Course Identification

Basic algebraic topology in the amount covered by preceding course "Algebraic Topology" (and hence also all its prerequisites):

Basic  group theory (up to the classification of finitely generated Abelian groups and notions of normal subgroups and conjugasy classes); 

Basic linear algebra (rank of a linear operator, classification of quadratic forms...) 

Basic analysis of functions of several variables (up to Implicit Function theorem and Taylor series),   

Basic topology of subsets of Euclidean spaces (notions of closed, open, compact, connected sets, continuous functions and their topological properties).

An introduction to the advanced methods and notions of algebraic topology: homology of local systems, obstruction theory, spectral sequences of filtered topological spaces, spectral sequences of fiber bundles and the multiplication in them, characteristic classes of vector bundles and their applications in topology of smooth manifolds.

One will be able to calculate homology and cohomology of not so easy topological spaces; calculate spectral sequences of filtered spaces and fiber bundles; construct homological obstructions to the extension of maps and sections of fiber bundles; calculate Stiefel--Whitney and Euler characteristic classes of vector bundles and apply them as obstructions to embeddings, immersions, cobordisms, etc. Also, the homological technique of spectral sequences is widely applied in algebraic geometry and complex analysis in the context of sheaf cohomology, so one will be better prepared to the reading of works on these topics.

A.Fomenko, D. Fuchs, Homotopical Topology