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.



Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology groups
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.


