at.algebraic topology - Killing the torsion in homotopy

I don't have an answer to this question, but for the analogous question for homology it looks like it can't be done. By the universal coefficient theorem, a construction like this for homology would give a construction for cohomology as well. To get a counterexample in cohomology, take an Eilenberg-MacLane space $K({mathbb Z},n)$ with $n$ even. This has rational cohomology a polynomial ring ${mathbb Q}[x]$ with $x$ of degree $n$. It follows from this that if you factor the torsion out of the integral cohomology ring you get a polynomial ring ${mathbb Z}[x]$ with $x$ of degree $n$, as one can see by looking at a map from ${mathbb C}P^infty$ to $K({mathbb Z},n)$ that induces an isomorphism on $H^n(--;{mathbb Z})$, using the fact that the integral cohomology of ${mathbb C}P^infty$ is a polynomial ring. However, there is no space whose integral cohomology ring is a polynomial ring on a generator of degree $n$ if $n > 4$, as one sees by looking at Steenrod squares and at Steenrod powers for the prime $p=3$. (This is Corollary 4L.10 in my book.)

In the context that Baez was talking about, rationalizing the homotopy groups is equivalent to rationalizing the homology groups, so it seems to be worth knowing that one can't kill torsion in homology, at least.