my professors said we need $X$ to be simply connected).