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. and D. Xu, A Guide to the Classification Theorem for Compact Surfaces. Springer. Berlin, 2013.
Goodman, S. E., Beginning Topology. AMS. Providence, R. I., 2009.
Griffiths, H. B., Surfaces. Cambridge University Press. Cambridge, 1976.
Hu, S.T., Elements of General Topology. Holden-Day. San Francisco, 1969.
Katok, A. and V. Climenhaga, Lectures on Surfaces: (almost) 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.
Margalef, J., E. Outerelo and J. L. Pinilla, Topología 5. Alhambra. Madrid, 1982.
Masa Vázquez, X. M., Topoloxía Xeral. USC. Santiago de Compostela, 1999.
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.
Scenario 1.
We will follow the general methodology for all the subjects of the degree that appears in the "Guide to the Faculty of Mathematics".
Scenario 2.
Depending on the type of attendance restrictions determined by the Faculty and provided that the USC provides the necessary means for this, classes that cannot be taught in person will be taught virtually. It will be done through institutional means (Virtual Campus, Teams, email), preferably synchronously, although subject to what the Faculty determines.
Scenario 3.
If the USC provides the necessary means for this, the classes of scenario 1 will be taught virtually through institutional means (Virtual Campus, Teams, email), preferably synchronously, although subject to what the Faculty determines.
Scenario 1.
We will use the general criterion of evaluation for all the subjects of the degree that appears in the "Guide to the Faculty of 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. 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.
Scenarios 2 and 3.
In Scenario 3, and possibly also in Scenario 2, both the continuous assessment and the final exam will be virtual. In the continuous evaluation exercises and questions will be proposed. The delivery times of the solutions will vary according to the difficulty, from immediate delivery to one week. In these scenarios, the final grade will be the sum of 50% of the continuous assessment grade with 50% of the final exam grade.
For the three scenarios, the same evaluation conditions and the continuous evaluation grade of the first opportunity will be maintained in the second opportunity.
In cases of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of the academic performance of students and the review of grades 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.
Contingency plan
The main changes in scenarios 2 and 3 are:
- Teaching will be taught virtually through institutional means (Virtual Campus, Teams, email), preferably synchronously, although subject to what the Faculty determines.
- The continuous evaluation and the final test become virtual, through the previous means.
- The final mark will be the sum of 50% of the mark of the continuous evaluation with 50% of the mark of the final exam.
Jesús Antonio Álvarez López
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813149
- jesus.alvarez [at] usc.es
- Category
- Professor: University Professor
Jose Manuel Carballes Vazquez
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813146
- xm.carballes [at] usc.es
- Category
- Professor: University Lecturer
| Monday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLE_02 | Galician | Classroom 08 |
| 12:00-13:00 | Grupo /CLE_01 | Galician | Classroom 07 |
| Tuesday | |||
| 09:00-10:00 | Grupo /CLE_02 | Galician | Classroom 08 |
| 10:00-11:00 | Grupo /CLE_01 | Galician | Classroom 07 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLIL_05 | Galician | Classroom 02 |
| 10:00-11:00 | Grupo /CLIL_03 | Galician | Classroom 03 |
| 11:00-12:00 | Grupo /CLIL_06 | Galician | Classroom 02 |
| Thursday | |||
| 09:00-10:00 | Grupo /CLIL_02 | Galician | Classroom 02 |
| 10:00-11:00 | Grupo /CLIL_01 | Galician | Boardroom |
| 10:00-11:00 | Grupo /CLIL_04 | Galician | Classroom 02 |
| 05.25.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 05.25.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 05.25.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 05.25.2021 10:00-14:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 07.14.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |