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.

Before studying manifolds in detail, we will introduce the notion of higher homotopy groups, which allow us to classify manifolds up to homotopy. However, since higher homotopy groups are extremely difficult to compute, we will introduce the notion of aspheric manifolds, i.e., those manifolds for which the higher homotopy groups are all trivial. We will be now ready to begin the detailed study of manifolds from a topological-geometric perspective. First, using the theory of handles (and Morse theory), we will classify closed surfaces. Next, we will study the "geography" of 3-manifolds, i.e., we will attempt to explicitly describe some 3-manifolds. Particular emphasis will be given to the relationship with their fundamental group. Some key terms include Heegaard splitting, Lens spaces and Seifert manifolds, bundles, prime-decomposition as well as the JSJ-decomposition.

Time permitting, some invariants of closed manifolds introduced by Gromov will be introduced.

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".

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 communicated to the interested students during the course.

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