### Origin ###

This question was asked by John Baez in [This Week's Finds in Mathematical Physics (Week 286)](http://math.ucr.edu/home/baez/week286.html). Therefore, please don't upvote this question (unless you really want to), but do upvote the answers.

### Background/Motivation ###

For a CW complex, which for simplicity will have $\pi_1 = 0$, you can do the operation of "rationalizing", which will change its homotopy $\pi_n \to \pi_n \otimes \mathbb Q$. This works by attaching enough cylinders so that each original cell is killed, but its subdivisions are born instead.

### Question ###
Does there exist a similar procedure of "killing the torsion" which would change the homotopy of 1-connected CW complex from $\pi_n$ to $\pi_n/\pi_n^{tors}$?

### Thoughts

One encounters problems if one just tries to kill off the cell: the procedure might have changed higher homology (this doesn't happen in rationalizing since cylinders are simple). So I suspect the answer is "No", but how to construct a counterexample?