TITLE: Homotopy type of a space
QUESTION [9 upvotes]: I was wondering what tools of algebraic topology are usually used to show that some things have the same homotopy type? Hatcher doesn't really talk about this in his book even though he defines the concept on page 3. Of course we can compute the homology or homotopy groups of a space, but just showing that they agree is not enough as far as I know.
For example, knowing that the Poincare conjecture is true, we know that every closed simply-connected 3-manifold is the 3-sphere. It follows that they must have the same homotopy type. Is this any easier to prove than Poincare itself? If so how? The reason I picked this example is that I know they are homotopy equivalent and I don't know an obvious map between the spaces.
EDIT: Dylan actually gave what's needed to finish off a proof. The map given by the generator of $\pi_3$ can easily be checked to induce isomorphisms on all homology groups. Now replace the $3$-manifold $M$ by a $2$-connected CW-model $Z$ by CW-approximation. Functoriality of CW-models then induces a map $f:S^3\to Z$ which induces isomorphisms on homology. The standard argument that replaces $Z$ by the mapping cylinder of $f$ and then applies Hurewicz on $H_n(M_f,S^3)$ shows that $\pi_n(M_f,S^3)=0$ for all $n$ on which implies that $M_f$ deformation retracts onto $S^3$ and they are homotopy equivalent. This gives the following chain of homotopy equivalences
$$S^3\simeq M_f\simeq Z\simeq M$$
so it follows that $M$ and $S^3$ has the same homotopy type.

REPLY [2 votes]: I agree with the point of the question that one does want to know how to construct homotopy equivalences. 
One basic  tool for this is a gluing theorem for homotopy equivalences  which I have advertised on this stackexchange answer. It gives conditions for a pushout of homotopy equivalences to be a homotopy equivalence. 
The proof uses no notions of homotopy groups, but the result was found by starting with  the well known fact that a homotopy equivalence $f: Y \to Z$ of spaces induces an isomorphism of homotopy groups, and then generalising replacing $(S^n,x)$ by a pair $(X,A)$ with the Homotopy Extension Property. The theorem first appeared in the 1968 edition of the book now titled "Topology and Groupoids" . Its origin is also in special cases due to J.H.C. Whitehead.  There is also a dual "cogluing theorem", for fibrations and pullbacks, rather than cofibrations and pushouts. The result has  also been set up in various model categories, but an advantage of the original proof is that it gives control of the homotopies involved.