We will be using the following two textbooks:
1) Homotopic topology, by A.Fomenko, D.Fuchs, and V.Gutenmacher.
Chapters 1 and 2: Homotopy and Homology,
Chapter 3: Spectral sequences,
Chapter 4: Cohomology operations,
Chapter 5: The Adams spectral sequence,
Index.
2) Algebraic
Topology by Alan Hatcher, Cambridge U Press.
Free download; printed version can be bought cheaply online.
Homotopy and homotopy equivalence. CW complexes. Cellular approximation.
Category theory, functors and adjointness.
Fundamental group and its computation. Coverings and their classification.
Fibrations and Serre fibrations. Relative homotopy groups.
Complexes and exact sequences. Homotopy sequence of a fibration.
Homotopy groups of CW complexes.
Weak equivalence and cellular approximation. Eilenberg-Maclane spaces.
Homology theory:
Chain complexes and chain maps. Homology of complexes.
Singular homology, homology of CW comlexes, computations.
Homotopy and homology, Hurewicz theorem.
Cohomology groups. Homology and cohomology with coefficients.
Kunneth formula. Multiplications in cohomology.
Applications of homology and cohomology.
Manifolds, Poincare duality.