B9054 - TOPOLOGIA GEOMETRICA

Learning outcomes

By the end of the course, students will have acquired the basic notions of geometric topology. Students will learn how to study the topology and geometry of manifolds through the algebraic properties of their fundamental groups and, viceversa, how to deduce algebraic properties of groups through their actions on topological spaces and the study of their associated geometric invariants.

Course contents

Geometric topology aims to study manifolds and the maps between them, for example, by embedding one manifold into another. In this course, we focus on the topology of low-dimensional manifolds, i.e., manifolds of dimension less than or equal to 3. The structure of the course is the following:

1) Preliminaries and some topics on homotopy theory

• CW-complexes and higher homotopy groups
• Euler characteristic and Whitehead Theorem
• Aspherical spaces

2) Introduction to manifolds

• Topological manifolds
• Smooth manifolds
• PL manifolds (and the connections with the other categories)
• Topological operations on manifolds (doubles, connected sum, twisted spheres)

3) Morse theory and handle decomposition

• Introduction to handle decomposition
• Morse functions
• Morse theory
• Classification of oriented closed connected surfaces

4) Interlude: fundamental groups of closed manifolds

• Realization problem: which finitely presented group can be the fundamental group of an oriented closed connected smooth manifold?

5) Heegaard splitting and 3-manifolds

• Heegaard splitting
• Fundamental groups of 3-manifolds
• Euler characteristic of 3-manifolds

6) Prime and irreducible 3-manifolds

• Fiber bundles and tubular neighbourhoods
• Irreducible manifolds and the Alexander Theorem
• Examples of irreducible manifolds
• Normal surfaces: towards the Prime Decomposition Theorem
• Prime Decomposition Theorem

7) Seifert manifolds

• Dehn filling
• Lens spaces
• Seifert manifolds

8) Overview on tori decompositions

• JSJ decomposition and the Geometrization Theorem

Prerequisites:

To fully appreciate the course content, students are advised to know the definition of a fundamental group, covering theory, and the definition of a smooth manifold.

Furthermore, we will use the notion of homology throughout the course. Students who have not had the opportunity to study homology in a previous course are welcome to learn this concept independently during the course (or before the beginning of the course) by reading the chapters listed below.

Some references that may be useful for learning the previous notions are the following:

• Allen Hatcher - "Algebraic topology" (available online). Chapters 1.1, 1.2, and 1.3 contain the theory of fundamental groups and coverings. Chapters 2.1 and 2.2 cover the notion of homology.
• John M. Lee - "Introduction to smooth manifolds." Chapter 1 contains the definition of a smooth manifold together with some examples. Chapter 2 describes smooth maps between smooth manifolds.

Readings/Bibliography

Books on 3-manifolds and geometric topology:

• Bruno Martelli - "An introduction to Geometric Topology" (available online);
• Allen Hatcher - "Notes on basic 3-manifolds topology" (available online);

• Jennifer Schultens - "Introduction to 3-manifolds".

Books on (algebraic) topology:

• Allen Hatcher - "Algebraic topology" (available online);
• Stefan Friedl - "Topology" (available online with the name "Full topology lecture notes").

Books on Morse theory:

• John Milnor - "Morse Theory".

Books on smooth and Riemannian manifolds:

• John M. Lee - "Introduction to smooth manifolds";
• John M. Lee - "Introduction to Riemannian manifolds".

Books on differential topology:

• Riccardo Benedetti - "Lectures on differential topology" (available online);
• John Milnor - "Topology from a differentiable point of view".

Finally on the Virtuale platform there will be the notes of the course (in Italian only).

Teaching methods

The course is organized in 48 hours of in-person teaching. Each lecture will contain new aspects of the theory as well as many examples/exercises. This will help the students to become more familiar with the new definitions.

Assessment methods

Students may choose to take the exam in one of two ways:

• A classical oral exam about the entire program of the course;
• An oral exam in the form of a seminar. The seminar, which lasts approximately 30 minutes (60 minutes if the seminar is delivered with another person), will focus on topics that complement and expand on what was covered during the course. A list of available seminars will be uploaded on the Virtuale platform at the beginning of the course. To pass the exam, a minimum threshold must be met. The final grade is calculated by adding points to a base score of 18. Points are awarded based on the following criteria: comprehension (3 points), exposition (3 points), organization of the seminar (3 points), use of appropriate technical terminology (3 points), and depth of analysis (1 point). A final score of 31 will be recorded as '30 e Lode'.

Teaching tools

Students can arrange meetings with the lecturer (scheduled in advance via email) in order to ask questions about the theory as well as about the topics for the seminar.

Students with learning disabilities (LD) or temporary or permanent disabilities are advised to promptly contact the relevant University office at https://site.unibo.it/studenti-con-disabilita-e-dsa/en . The office will recommend some adjustments, which must be submitted to the lecturer for approval 15 days in advance. The lecturer will evaluate their compatibility, taking into account the course's learning goals.

Office hours

See the website of Marco Moraschini