Throughout, let $X$ be a *connected finite* CW-complex. If the universal covering of $X$ is contractible, then $\pi\_n(X)=0$ for all $n \geq 2$. In this case $X$ is a model for $B\pi\_1(X)$.


I am wondering whether this is the only reason why higher homotopy groups vanish above a certain degree. More precisely:



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> **Question:** Let $k \geq 3$ be an integer. Can it happen that $\pi\_n(X) = 0$ for all $n \geq k$ and $\pi\_{k-1}(X)\neq 0$?
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Vanishing of higher homotopy groups of finite complexes
No, it cannot happen. In [a paper by McGibbon and Neisendorfer](https://doi.org/10.1007/BF02566349 "McGibbon, C.A., Neisendorfer, J.A. On the homotopy groups of a finite dimensional space. Commentarii Mathematici Helvetici 59, 253–257 (1984). zbMATH review at https://zbmath.org/?q=an:0538.55010"), it is proven that if *X* is a 1-connected space and its mod-p-homology is non-zero in some degree, but zero in all higher degrees, then the $\pi\_n X$ contain a subgroup of order p for infinitely many *n*. This can be applied to the universal cover of your *X*.


