To elaborate on Tyler's comment: One more step is needed to go from trivial action of the fundamental group on the higher homotopy groups of the two spaces to trivial action on the relative homotopy groups of the mapping cylinder. This can be done by an obstruction theory argument as in Proposition 4.74 of my book.

As there seems to be no standard term, this is a good chance to invent a new one. Here's a candidate based on only a couple minutes' thought: polydendron. Sort of a variant on polyhedron bringing in the tree idea. A google search produces hits related to polymers, but nothing in math among the top items at least. (There are things called dendrimers, so that could be another possibility. Or just dendron, perhaps.)

Just to clarify: The original question seems to be about the distinction between a space being contractible and the possibly stronger condition of deformation retracting to a point. For nice spaces (manifolds, CW complexes, ...) the two conditions are in fact equivalent. A textbook reference is Corollary 0.20 in Chapter 0 of my algebraic topology book. (See also Example 0.15 and Proposition 0.16.) In the exercises at the end of the chapter there are some examples of weird spaces that are contractible but do not deformation retract to any point.

Jason: I would like very much to see the proof using continued fractions that you refer to.

Nice pictures! I taught an undergraduate number theory course from this point of view (The Farey diagram) last semester, the notes for which are available here: math.cornell.edu/~hatcher/TN/TNpage.html See Chapters 1 and 2 in particular. Conway's topographs also form an integral part of the story. (My apologies for the shameless self-promotion!) When I revise the notes I'll have to add the nice fact discussed in the original post above, which was new to me. Thanks to all for the great answers!

Please correct me if I'm wrong, but it looks like what Zabrodsky does is a little weaker than what I interpreted the original question to ask, which was for a map that induced a surjection on homotopy groups with kernel the torsion subgroups. Zabrodsky only asks for a map that induces an isomorphism on rational homotopy groups and kills the torsion, so after factoring out torsion in the domain it would be injective but not necessarily surjective. Tyler's example from January 5 is a counterexample to the stronger requirement but not to Zabrodsky's weaker requirement.

You’re right, the “just a little more work” takes care of it.

I happened to be looking at M.A. Armstrong's very nice book "Groups and Symmetry" today and noticed that he has a proof along similar lines, except that he first minimizes the number of generators and then the height. Minimizing the number of generators seems to be necessary in order to obtain a splitting where each modulus divides the next. (Perhaps minimizing the number of generators is implicit in the "By induction on $n$" at the beginning of Greg's proof.)

Baez is talking about a different version of ${\mathbb C}P^\infty$ from the one topologists usually consider. Namely he takes nonzero rational functions with coefficients in ${\mathbb C}$ modulo scalar multiplication. The rational functions form an infinite dimensional vector space over ${\mathbb C}$, but of uncountable dimension since all the functions $1/(z+a)$ are linearly independent as $a$ ranges over ${\mathbb C}$. This gives a fatter version of ${\mathbb C}P^\infty$ that's actually an abelian group.

The H-space structure in a $K(A,n)$ is unique up to homotopy since homotopy classes of maps $K(A,n)\times K(A,n) \to K(A,n)$ correspond bijectively with homomorphisms $A\times A \to A $, and the H-space condition says the homomorphism restricts to the identity on each factor so it is just the addition operation in the abelian group $A$.

As a footnote, the construction does not extend to the quaternionic case since commutativity of multiplication of coefficients is needed in order for the multiplication of polynomials to be well defined modulo scalar multiplication. In the complex case, if you only factor out by scalar multiplication by numbers that are a real number times a p-th root of unity, you get an H-space structure on an infinite-dimensional lens space, a $K({\mathbb Z}_p,1)$.