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

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.

• Consider the rationalization of the $n$-sphere, call it $X_n$, defined as the mapping telescope of a sequence of maps $S^n \to S^n \to S^n \to \cdots$ in which the $i$-th map has degree $i$. The space $X_n$ has just one non-trivial reduced homology group. Moreover, the homotopy groups of $X_n$ can also be computed knowing that sequential homotopy colimits commute with homotopy groups. It follows that $X_n$ has non-trivial homotopy groups in at most two degrees, $n$ and perhaps $2n-1$, which correspond precisely to the homotopy groups of the $n$-sphere which are infinite. Jun 20, 2015 at 18:11
• Alternatively, fix an odd natural number $n$, and consider the Eilenberg–MacLane space $K(\mathbb{Q},n)$; it has the same homotopy type as $X_n$ above. One can prove that $K(\mathbb{Q},n)$ has reduced homology concentrated in degree $n$ by applying inductively the Serre spectral sequence for the fibre sequences $K(\mathbb{Q},i-1) \to \ast \to K(\mathbb{Q},i)$. Jun 20, 2015 at 18:27
• Thnaks for this comment. But in fact I was thinking about ${finite}$ CW-complexes, so David's answers was what I was looking for. Jun 21, 2015 at 19:03

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.