Teaching plan for the course unit

 General information

Course unit name: Geometry and Topology of Manifolds

Course unit code: 568176

Coordinator: Carles Casacuberta Vergés

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 Estimated learning time Total number of hours 150

 Face-to-face learning activities 60
 -  Lecture (Lectures.) 30 -  Lecture with practical component (Problem-solving sessions.) 30
 Supervised project 20
 Independent learning 70

 Competences to be gained during study

 — Capacity to understand concepts and rigorous proofs of fundamental mathematical theorems. — Capacity to use the results and suitable techniques acquired to solve problems. — Capacity to develop logical thinking and find errors in wrong arguments. — Capacity to present mathematical ideas orally and in written form. — Ability to read and understand mathematical research articles. — Capacity to summarize the content of specialized seminars and colloquia.

 Learning objectives
 Referring to knowledge — To discover properties of smooth manifolds that only depend on the underlying topology. — To define orientability on topological manifolds and on smooth manifolds. — To know how to use differential forms on smooth manifolds and why they are useful. — To prove Stokes’ theorem on smooth manifolds and related results. — To prove that singular cohomology with real coefficients coincides with de Rham cohomology on smooth manifolds.  Referring to abilities, skills — To calculate with differential forms on smooth manifolds. — To integrate differential forms along smooth chains.  — To calculate singular homology and singular cohomology of topological spaces. — To calculate de Rham cohomology of smooth manifolds. — To solve problems with combined techniques from differential geometry and topology. — To present solutions of exercises orally and in written form. — To use bibliographic resources for further learning.

 Teaching blocks

1. Singular cohomology

1.1. Homology and cohomology

1.2. Homotopy invariance

1.3. The Mayer-Vietoris exact sequence

1.4. Cellular complexes

2. Manifolds

2.1. Manifolds with boundary and without boundary

2.2. Smooth structures

2.3. Tangent bundle and cotangent bundle

3. Cohomology of differential forms

3.1. Differential forms on manifolds

3.2. De Rham cohomology

3.3. Integration of forms with compact support

3.4. Stokes’ theorem

3.5. De Rham’s theorem

 Teaching methods and general organization

 — Lectures (2 hours per week) — Problem sessions (2 hours per week) — Oral and written presentation of exercises — Bibliographic searches — Independent learning

 Official assessment of learning outcomes

 The continuous assessment consists of written resolution of exercises and classroom presentations, which are worth 75% of the final grade. A written examination will account for the remaining 25%. A minimum mark of 3/10 is required to pass the exam.   Examination-based assessment The single examination assessment will consist of a written examination. It will not be necessary to renounce to the continuous assessment in order to be entitled to examination-based assessment. In cases where both grades are available, the final grade will be their higher of the two scores. Reassessment will consist of a written examination. All students are entitled to reassessment. The final grade will be the highest score between single examination assessment and the reassessment marks. The final grade will be "Absent" if no examination is sit.