Algebraic Topology

Recommended Prerequisites

Group theory. General topology.

Teaching Methods

Classes are expository, but also demanding the participation of students, who are expected to present proofs of theorems and solve proposed exercises.

Learning Outcomes

In this curricular unit the basic theory of algebraic topology is introduced, namely the concepts of homology and homotopy. The concepts are applied to important classes of topological spaces, such as surfaces or CW-complexes and spaces obtained as quotients or coverings of other spaces.

The following generic competences are developed: knowing mathematical results, ability to generalize and abstract; logic argumentation; written and oral rigorous and clear expression; ability to do research; ability to do autonomous learning;  imagination, creativity and critical thinking.

Work Placement(s)

No

Syllabus

Quotient topology. Surface classification. Group actions. CW-complexes. Fundamental group. Van Kampen theorem. Coverings. Homology groups. Mayer-Vietoris sequence. Homotopy groups.

Head Lecturer(s)

Gonçalo Gutierres da Conceição

Assessment Methods

Continuous assessment
Two mid-term exams : 100.0%

Final assessment
Exam: 100.0%

Bibliography

A. Salgueiro, Topologia Algébrica, Departamento de Matemática da FCTUC, 2009.

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