Algebraic topology

computer science
mathematics
physics

elective courses

Algebra I 1000-113bAG1a
Linear algebra and geometry I 1000-111bGA1a
Linear algebra and geometry II 1000-112bGA2a
Mathematical analysis II.1 1000-113bAM3a
Mathematical analysis II.2 1000-114bAM4a
Topology I 1000-113bTP1a

Preliminary knowledge of basic properties of homotopy and covering spaces, discussed in the course Topology II (1000-134TP2) would be helpful. Topics concerning homology will be discussed independently.

Homotopy groups. Fibrations and cofibrations. Long exact sequence of homotopy grooups of fibration. Axioms for generalized (co-)homology. Singular (co-)homology. Degree of self-maps of spheres. Cellular (co-)homology. De Rham cohomology. Multiplicative structure in singular (co-)homology. Orientation of topological manifolds and duality theorems. Intersection number and linking number.

1. Homotopy - basic properties (summary). Homotopy extension property (cofibrations) and homotopy lifting property (fibrations). Homotopy groups and Long exact sequence of homotopy grooups of fibration. The Hopf fibration. The Eilenberg-MacLane spaces.

2. Axioms for (co-)homology. Singular (co-)homology. Acyclic models. Mayer-Vietoris exact sequence. De Rham cohomology and de Rham theorem.

3. Homotopy classification of self-maps of spheres via Brouwer degree.

4. CW-complexes and cellular (co-)homology.

5. The Eilenberg-Zilber theorem and multiplicative structures in singular (co-)homology. The Hopf invariant.

6. Geometric and homological orientations of manifolds. Duality theorems (Poincare, Alexander, Lefschetz). Geometric ans homological interpretation of the interesection number and the linking number. The Lefschetz fixed-point theorem.

1. G. Bredon, Topology and Geometry, Graduate Texts in Mathematics 139, Springer Verlag, New York 1993

2. Fulton, W. Algebraic Topology. A First Course. GTM 153. Springer

3. Greenberg, M.J., Harper, J.R. Algebraic Topology. A First Course.

4. Hatcher, A. Algebraic Topology, Cambridge University Press, Cambridge 2002

5. May J.P. , A Concise Course in Algebraic Topology. Chicago Lecture Notes in Mathematics, The University of Chicago and London, 1999

6. E. Spanier, Algebraic Topology, McGraw-Hill

At the end of the course student will be able

- to formulate notions and theorems included in the course and explain them on geometric examples;

- to prove selected theorems and calculate some homological invariants;

- to explain connection between geometric and homological invariants of manifolds.

the final grade will be based on students' performance during the semester and a written exam