General information 
Course unit name: Geometry and Topology of Manifolds
Course unit code: 568176
Academic year: 20182019
Coordinator: Carles Casacuberta Vergés
Department: Department of Mathematics and Computer Science
Credits: 6
Single program: S
Estimated learning time 
Total number of hours 150 
Facetoface learning activities 
60 
 Lecture (Lectures.) 
30 

 Lecture with practical component (Problemsolving sessions.) 
30 
Supervised project 
20 
Independent learning 
70 
Competences to be gained during study 

Learning objectives 
Referring to knowledge — To discover properties of smooth manifolds that only depend on the underlying topology.
Referring to abilities, skills — To calculate with differential forms on smooth manifolds.

Teaching blocks 
1. Singular cohomology
1.1. Homology and cohomology
1.2. Homotopy invariance
1.3. The MayerVietoris 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 

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.
Examinationbased assessment The single examination assessment will consist of a written examination.

Reading and study resources 
Consulteu la disponibilitat a CERCABIB
Book
Bott, R. ; Tu, L. W. Differential forms in algebraic topology. : Springer, 1982.
Bredon, G. E. Topology and geometry. New York : Springer, 1993.
Do Carmo, M. P. Differential forms and applications. Berlin : Springer, 1994.
Hatcher, A. Algebraic topology. Cambridge : Cambridge University Press, 2002.
Jänich, K. Vector analysis. New York : Springer, 2001.
Morita, S. Geometry of differential forms. Providence : American Mathematical Society, 2001.
Vick, J. W. Homology theory : an introduction to algebraic topology. New York : Springer, 1994.