= Rational homotopy group
{title2=$\pi_n(X)\otimes\mathbb Q$}
{wiki}
For $n\geq2$, the <rational homotopy group> of a based <topological space> $X$ is the <vector space>
$$
\pi_n(X)\otimes_{\mathbb Z}\mathbb Q.
$$
Here $\pi_n(X)$ is an <abelian group>, so this <tensor product> is defined. For a <finitely generated abelian group> $\pi_n(X)\cong\mathbb Z^r\oplus T$ with $T$ finite, the result is $\mathbb Q^r$: finite-order classes disappear and the free generators acquire rational <coefficients>. More generally every <torsion element> maps to zero, since $ma=0$ implies $a\otimes q=(ma)\otimes(q/m)=0$ for $m\ne0$.
For example, $\pi_3(S^2)\cong\mathbb Z$ gives $\pi_3(S^2)\otimes\mathbb Q\cong\mathbb Q$, whereas $\pi_4(S^3)\cong\mathbb Z/2$ becomes zero. The general <tensor product> definition also handles <groups> that are not finitely generated: for an <Eilenberg–MacLane space> $K(\mathbb Q,n)$, its degree-$n$ <rational homotopy group> is still $\mathbb Q$.
For a <simply connected> <CW complex>, these <vector spaces> are the <homotopy groups> of its <rationalization of a topological space>. An arbitrary nonabelian <fundamental group> cannot be tensored with $\mathbb Q$ by this <abelian group> formula; the nilpotent-group localization used more generally in <rational homotopy theory> is a separate construction.
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