Algebraic Topology

Algebraic and Geometric Topology. Part 1



1. The fundamental group and coverings
   
   1.1. The Seifert--Van Kampen Theorem 
   1.2. The covering automorphisms group 
   1.3. The classification of coverings

2. Homotopy groups and fibrations

    2.1. Homotopy groups of spheres and the mapping degree  
    2.2. The Freudenthal suspension theorem 
    2.3. The long exact sequence of a fibration 
    
3. Low dimensional manifolds
   
    3.1. Classification of compact manifolds of dimensions 1 and 2  
    3.2. Spherical manifolds of dimension 2 and 3
    3.3. Aspherical manifolds of dimension 2 and 3  

4. Knots, links, and braids
   
    4.1  The genus, the signature, and the group of a knot
    4.2  The Alexander, Jones, and HOMFLY-PT polynomials 
    4.3  Braid groups and mapping class groups of punctured disks 
   
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