TITLE: Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology groups
QUESTION [18 upvotes]: Let $X$ be a simply connected finite CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily contractible?
I do not really believe that this is true; but I was also not able to construct a counterexample.

REPLY [28 votes]: By results of J.P. Serre (for $p=2$) and Y. Umeda (for odd $p$) we know that a 1-connected finite CW-complex $X$ with non-trivial cohomology mod $p$ has infinitely many non-trivial homotopy groups mod $p$.
In fact  C.A.  McGibbon  and J.A.  Neisendorfer have proved the
existence of $p$-torsion elements in infinitely many dimensions for finite dimensional spaces using H. Miller's theorem on the Sullivan conjecture.