Topological results from geometry Hi people,
I'm interested in results, such as the Gauß-Bonnet theorem, Fàry-Milnor theorem or classification theorems for manifolds, which give topological properties from geometric considerations. Can anyone recommend some good texts? In particular I'd like to see a nice proof of Fàry-Milnor and of the theorem of turning tangents (total curvature of an imbedded plane curve is $2\pi$).
thanks
 A: About the Fary–Milnor theorem.  Milnor's original proof is already very nice (see here).  I also very much like this proof by Alexander & Bishop (see also a version of this proof in my book).
A: A nice topic to read about is Chern-Weil theory. This is the generalisation of Gauss-Bonnet to higher dimensions and to vector bundles other than the tangent bundle. Put very briefly, topological invariants of a vector bundle over a manifold (its characteristic classes - certain classes in the cohomology of the base) can be computed using the curvature tensor of any choice of connection in the bundle. 
The prototype is Gauss-Bonnet in which, as you know, the Euler characteristic of a (compact orientable) surface is equal to a fixed constant times the integral of the scalar curvature of any Riemannian metric on the surface.
A: This topic being quite large, I cannot insist enough to recommand you to take a look to Marcel Berger's Panoramic view of Riemannian geometry. The Bonnet-Myers theorem, the sphere theorems (for the recent development on this one, I think the web page of Simon Brendle contains a survey) are two celebrated examples of the topological consequences of geometric properties in the setting or Riemannian geometry.
A: I think you should read something about the Ricci flow and Perelmann s work (for 3mfs), or Seiberg Witten/Yang-Mills theory (for 4-mfs). These theories give you very deep results in topology. But the hole theory is geometric.
A: You might want to look up some things about index theorems (particularly Atiyah-Singer).  They tend to relate topological and geometric data, so you can put geometric data in and topological data out.
A: Here is an example where topological objects are constructed from geometrical data through representation theory. Let G/P be a flag variety of a complex Lie group G. Let G0 be a real form of G, and D be an open orbit of G0 in G/P. The Dolbeault cohomology spaces H^n(D, L) of line bundles over D carry irreducible representations of G0 which can
be constructed from geometrical data of the orbit. Here is a review article on the subject. When G0 is compact, this construction reduces to the famous Bott-Borel-Weil theorem.
