Topology of Manifolds
Code: 100114 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Mathematics | OT | 4 |
Contact
- Name:
- Natalia Castellana Vila
- Email:
- natalia.castellana@uab.cat
Teachers
- Natalia Castellana Vila
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
The prerequisites are the first and second year courses as well as the Topology course. It is recommended to have taken the subject of Differential Geometry.
Objectives and Contextualisation
In this course we introduce the most basic algebraic invariants that we can associate to a topological space (particularly to a manifold) and that provide us with a first approximation to the global properties of these objects.
With special emphasis on the fundamental group, homology groups and cohomology algebras, as well as orientabiity.
Competences
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Apply critical spirit and thoroughness to validate or reject both one's own arguments and those of others.
- Demonstrate a high capacity for abstraction.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
- Effectively use bibliographies and electronic resources to obtain information.
- Formulate hypotheses and devise strategies to confirm or reject them.
- Generate innovative and competitive proposals for research and professional activities.
- Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
Learning Outcomes
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Apply critical spirit and thoroughness to validate or reject both one's own arguments and those of others.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
- Devise demonstrations of mathematical results in the field of geometry and topology.
- Effectively use bibliographies and electronic resources to obtain information.
- Generate innovative and competitive proposals for research and professional activities.
- Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- Understand abstract language and in-depth demonstrations of some advanced theorems of geometry and topology.
Content
The course will address the topics below.
- Definition and examples of homotopy of applications and spaces.
- Topological and differentiable varieties.
- Chain complexes.
- Homology and cohomology.
- Orientability.
- Fundamental group and covering spaces.
In addition, these topics will provide the following remarkable results:
- Classification of related compact surfaces.
- Brouwer's fixed point theorem.
- Jordan-Brouwer separation theorem.
- Topological invariance of the dimension of a variety.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Problem Sessions | 15 | 0.6 | 2, 1, 3, 6, 10, 8, 7 |
| Theory classes | 30 | 1.2 | 2, 1, 3, 6, 10, 8, 7 |
| Type: Supervised | |||
| Seminars | 6 | 0.24 | 2, 1, 3, 6, 10, 8, 7 |
| Type: Autonomous | |||
| Assimilations of theoretical results | 45 | 1.8 | 6, 10, 8, 5 |
| Homework | 15 | 0.6 | 2, 11, 3, 10, 8, 5 |
| Solving problems | 30 | 1.2 | 11, 4, 10, 8 |
In the lectures the concepts, arguments and basic results of the subject are exposed, where one expects an active participation of the students. This is complemented by problem sessions, seminars and participatory oral presentations from students, according to the contents of the course, where the students achieve the knowledge and the capacities to use these materials in readings or studies of close or more advanced topics.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Exam | 40% | 4 | 0.16 | 11, 1, 3, 4, 9, 7, 5 |
| Homework assignments | 30% | 4 | 0.16 | 11, 3, 4, 10, 9, 5 |
| Oral presentation, type P. | 30% | 1 | 0.04 | 2, 11, 1, 3, 6, 9, 8, 7, 5 |
The qualification is the weigthed mean of the following marks:
- exercises (30%),
- exam (40%)
- ellaboration of a work on a proposed subject (30%)
A minimum mark of 3.5 for each evaluated activity is required. If necessary, the exam and the work will be reevaluated. The "matrícula d'honor" will be decided before reevaluations.
The one day assessment (avaluació unica) will take place on the same day as the final course presentations. The one day assessment will consist of the delivery of exercices (previously assignes), the final presentation and an exam,
Disclaimer: I have made my best to translate into English the Catalan version. In the unlikely case of differences between versions, we'll follow the criteria in the Catalan original version.
Bibliography
Main references:
- W. Fulton, Algebraic topology. A first course. Graduate Texts in Mathematics, 153. Springer-Verlag, New York, 1995.
- A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)
- J. M. Lee, Introduction to Topological Manifolds, , Graduate Texts in Mathematics 202, Springer-Verlag, New York, 2011.
Additional refs:
- L. W. Tu, An introduction to manifolds. Universitext. Springer, New York, second edition, 2011.
- R. Bott and L.W. Tu, Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, NewYork-Berlin, 1982.
Online resources:
- Clara Löh, Algebraic Topology, https://loeh.app.uni-regensburg.de/teaching/topologie1_ws2122/lecture_notes.pdf
- Emily Riehl, Categorical homotopy theory, https://math.jhu.edu/~eriehl/cathtpy/
Software
No specific software will be used.
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PAUL) Classroom practices | 1 | Catalan/Spanish | first semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan/Spanish | first semester | afternoon |
| (TE) Theory | 1 | Catalan | first semester | morning-mixed |