My examples of "Proofs from the Geometry/Topology book" (needs a lot of expansion)

1. Modular forms and quadratic forms which represent the same numbers equally often. The Milnor examples.
2. Sunada method (Note nonarithmetic surfaces of constant curvature which are isospectral)
3. Atiyah-Hirzebruch theorem on vanishing of A-genus in the presence of a circle action.
4. Counting hyperbolic manifolds./Hidden symmetry (see also Sunada)
5. Counterexamples to equivariant Borel/BCHSW theorem/Chang-W theorems on nonrigidity in two settings.
6. Homology propagation, Zabrodsky mixing, continuous versus discrete symmetry (for circle actions on the sphere).
7. Gromov-Lawson, almost flat bundle argument and Lusztig's thesis
8. Surjecting onto free groups, deficiencies, euler characteristics of 4-manifolds, together with L^2 variants. (Golod Shaferevich?) Combine with information about deficiencies...
9. Whyte quasi-isometries. In fact, the who q.i. idea, and also proofs of Novikov and rigidity by rescaling.
10. Margulis, property T -- nonamenability and tilings, measure transport idea. Nonembedding in Hilbert space.
11. Closed geodesics on a manifold (Lusternick Fet) + on manifold with unsolvable word problem.
12. Exotic spheres.
13. The whole approach to Hopf conjecture via L^2 + Kahler hyperbolicity
14. Torelli isn't finite type
15. Brouwer and Lefshetz f.p.theorems, Smith theorem (a la Borel)
16. Hopf's theorem on cohomology of Lie groups. Application to TSC.
17 Bott periodicity and application to Hopf invariant with application to division algebras
18. Poincare recurrence, ergodic theorem, rotation numbers, ....
19. Dehn function of heisenberg + wirtinger inequality and minimal surfaes....
20. Milnor counterexamples to Hauptvermutung.
21 Some double suspensions or non-manifolds x R which are manifolds.
22. Some interesting nonlinear actions on the sphere from h-cobordism. Various sources.
23. Hilbert cube is homogeneous. Q x T = Q.
24. A homeomorphism of R^2 with dense orbits. (It's the random measure preserving tranformation on an open disk.)