Topologia Algébrica

Objectives

To know how to compute homology and cohomology using homotopy equivalences, exact sequences, cell decompositions, the Künneth formula and the universal coefficient theorems. To understand the basic applications of homology theory to the topology of euclidean spaces and manifolds.

Syllabus

Basic notions: Homotopy type. The homotopy extension property. Criteria for homotopy equivalence. Definition and basic properties of CW complexes. Homology: Singular and simplicial homology. Calculations. Homotopy invariance. Excision. Mayer–Vietoris sequence. The Euler characteristic. Relation with the fundamental group. The universal coefficient formula. Separation theorems and invariance of domain. Simplicial approximation. The Lefschetz fixed point Theorem. Cohomology: The universal coefficient theorem for cohomology. Definition and properties of the cross and cup products. The Kunneth formula for homology. The cap product. Manifolds and duality: Orientations. Cohomology with compact support. Poincaré duality. Alexander and Lefschetz duality. Possible additional topics: (Co)homology with local coefficients. Homology of H-spaces and Lie groups. The Leray–Hirsch theorem and applications.

Prerequisites

Knowledge of general topology; Groups, Rings and Modules.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.