# Topology and Manifolds

## Objectives

Students must:

a) Understand and know how to prove the fundamental results about topological spaces, continuous functions, compactness, connectedness, separation/countability axioms, quotient topology. Uryshon Metrization and Tietze Extension theorem will be addressed.

b) Understand the notion of manifold without boundary. Know how to orient manifolds, compute tangent spaces to manifolds. Understand the notion of vector field, differential form and tensor product in manifolds.

## General characterization

##### Code

12951

##### Credits

6.0

##### Responsible teacher

João Pedro Bizarro Cabral

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

`Linear algebra, differential and integral calculus of several variables, topology in metric spaces.`

### Bibliography

James R. Munkres: Topology. Prentice Hall (2000)

Armstrong, Mark Anthony: Basic topology. Corrected reprint of the 1979 original. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1983.

Andrew Mclnerney: Firsts steps in Differential Geometry. Springer, 2013.

Vladimir A. Zorich: Mathematical Analysis II – second edition. Springer, 2016

### Teaching method

The course will have 4 hours of theoretical-practical classes per week. There will be specific time in classes for students to pose questions or doubts about exercises. The autonomous learning by the students will be centered around solving advanced exercise sheets, some of which containing proofs of fundamental theorems.

The students must submit weekly series of exercises. Each student will make at least an oral presentation of 1 hour (with a complete proof of a theorem, explanation of section of a book or paper). The final grade will be the simple average of all grades of exercises and presentations.

A student will obtain frequency if it submits all series of exercises, with the possible exception of one.

### Evaluation method

The students must submit weekly series of exercises. Each student will make at least an oral presentation of 1 hour (with a complete proof of a theorem, explanation of section of a book or paper). The final grade will be the simple average of all grades of exercises and presentations.

A student will obtain frequency if it submits all series of exercises, with the possible exception of one.

## Subject matter

The topics to address will depend on the background of the students. The minimal objectives must be taken into account.

1)Topological spaces. Basis and sub-basis. Products of topological spaces. Subspace topology. Closed sets and limit points. Continuous functions. Homeomorphisms.

2)Connectedness. Arc connectedness. Components.

3)Compactness. Lebesgue numbers. Compactness in metric spaces. Local compactness. Compactification by a point.

4)Axioms of countability. Separation axioms. Urysohn''s Lemma.Tietze extension theorem.

5)Manifolds without boundary. Maps between manifolds. Orientation of Manifolds. Partitions of the unity. Immersions of compact manifolds in .

6)Tangent space of a manifold, geometric and analytic notion. Differential. Vector fields. Integral curves. Flow of a vector field.

7) Differential forms. Exterior product. Poincaré lemma. Tensor product.