- Docente: Roberto Pagaria
- Credits: 6
- SSD: MATH-02/B
- Language: Italian
- Moduli: Roberto Pagaria (Modulo 1) Andrea Bianchi (Modulo 2)
- Teaching Mode: In-person learning (entirely or partially) (Modulo 1); In-person learning (entirely or partially) (Modulo 2)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 6061)
-
from Sep 21, 2026 to Dec 15, 2026
Learning outcomes
Upon completion of the course, students will understand the basics of algebraic topology, particularly homology theories. They will be able to apply this formalism to other areas of mathematics.
Course contents
Simplicial and cellular complexes. Homotopy of continuous functions. Higher homotopy groups.
Simplicial homology of a simplicial complex and singular homology of a topological space.
Excision theorem, exact pair and Mayer-Vietoris sequences. Axioms of homology. An introduction to Hurewicz's theorem.
Cohomology, its relationship to homology. Cup product in cohomology and the cap product between homology and cohomology. Poincaré duality for oriented topological varieties. Axioms of cohomology.
Universal coefficient theorems. Ext and Tor groups.
Applications: Classical theorems of topology, domain invariance, fixed-point theorems. Jordan's separation theorem.
Readings/Bibliography
A. Hatcher: Algebraic Topology.
Rotman: An introduction to Algebraic topology
Teaching methods
Lectures at the blackboard and exercise sessions
Assessment methods
Oral exams and exercises given during the course.
Office hours
See the website of Roberto Pagaria
See the website of Andrea Bianchi