Kronheimer, P.B.; Mrowka, T.S., Witten’s conjecture and Property P, Geom. Topol. 8, 295-310 (2004). ZBL1072.57005.

This paper is rather deep in the field of 3-manifold topology, using most of the major developments of low-dimensional topology from the previous 30 years. The main theorem states that a homotopy 3-sphere cannot occur as non-trivial surgery on a knot. Of course, this also follows now from the geometrization theorem and the knot complement problem. However, at the time of publication the geometrization theorem had not been vetted or published. Tracing back the proofs of theorems that this relies on involves the fields of

• Riemannian geometry (e.g. used in instanton homology)

• Algebraic geometry (featuring heavily in the proof of the cyclic surgery theorem)

• Complex analysis (used in Thurston's proof of geometrization of Haken 3-manifolds, as well as pseudo-holomorphic curves I suppose)

• Dynamics (used in Thurston's proof again, e.g. in Sullivan rigidity, a generalization of Mostow rigidity)

• Analysis and PDEs (for gauge theory)

• Mathematical Physics, in the guise of gauge theory, but specifically the work on Witten's conjecture of the equivalence between Seiberg-Witten and Donaldson invariants. This conjecture was motivated by ideas from string theory, so is not rigorous mathematics.

• and of course Topology, with quite a few specialties involved (foliations, symplectic and contact structures, 3- and 4-dimensional manifolds, Kleinian groups, Morse Theory).

I should also comment that there are now shorter proofs of this and related theorems independent of the Poincaré conjecture available that don't use quite as much gauge theory. And one can substitute Perelman's proof of geometrization for Thurston's, which substitutes Riemannian geometry and PDEs for complex analysis and dynamics.