Structure sets for three dimensional surgery Is  there  a  treatment  in the  literature of  the  structure  sets  relating  simple  homotopy  equivalences  to homeomorphisms  in the  three  dimensional  case? I am  aware  that  due to  the  geometrization  conjecture, homotopy  equivalences and homeomorphism agree. I  am  also aware of  Jahren and  kwasik's paper Three-dimensional surgery theory, UNil-groups and the Borel
conjecture  in Topology 42 (2003) 1353–1369. 
 A: It is well known that the surgery sequence completely breaks down in dimension 3, so looking at structure sets is not really the right thing to do. For instance, in the classification of high-dimensional lens spaces, a given simple homotopy type (as in Mike Miller's comment) will contain many homeomorphism types (detected by `multisignatures'). But geometrization says that this doesn't hold in dimension 3. This represents a failure of the L-group $L_4$ to act on the structure set of (say) a lens space. 
Likewise, there are normal maps whose corresponding surgery problems with finite fundamental group have trivial surgery obstructions, but don't have a solution. Again this is from geometrization, which tells you that the finite fundamental groups of 3-manifolds are those of spherical space forms. You can learn about this in the book A survey of the spherical space form problem by Davis and Milgram. 
Such failures are traceable to either of these facts: (1) surgery along circles in a 3-manifold generally changes the fundamental group, or (2) it's not so easy to find embedded spheres in 4-manifolds. The best one can usually do is surgery up to homology equivalence. This is basically folklore; it is briefly discussed in the Kirby-Taylor article "A survey of 4-manifolds through the eyes of surgery".
There are some consistencies with surgery theory and 3-manifold theory; for instance I seem to recall that (with a few exceptions) the fundamental groups of Haken manifolds have vanishing $L_3$, which is consistent with Waldhausen's theorem that such manifolds are classified by their fundamental group.
Edit: There is more about what surgery theory looks like in dimension 3 in an article of Jahren-Kwasik (Three-dimensional surgery theory, UNil-groups and the Borel conjecture, Topology 42 (2003) 1353–1369)
