Geometry II

Academic Year 2026/2027 - Teacher: SANTI DOMENICO SPADARO

Required Prerequisites

A good knowledge of general topology. The main concepts will be reviewed during classes.

Attending is strongly advised.

Set-theoretic preliminaries: Zermelo's well-ordering theorem and the order topology on the first uncountable ordinal.
Fundamentals of paracompact spaces. Partitions of unity.
Products of paracompact spaces, Michael's Line.
The Theorems of Michael and Stone.
Introduction to topological manifolds.
Homotopy of continuous functions and topological spaces.
Paths and homotopy of paths.
Introduction to the fundamental group.
An outline of covering spaces. The fundamental group of the circle.
Applications of the fundamental group of the circle (the Brouwer Fixed Point Theorem and the Borsuk-Ulam theorem in dimension 2. The Fundamental Theorem of Algebra).
The fundamental group of the sphere.

	Subjects	Text References
1	Set-theoretic preliminaries.	Course notes.
2	Paracompact spaces.	Course notes.
3	Partitions of unity.	Course notes.
4	Product of paracompact spaces.	Course notes.
5	The Theorems of Michael and Stone.	Course notes.
6	Introduction to topological manifolds.	Course ntoes.
7	Homotopy of continuous functions and topological spaces.	Course notes.
8	Paths and path homotopy.	Course notes.
9	Introduction to the fundamental group.	Course notes.
10	Covering spaces. The fundamental group of the circle.	Course notes.
11	Applications of the fundamental group of the circle.	Course notes.
12	The fundamental group of the sphere.	Course notes.