Does ''homology vanishes eventually'' imply ''homotopy vanishes eventually''?

Question

Let $X$ be a connected CW complex.

Assume there is an integer $N\geq 0$ such that the singular homology $H_n(X)=0$ vanishes for all $n\geq N$.

Is there an integer $M\geq 0$ such that $\pi_m(X)=0$ for all $m\geq M$?

Answer 1

Try $X=S^2$ (and look at the table on this page)