1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2019/2020
4. Mathematics - Second-cycle studies leading to MS degree
5. Topology

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Topology - course description

Aim of the course

The basic notions of algebraic and geometric topology.

Prerequisites

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General topology, group theory.

Scope

Lecture
The Fundamental group
1. Homotopy (2 h)
2. Retractions (1 h)
3. Construction of the fundamental group (3 h)
4. The Fundamental group of the Cartesian product (1 h)
5. Symplices and symplicial complexes (2 h)
6. Calculating of the fundamental groups (2 h)
7. The fundamental group of the circle, the torus, the sphere, the projective plane (2 h)
The Jordan theorem (proof), the Schoenfliesa theorem (3 h)
Topology in art – Alexander’s sphere, Wady’s leaks, art of M.C. Escher (2 h)
Classification Theorem for Surfaces
1. Surfaces (1 h)
2. Polytopes (1 h )
3. Triangulation of surfaces (1 h)
4. The proof of Classification Theorem for Surfaces (2 h )
The Borsuka-Ulama theorem
1. The various forms of the Borsuk-Ulam theorem (2 h)
2. The Tucker lemma and the proof of the Borsuk-Ulam theorem (2 h)
3. Applications of the Borsuk-Ulam theorem (2 h)
4. The Brouwer fixed-point theorem (2 h)
Degree of mappings. (2 h)

Class
Topologies
1. Basic exercises on topologies (1 h)
2. Examples of topologies (1 h)
Homotopy
1. Exercises on homotopy and equivalence relations (2 h)
2. Exercises dealing with the construction of fundamental group (3 h)
3. Exercises on retractions (1 h)
4. Exercises on the fundamental group (3 h)
Classification Theorem for Surfaces
1. -Exercises on classification of surfaces (2 h)
2. Exercises on triangulations of surfaces (1 h)
The Borsuka-Ulama theorem
1. Proofs of various versions of The Borsuka-Ulama theorem (4 h)
2. Exercises which use The Borsuka-Ulama theorem (2 h)
3. Proof of the Sperner lemma (2 h)
Presentations and class tests (6 h)

 

Teaching methods

Lectures and disccusions.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Exams and talks.

Recommended reading

1. Roman Duda, Wprowadzenie do topologii I, II, PWN, 1986.
2. Jiri Matousek, Using the Borsuk-Ulam theorem, Springer, 2003

Further reading

1. Jerzy Mioduszewski, Wykład z topologii, Wydawnictwo Uniwersytetu Śląskiego, 1994

2. Allen Hatcher, Algebraic Topology, www.math.cornell.edu/~hatcher/

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 20-09-2019 11:17)