Differential Topology

Differential topology is the study of the topology of manifolds using invariants defined by means of a differentiable structure. The methods of differential topology are widely used in topology and in differential and algebraic geometry. The program carried out in the academic year 2023-24 is reported below. In the current academic year the course may undergo changes.

Program for the a.y. 2023/24:

- vector bundles on manifolds;
- bundles as a pull-back of the tautological bundle on the Grassmannian
- Thom's isomorphism, Thom's class
- the Poincarè dual of a submanifold
- transversality and transverse intersections of submanifolds
- statement of Sard's theorem
- stability and genericity of transversality
- the Euler class of a real and oriented vector bundle;
- the Euler class and cross sections of a bundle;
- Poincarè-Hopf theorem on the zeroes of a vector field;
- connections on vector bundles and curvature;
- invariant polynomials and the construction of characteristic classes;
- Chern classes and characters and their fundamental properties;
- Pontryagin classes;
- Chern and Pontryagin classes of the complex projective space;
- Pontryagin classes and cobordism (a hint);
- Euler's class as the Pfaffian of curvature;
- the generalized Gauss-Bonnet theorem;
- computation of the Betti numbers of a complex algebraic hypersurface.