Abstract: Obstruction theory is a machinery originated in homotopy theory that can be a precious instrument in your toolbox. Among the questions that can be addressed with it are the classification of bundles, the existence of certain structures (e.g. almost complex, spin structures etc..), the existence of sections of bundles and the classification of maps X→Y up to homotopy. Moreover, a large amount of topological invariants admit a geometric interpretation in terms of obstructions. Just to name a few: the Euler/Chern/Steifel-Whitney classes and the Kirby-Siebenmann invariant.One of the aims of this talk is to provide (some of) the basis to face the proofs (not presented in the talk) of many theorems that are often taken as common knowledge:
1) A closed, connected manifold has zero Euler characteristic iff it has a non-vanishing vector field,
2) A manifold is orientable iff the first Steifel-Whitney class is trivial,
3) An orientable manifold is spin iff the second Steifel-Whitney class is trivial,
4) Orientable 3 manifolds are parallelizable,
5) complex line bundles are classified by the first Chern class,
6) SU(2) bundles over 4 manifolds are classified by the second Chern class.