# 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.

• 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. – Mike Miller 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.
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.