ECTS credits ECTS credits: 4.5
ECTS Hours Rules/Memories Student's work ECTS: 74.2 Hours of tutorials: 2.25 Expository Class: 18 Interactive Classroom: 18 Total: 112.45
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Geometry and Topology
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
- Study of the concepts of compactness and connectedness in topological spaces.
- Formalization of the idea of continuous deformation.
- Introduction of the fundamental group and its computation for simple spaces.
- Use of the fundamental group to classify compact surfaces.
1.- Compactness. (3 teaching hours).
- Compact spaces.
- Local compactness.
- Compactification.
2.- Connectedness. (5 teaching hours).
- Connected spaces. Components.
- Pathwise connectedness.
- Local connectedness.
3.- Homotopy. (5 teaching hours).
- Homotopy of maps.
- Homotopy Type. Deformation.
- Homotopy of paths.
4.- Fundamental Group. (10 teaching hours).
- Loops. Fundamental Group.
- Computation of the Fundamental Group.
- The Fundamental Group of quotients of polygonal regions.
- First Homology Group.
5.- Compact Surfaces. (5 teaching hours).
- Surfaces and surfaces with boundary.
- Plane models and schemes.
- Equivalence of schemes.
- Classification. The Euler characteristic.
Basic:
- Kosniowski, C. , Topología algebraica. Editorial Reverté. Barcelona, 1989.
- Massey, W. S., Introducción a la topología algebraica. Editorial Reverté. Barcelona, 1972.
- Munkres, J. R., Topología. Prentice-Hall. Madrid, 2002.
Complementary:
- Armstrong, M. A., Topología básica. Editorial Reverté. Barcelona, 1987.
- Carlson, S. C., Topology of Surfaces, Knots, and Manifolds: A First Undergraduate Course. John
Wiley & Sons. New York, 2001.
- Dugundji, J., Topology. Allyn and Bacon. Boston, 1966.
Gallier, J. e D. Xu, A Guide to the Classification Theorem for Compact Surfaces. Springer. Berlín, 2013.
- Goodman, S. E., Beginning Topology. AMS. Providence, R. I., 2009.
- Griffiths, H. B., Surfaces. Cambridge University Press. Cambridge, 1976.
- Hatcher, A., Algebraic Topology. Cornell University, 2001. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
- Hu, S.T., Elements of General Topology. Holden-Day. San Francisco, 1969.
- Katok, A. e V. Climenhaga, Lectures on Surfaces: (almost) everything you wanted to know about them. AMS. Providence, R. I., 2008.
- Kinsey, L. C., Topology of Surfaces. Springer. New York, 1993.
- Lee, J. M., Introduction to Topological Manifolds. Springer. New York, 2000.
- Lima, E. L., Grupo fundamental e espaços de recobrimento. IMPA. Rio de Janeiro, 1998.
- Macho Stadler, M., Topología. EHU, Leioa, 2015. https://www.ehu.eus/~mtwmastm/Topologia1415.pdf
- Macho Stadler, M., Topología Algebraica. EHU, Leioa, 2006. https://www.ehu.eus/~mtwmastm/TopoAlg0607.pdf
- Margalef, J., E. Outerelo e J. L. Pinilla, Topología 5. Alhambra. Madrid, 1982.
- Masa Vázquez, X. M., Curso de Topoloxía. USC. Santiago de Compostela, 2019.
- Muñoz, V., González-Prieto, Á., Rojo, J. Á. Geometry and Topology of Manifolds. Surfaces and Beyond. AMS. Providence, R.I., USA, 2020.
- Petit, J.-P., El Topologicón. Saber sin Fronteras, 2006. http://www.savoir-sans-frontieres.com/JPP/telechargeables/ESPANOL/El_To…
- Willard, S., General Topology. Addison-Wesley. Reading, 1970.
General:
- The general ones of the degree; in particular: the understanding and use mathematical language (CE1) and the knowledge of how to abstract the structural properties and be able to prove them with proofs or refute them with counterexamples (CE3, CE4).
- The general ones of the module; in particular: the knowledge and use of the concepts, methods and basic results of Topology, to acquire intuition in the study of abstract topological spaces and to have examples that illustrate diverse properties.
Specific:
- The generalization to topological spaces of concepts already known in Euclidean spaces.
- Understanding, recognition and use of the notions of compactness and connectedness.
- Development of the ability to intuitively recognize homotopic equivalence.
- Computation and use of the fundamental group.
- Topological recognition of compact surfaces and their classification.
Transversal:
- The transversal ones of the module: the practice of formal mathematical writing.
- The use of logical reasoning to solve problems.
- The transformation of topological problems into algebraic problems.
We will follow the general methodology for all the subjects of the degree that appears in the Memory of the Degree in Mathematics.
We will use the general criterion of evaluation for all the subjects of the degree that appears in the Memory of the Degree in Mathematics, granting to the continuous assessment a weight of 30% in the final grade, which will be given by control exams and the solution of problems (continuously in the interactive seminar/laboratory classes). In particular, the final grade will be the maximum of the final exam grade and the sum of 30% of the continuous assessment grade with 70% of the final exam grade. The exams of each group will be independent.
In the second opportunity, the grade of the continuous evaluation of the first opportunity will be maintained.
For cases of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of students' academic performance and the review of qualifications will apply.
According to the grade memory, the working time necessary to pass the subject is 112.5 hours distributed in the following way:
PRESENTIAL WORK IN THE CLASSROOM
Classes in large group: 28 h.
Small group classes: 14 h.
Sessions in very small groups: 2 h.
Total face-to-face work: 44 h.
PERSONAL WORK
Autonomous study: 45 h.
Writing of exercises or other works: 15 h.
Recommended readings or similar: 7.5 h.
Total personal work: 67.5 h
- Having previously studied the subjects "Topoloxía two espaces euclidianos", "Topoloxía xeral" and "Estruturas alxébricas".
- Attending classes and actively participate in the continuous assessment program.
Jesús Antonio Álvarez López
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813149
- jesus.alvarez [at] usc.es
- Category
- Professor: University Professor
Jose Manuel Carballes Vazquez
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813146
- xm.carballes [at] usc.es
- Category
- Professor: University Lecturer
Alejandro Omar Majadas Moure
- Department
- Mathematics
- Area
- Geometry and Topology
- alejandro.majadas [at] usc.es
- Category
- Xunta Pre-doctoral Contract
| Tuesday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLE_01 | Galician | Classroom 03 |
| 11:00-12:00 | Grupo /CLE_02 | Galician | Classroom 06 |
| Wednesday | |||
| 09:00-10:00 | Grupo /CLIL_04 | Spanish, Galician | Classroom 07 |
| 10:00-11:00 | Grupo /CLIL_06 | Galician, Spanish | Classroom 07 |
| 11:00-12:00 | Grupo /CLIL_05 | Galician, Spanish | Classroom 07 |
| Thursday | |||
| 11:00-12:00 | Grupo /CLIL_03 | Galician | Classroom 07 |
| 12:00-13:00 | Grupo /CLIL_01 | Galician | Classroom 07 |
| 13:00-14:00 | Grupo /CLIL_02 | Galician | Classroom 07 |
| 06.03.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 07.03.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |