AGT Course

Algebraic and Geometric Topology. Part 1



1. Extension and lifting problems in topology
   
   1.1. Homotopies and homotopy equivalences 
   1.2. The identity map extension problem 
   1.3. The identity map lifting problem 

2. Homotopy groups of spheres and Hopf fibrations

    2.1. The n-th homotopy group of the n-sphere
    2.2. The degree of a self map of the n-sphere
    2.3. The Hopf fibrations and their applications
    
3. Low dimensional manifolds
   
    3.1. Compact manifolds of dimensions 1 and 2  
    3.2. The Poincaré conjecture in dimension 3 
    3.3. Spherical manifolds of dimension 3 

4. Knots, links, and braids
   
    4.1  The genus and signature of knots
    4.2  The Alexander and Jones polynomials of links
    4.3  The braid groups and 
   
5. High dimensional manifolds
   
    5.1. Embeddings of manifolds into Euclidean spaces 
    5.2. The Stiefel and Grassmann manifolds
    5.3. The generalized Poincaré conjecture

Algebraic and Geometric Topology. Part 2
 

1. The fundamental group and coverings 

    1.1  The Seifert--Van Kampen theorem 
    1.2  The automorphism group of a covering 
    1.3  The classification of coverings

2. Homology and cohomology groups
 
    2.1  Homology groups - singular and cellular
    2.2  Cohomology groups and rings
    2.3  The Poincaré duality for manifolds

3. Homotopy groups of general spaces
 
     3.1  Homotopy groups and the Hurewicz theorem  
     3.2  The Freudenthal suspension theorem  
     3.3  The long exact sequence of a fibration
 
4. Homotopy theory of CW complexes
 
    4.1  Aspherical CW complexes and the Borel conjecture
    4.2  Eilenberg-McLane spaces and the Postnikov systems
    4.3  The Whitehead Theorem 
 
5. Smooth manifolds and vector fields
 
    5.1  Intersection numbers and transversality
    5.2  The Lefschetz fixed point theory
    5.3  The Poincare-Hopf Theorem


Algebraic and Geometric Topology. Part 3

1. Vector bundles

2. Characteristic classes

3. Topological K-theory

4. Cobordism theory

5. Manifolds and modular forms