Algebraic Topology

The course has two parts: Part 1 concerning the study of covering spaces, both from a topological and homotopical perspective, and Part 2 concerning the study of singular homology.

Extra attention will be paid to the construction of examples and the computation of algebraic invariants.

PART 1 : Topological and homotopical theory of covering spaces

1. Surfaces and manifolds (4 CLE + 1 CLIL)
Attaching of spaces. Closed surfaces. Projective and lens spaces. Topological and differentiable manifolds.

2. Group actions and orbit spaces (4 CLE + 1 CLI)
Topological groups and Lie groups. Group actions. Orbit spaces. Examples: closed surfaces of non-negative constant curvature.

3. Properly discontinuous actions (2 CLE + 1 CLIL)
Properly discontinuous actions. Discrete groups of isometries.

4. Covering spaces (6 CLE + 2 CLIL)
Definition of covering space. The fundamental example: fundamental: action of a discrete subgroup of a topological group. Deck automorphism group of a covering. Regular covering. Application: quotient manifolds and closed surfaces of negative constant curvature.

5. Homotopical theory of covering spaces (6 CLE + 2 CLIL)
Homotopy of maps. Deformation retracts and contractible spaces. Homotopy of paths. Homotopy lifting property. Fundamental group of a covering. Classification of covering spaces. Universal covering. Existence of universal covering spaces. Applications.

PARTE 2 : Singular homology

6. Singular homology (5 CLE + 2 CLIL)
Simplexes. Simplicial complexes and maps. Singular chain complex. Singular homology. Effect of continuous maps and topological invariance. Homology of a contractible spaces. Chain homotopy. Homotopical invarance. Relation with the fundamental group.

7. Mayer-Vietoris sequence (6 CLE + 3 CLIL)
Mayer-Vietoris exact sequence. First application: homology of spheres and degree of maps between spheres. Second application: suspensions and homology of suspensions. Third application: homology of closed surfaces. Fourth application: homology of real and complex projective spaces.

8. Relative homology and excision (5 CLE + 1 CLIL)
Barycentric subdivision. Relative homology. Long exact sequence of a pair. Subdivisión baricéntrica. Homoloxía relativa. Sucesión exact larga dun par. Excision for singular homology. Interpretation of relative homology groups. Mayer-Vietoris revisited.

9. Applications (4 CLE + 1 CLIL)
Brouwer Theorem and Borsuk-Ulam Theorem. Poincaré Theorem. Poincaré index of a closed curve in the plane: Poincaré formula. Cauchy Theorem and D'Alembert Theorem. Jordan-Brouwer Separation Theorem. Invariance of domain.

Basic bibliography
Armstrong M. A., Topología básica. Editorial Reverté. Barcelona, 1987.
Croom F.H., Basic Concepts of Algebraic Topology. Springer-Verlag, New York, 1978.
Hatcher A., Algebraic topology. Cambridge University Press, Cambridge, 2002.
Munkres J. R., Elements of Algebraic Topology. Addison-Wesley, Menlo Park,1984.
Vick J. W., Homology Theory. Springer-Verlag, New York, 1994.

Complementary bibliography
Bourbaki N., Éléments de Mathématique. Topologie générale, chapitres 1 à 4. C.C.L.S, Paris, 1971.
Bredon G. E., Topology and Geometry. Springer-Verlag, Berlin, 1993.
Conlon L., Differentiable Manifolds. Birkhäuser, Boston, 2009.
Dubrovin B.A., Fomenko A.T., Novikov S.P., Modern Geometry. Methods and Applications. Springer-Verlag, New York, 1985
Dugundji J., Topology. Allyn and Bacon. Boston, 1966.
Eilenberg S., Steenrod N., Foundations of Algebraic Topology. Princeton University Press, Princeton, 1951.
Godbillon C., Éléments de Topologie Algébrique. Hermann, Paris, 1971.
Greenberg M. J., Harper, J. R., Algebraic Topology: a first course. Benjamin, Massachusetts, 1981.
Kosniowski C., Topología Algebraica. Editorial Reverté, Barcelona, 1986.
Lee J.M., Introduction to Topological Manifolds. Springer-Verlag, Berlin, 2000.
Massey, W. S., Introducción a la Topología Algebraica. Editorial Reverté, Barcelona, 1972.
Massey, W. S., A Basic Course in Algebraic Topology. Springer-Verlag, New York, 1991.
Massey W.S., Singular Homology Theory. Springer-Verlag, New York,1980.
May J.P., A Concise course in algebraic topology. University of Chicago Press, Chicago, 1999.
Spanier E., Algebraic Topology. Springer-Verlag, Berlin, 1995.
Steenrod N., The Topology of Fibre Bundles. Princeton University Press, Princeton,1951.

In addition to achieve the general and transverse competences taken up in the memory of the degree,

- To know the concept of covering space and its homotopical properties, and to be able to construct regular and non regular covering space of some topological spaces.

- To know basic concepts of Algebraic Topology.

- To apply skills gained in the previous studies of topology, geometry, and algebra to effectively compute the fundamental group and the homology groups of some topological spaces.

- To be able to apply some homological methods to solve topological and geometrical problems.

- To know examples and counterexamples illustrating learned properties.

3 lectures and 1 problem-based learning session per week. A periodic control of training will be done by means of the resolution of exercices and problems.

Continual assessment (40% final grade) based upon the participation of each student at class and final assessment (60% final grade) by means of a written test fixed in the calendar of the Faculty. A positive continual assessment will exempt from the total or partial realization of the written test.

58 hours of lectures
92 further hours' study