Is it possible to find a finite (connected) CW complex $K$ such that $\pi_i(K)$ is finite $\forall i\geq 1$?
Can I ask for $K$ to be simpliy connected as well or to have only a finite number of non zero groups?
I know that Serre proved that a simply connected finite CW complex will have infinite non zero homotopy groups but I would like to see a concrete example of something that doesn't have any $\mathbb{Z}$ "components".
I am sure it is probably a well known example but I was not able to find anything on my own.
By the theory of Serre classes, if $X$ is a simply connected space whose reduced homology groups are all finite, then all the homotopy groups of $X$ are finite as well. So, for instance, you could take $X$ which is obtained from $S^2$ by attaching a 3-cell along a map of degree $n$ for some $n>1$. Then the only nontrivial reduced homology group of $X$ is $H_2(X)\cong\mathbb{Z}/(n)$, so $X$ is not contractible but all of its homotopy groups are finite.
However, if a finite CW-complex $X$ has only finitely many nontrivial homotopy groups and they all are finite, then $X$ is contractible. First, note that the universal cover of $X$ is also a finite CW-complex (since $\pi_1(X)$ is finite) and so must be contractible since it has only finitely many nontrivial homotopy groups. So, $X$ is a $K(G,1)$ for some finite group $G$. However, a $K(G,1)$ for a finite nontrivial group $G$ has infinite cohomological dimension so cannot be a finite (or even finite-dimensional) CW-complex.