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.

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    $\begingroup$ I am not sure I understand your first claim: there are lens spaces which are homotopy equivalent but not homeomorphic. If you have no lens spaces in your prime decomposition, then okay, you're fine. But it is a result of Turaev (assuming geometrization) that simple homotopy type and homeomorphism type of 3-manifolds coincide. $\endgroup$
    – mme
    Nov 13 '18 at 14:46

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)


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