Content:

• Homology and cohomology of manifolds proving Poincare duality isomorphism,
• Cohomology operations introducing in particular Steenrod squares,
• Vector bundles, and
• Characteristic classes, in particular we will introduce Stiefely-Whiteny classes using Steenrod squares.

Introduced methods will be illustrated by applications on different classical problems of algebraic topology.

The books that can be used for these topics are:

1. G. Bredon: Topology and Geometry, Springer GTM 139,
2. J. Milnor, J.D. Sasheff: Characteristic classes, (AM-76) Princeton University press
3. Allen Hatcher: Vector Bundles & K-theiry, online book
   http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html