Matematica | HOMOLOGY AND COHOMOLOGY

cod. 0512300047

HOMOLOGY AND COHOMOLOGY

	0512300047
	DIPARTIMENTO DI MATEMATICA
	EQF6
	MATHEMATICS
	2022/2023

PERCORSO SHARED STUDY PATHWAY Cohort 2020/2021

	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	6	48	LESSONS	OPTIONAL SUBJECTS

EXAMS

	Exam	Date	Session
	OMOLOGIA E COOMOLOGIA	03/04/2023 - 14:00	SESSIONE ORDINARIA

	Objectives
	THE AIM OF THE COURSE IS PROVIDING THE FOUNDATIONS OF HOMOLOGY THEORY AND ITS APPLICATIONS, WITH A SPECIAL EMPHASIS ON APPLICATIONS IN ALGEBRAIC TOPOLOGY. - KNOWLEDGE AND UNDERSTANDING: AT THE END OF THE COURSE, THE STUDENT WILL KNOW THE RUDIMENTS OF HOMOLOGY THEORY; INCLUDING SOME OF ITS SEVERAL APPLICATIONS. HE/SHE WILL UNDERSTAND THE ROLE OF THIS LANGUAGE IN MODERN MATHEMATCIS AND WILL BE ABLE TO AUTONOMOUSLY UNDERTAKE THE STUDY OF MORE ADVANCED TOPICS IN HOMOLOGICAL ALGEBRA AND ALGEBRAIC TOPOLOGY, INCLUDING THOSE NOT LISTED IN THE PROGRAM OF THE COURSE. - APPLYING KNOWLEDGE AND UNDERSTANDING: THE AIM OF THE COURSE IS MAKING THE STUDENT ABLE TO APPLY NOTIONS AND TECHNIQUES FROM HOMOLOGY IN BOTH A GEOMETRIC AND INTERDISCIPLINARY SET UP, WITH A SPECIAL EMPHASIS ON ALGEBRA. AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO APPLY HOMOLOGICAL TECNIQUES TO TOPOLOGICAL SPACES VIA SINGULAR HOMOLOGY AND COHOMOLOGY. HE/SHE WILL ALSO BE ABLE TO STUDY SOME SIMPLE ALGEBRAIC STRUCTURES LIKE GROUPS, LIE AND ASSOCIATIVE ALGEBRAS, VIA THE ASSOCIATED HOMOLOGIES.

THE AIM OF THE COURSE IS PROVIDING THE FOUNDATIONS OF HOMOLOGY THEORY AND ITS APPLICATIONS, WITH A SPECIAL EMPHASIS ON APPLICATIONS IN ALGEBRAIC TOPOLOGY.

- KNOWLEDGE AND UNDERSTANDING: AT THE END OF THE COURSE, THE STUDENT WILL KNOW THE RUDIMENTS OF HOMOLOGY THEORY; INCLUDING SOME OF ITS SEVERAL APPLICATIONS. HE/SHE WILL UNDERSTAND THE ROLE OF THIS LANGUAGE IN MODERN MATHEMATCIS AND WILL BE ABLE TO AUTONOMOUSLY UNDERTAKE THE STUDY OF MORE ADVANCED TOPICS IN HOMOLOGICAL ALGEBRA AND ALGEBRAIC TOPOLOGY, INCLUDING THOSE NOT LISTED IN THE PROGRAM OF THE COURSE.

- APPLYING KNOWLEDGE AND UNDERSTANDING: THE AIM OF THE COURSE IS MAKING THE STUDENT ABLE TO APPLY NOTIONS AND TECHNIQUES FROM HOMOLOGY IN BOTH A GEOMETRIC AND INTERDISCIPLINARY SET UP, WITH A SPECIAL EMPHASIS ON ALGEBRA. AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO APPLY HOMOLOGICAL TECNIQUES TO TOPOLOGICAL SPACES VIA SINGULAR HOMOLOGY AND COHOMOLOGY. HE/SHE WILL ALSO BE ABLE TO STUDY SOME SIMPLE ALGEBRAIC STRUCTURES LIKE GROUPS, LIE AND ASSOCIATIVE ALGEBRAS, VIA THE ASSOCIATED HOMOLOGIES.

	Prerequisites
	IT IS REQUIRED A BASIC KNOWLEDGE OF THE GEOMETRY AND ALGEBRA COURSES OF THE FIRST TWO YEARS OF THE BATCHELOR DEGREE IN MATHEMATICS, IN PARTICULAR THE NOTIONS OF VECTOR SPACE, LINEAR MAP, GROUP, ABELIAN GROUP, GROUP HOMOMORPHISM, RINGS, INCLUDING THEIR BASIC PROPERTIES.

	Contents
	1. MULTILINEAR ALGEBRA MODULES OVER A RING. LINEAR MAPS AND HOMOORPHISM THEOREMS. FREE MODULES. MULTILINEAR MAPS. TENSOR PRODUCTS. SYMMETRIC AND EXTERIOR ALGEBRAS. 2. (CO)CHAIN COMPLEXES (CO)CHAIN COMPLEXES. (CO)HOMOLOGY. EXACT SEQUENCES. COCHAIN MAPS. QUASI-ISOMORPHISMS. ALGEBRAIC HOMOTOPIES. HOMOTOPIC MAPS. HOMOTOPY EQUIVALENCES. CONTRACTIONS. SNAKE LEMMA. 3. ALGEBRAIC APPLICATIONS (CO)SIMPLICIAL SETS AND MODULES. GROUP (CO)HOMOLOGY. LIE ALGEBRAS. CHEVALLEY-EILENBERG (CO)HOMOLOGY. ASSOCIATIVE ALGEBRAS. HOCHSCHILD (CO)HOMOLOGY. 4. GEOMETRIC APPLICATIONS SINGULAR (CO)CHAINS OF A TOPOLOGICAL SPACE. SINGULAR (CO)HOMOLOGY. GEOMETRIC HOMOTOPIES. OMOTOPIC MAPS. DEFORMATIONS RETRACTS AND CONTRACTIBLE SPACES. MAYER-VIETORIS SEQUENCE. CUP PRODUCT. DE RHAM COHOMOLOGY OF AN OPEN SUBSET OF R^N.

Contents

1. MULTILINEAR ALGEBRA

MODULES OVER A RING. LINEAR MAPS AND HOMOORPHISM THEOREMS. FREE MODULES. MULTILINEAR MAPS. TENSOR PRODUCTS. SYMMETRIC AND EXTERIOR ALGEBRAS.

2. (CO)CHAIN COMPLEXES

(CO)CHAIN COMPLEXES. (CO)HOMOLOGY. EXACT SEQUENCES. COCHAIN MAPS. QUASI-ISOMORPHISMS. ALGEBRAIC HOMOTOPIES. HOMOTOPIC MAPS. HOMOTOPY EQUIVALENCES. CONTRACTIONS. SNAKE LEMMA.

3. ALGEBRAIC APPLICATIONS

(CO)SIMPLICIAL SETS AND MODULES. GROUP (CO)HOMOLOGY. LIE ALGEBRAS. CHEVALLEY-EILENBERG (CO)HOMOLOGY. ASSOCIATIVE ALGEBRAS. HOCHSCHILD (CO)HOMOLOGY.

4. GEOMETRIC APPLICATIONS

SINGULAR (CO)CHAINS OF A TOPOLOGICAL SPACE. SINGULAR (CO)HOMOLOGY. GEOMETRIC HOMOTOPIES. OMOTOPIC MAPS. DEFORMATIONS RETRACTS AND CONTRACTIBLE SPACES. MAYER-VIETORIS SEQUENCE. CUP PRODUCT. DE RHAM COHOMOLOGY OF AN OPEN SUBSET OF R^N.

	Teaching Methods
	TEACHING WILL CONSIST MAINLY OF FRONT LECTURES. HOWEVER, DURING THE LECTURES, THEY WILL BE ASSIGNED EXERCISES AND PROBLEMS TO BE SOLVED BY THE STUDENTS AS "HOMEWORK", WITH THE AIM OF PROMOTING AN "ACTIVE" (HENCE MORE EFFECTIVE) LEARNING PROCESS, AND THE AUTONOMY OF JUDGMENT ON THE SUBJECT OF THE COURSE.

	Verification of learning
	THE STUDENT CAN TAKE THE EXAM IN ONE OF THE TWO FORMS DESCRIBED BELOW AT HIS/HER CHOICE: EXAM FORM 1: THE STUDENT DOES THE HOMEWORK AND HAND IN THEM DURING THE COURSE. AFTER THE COURSE AND WITHIN THE ACADEMIC YEAR IN WHICH THE COURSE TOOK PLACE, HE/SHE GIVE A SEMINAR ON A TOPIC NOT DISCUSSED IN THE COURSE AND AGREED WITH THE TEACHER. IN THE FINAL ASSESSMENT, EXPRESSED IN THIRTIES, THE ASSESSMENT OF THE HOMEWORKS, WILL COUNT FOR 50% AND THE ASSESSMENT OF THE SEMINAR WILL COUNT FOR THE REMAINING 50%. EXAM FORM 2: A SINGLE TRADITIONAL ORAL INTERVIEW ON THE TOPICS DISCUSSED IN THE COURSE. THE CUM LAUDE MAY BE GIVEN TO STUDENTS WHO PROVE TO BE ABLETO APPLY THEIR KNOWLEDGE AUTONOMOUSLY EVEN IN CONTEXTS OTHER THAN THOSE PROPOSED IN THE COURSE.

Verification of learning

THE STUDENT CAN TAKE THE EXAM IN ONE OF THE TWO FORMS DESCRIBED BELOW AT HIS/HER CHOICE:

EXAM FORM 1:

THE STUDENT DOES THE HOMEWORK AND HAND IN THEM DURING THE COURSE. AFTER THE COURSE AND WITHIN THE ACADEMIC YEAR IN WHICH THE COURSE TOOK PLACE, HE/SHE GIVE A SEMINAR ON A TOPIC NOT DISCUSSED IN THE COURSE AND AGREED WITH THE TEACHER.

IN THE FINAL ASSESSMENT, EXPRESSED IN THIRTIES, THE ASSESSMENT OF THE HOMEWORKS, WILL COUNT FOR 50% AND THE ASSESSMENT OF THE SEMINAR WILL COUNT FOR THE REMAINING 50%.

EXAM FORM 2:

A SINGLE TRADITIONAL ORAL INTERVIEW ON THE TOPICS DISCUSSED IN THE COURSE.

THE CUM LAUDE MAY BE GIVEN TO STUDENTS WHO PROVE TO BE ABLETO APPLY THEIR KNOWLEDGE AUTONOMOUSLY EVEN IN CONTEXTS OTHER THAN THOSE PROPOSED IN THE COURSE.

	Texts
	LECTURE NOTES WRITTEN BY THE INSTRUCTOR. TO GO DEEPER INTO THE ALGEBRAIC ASPECTS: C. A. WEIBEL, AN INTRODUCTION TO HOMOLOGICAL ALGEBRA, CAMBRIDGE UNIVERSITY PRESS. TO GO DEEPER INTO THE GEOMETRIC ASPECTS: J. M. LEE, INTRODUCTION TO TOPOLOGICAL MANIFOLDS, GRADUATE TEXT IN MATHEMATICS, SPRINGER.