Luca VITAGLIANO | HOMOLOGY AND COHOMOLOGY
Luca VITAGLIANO HOMOLOGY AND COHOMOLOGY
cod. 0512300047
HOMOLOGY AND COHOMOLOGY
| 0512300047 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF6 | |
| MATHEMATICS | |
| 2025/2026 |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/03 | 6 | 48 | LESSONS |
| Objectives | |
|---|---|
| THE AIM OF THE COURSE IS TO PROVIDE THE FUNDAMENTALS OF THE HOMOLOGY THEORY AND OF ITS APPLICATIONS, WITH A SPECIAL EMPHASIS ON THE APPLICATIONS IN ALGEBRAIC TOPOLOGY. - KNOWLEDGE AND UNDERSTANDING: AT THE END OF THE COURSE, THE STUDENT WILL KNOW THE FUNDAMENTALS OF HOMOLOGY THEORY, INCLUDING SOME OF ITS NUMEROUS APPLICATIONS. THEY WILL UNDERSTAND THE ROLE OF THIS LANGUAGE IN MODERN MATHEMATICS AND WILL BE ABLE TO INDEPENDENTLY STUDY ADVANCED TOPICS IN HOMOLOGICAL ALGEBRA AND ALGEBRAIC TOPOLOGY, INCLUDING TOPICS WHICH ARE NOT PART OF THE COURSE PROGRAM. - APPLYING KNOWLEDGE AND UNDERSTANDING: THE AIM OF THE COURSE IS TO ENABLE THE STUDENT TO APPLY HOMOLOGICAL NOTIONS AND TECHNIQUES 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 TECHNIQUES TO THE STUDY OF TOPOLOGICAL SPACES VIA SINGULAR HOMOLOGY AND COHOMOLOGY. THEY WILL ALSO BE ABLE TO STUDY SOME SIMPLE ALGEBRAIC STRUCTURES LIKE GROUPS, LIE ALGEBRAS AND ASSOCIATIVE ALGEBRAS, VIA THE ASSOCIATED HOMOLOGIES. - AUTONOMY OF JUDGEMENT: AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO ORIENT THEMSELVES AMONG THE NUMEROUS RAMIFICATIONS OF HOMOLOGY, INCLUDING ITS APPLICATIONS IN A GEOMETRIC, ALGEBRAIC, AND INTERDISCIPLINARY SET UP. THEY WILL ALSO BE ABLE TO ASSESS WHICH KIND OF INFORMATION ONE CAN EXTRACT FROM EITHER A GEOMETRIC OR AN ALGEBRAIC STRUCTURE USING HOMOLOGICAL TOOLS. - COMMUNICATION SKILLS: SOLVING THE WEEKLY HOME-WORKS, AND DISCUSSING THEIR SOLUTIONS WITH THE INSTRUCTOR, THE STUDENT WILL LEARN TO COMMUNICATE IN A CLEAR, CONCISE AND RIGOROUS WAY, BOTH ORALLY AND IN WRITING, THE PROOFS OF SIMPLE (HOMOLOGICAL) ASSERTIONS THAT THEY HAVE INDEPENDENTLY ELABORATED. MOREOVER, PREPARING A FINAL TALK, THE STUDENT WILL LEARN HOW TO PRESENT A MONOGRAPHIC TOPIC IN ORGANIC AND COMPLETE WAY, BUT ALSO IN A WAY ACCESSIBLE TO THEIR PEERS, AND USING BOTH A “LITERATE REGISTER” AND A MORE INFORMAL REGISTER. - LEARNING SKILLS: AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO STUDY ON THEIR OWN ANY TEXTBOOK ON HOMOLOGICAL ALGEBRA OR ALGEBRAIC TOPOLOGY, INCLUDING TEXTBOOKS WHICH ARE ADVANCED BOTH IN THE LANGUAGE AND THE CONTENTS, AND THEY WILL BE ABLE TO FILL POSSIBLE GAPS IN THE PROOFS OF THE SIMPLEST ASSERTIONS ALONG THE TEXT THAT THEY ARE STUDYING. THEY WILL ALSO BE ABLE TO CARRY OUT BIBLIOGRAPHIC SEARCHES WITH THE AIM OF FINDING MATERIAL ON COMPLEX AND SPECIALISTIC TOPICS IN HOMOLOGICAL ALGEBRA AND ALGEBRAIC TOPOLOGY, TO PREPARE A SEMINAR AND/OR AN ESSAY AND/OR A LONGER TEXT (E.G. A GRADUATION THESIS). |
| 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 (11 HRS) MODULES OVER A RING. LINEAR MAPS AND HOMOORPHISM THEOREMS. FREE MODULES. MULTILINEAR MAPS. TENSOR PRODUCTS. SYMMETRIC AND EXTERIOR ALGEBRAS. 2. (CO)CHAIN COMPLEXES (11 HRS) (CO)CHAIN COMPLEXES. (CO)HOMOLOGY. EXACT SEQUENCES. COCHAIN MAPS. QUASI-ISOMORPHISMS. ALGEBRAIC HOMOTOPIES. HOMOTOPIC MAPS. HOMOTOPY EQUIVALENCES. CONTRACTIONS. SNAKE LEMMA. 3. ALGEBRAIC APPLICATIONS (11 HRS) (CO)SIMPLICIAL SETS AND MODULES. GROUP (CO)HOMOLOGY. LIE ALGEBRAS. CHEVALLEY-EILENBERG (CO)HOMOLOGY. ASSOCIATIVE ALGEBRAS. HOCHSCHILD (CO)HOMOLOGY. 4. GEOMETRIC APPLICATIONS (15 HRS) 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. THE HOMEWORKS WILL BE ASSIGNED WEEKLY AND WILL BE DEVIDED IN "MANDATORY EXERCISE" AND "NON-MANDATORY EXERCISES". THE INSTRUCTURE WILL BE WILLING TO CHECK THE STUDENT'S SOLUTION DURING THEIR OFFICE HOURS. |
| 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, EVERY WEEK, AND THEY DISCUSS THEIR SOLUTIONS WITH THE INSTRUCTOR DURING THE OFFICE HOURS. AFTER THE COURSE AND WITHIN THE ACADEMIC YEAR IN WHICH THE COURSE TOOK PLACE, HE/SHE GIVE A SEMINAR (DURATION 35 MINUTES) ON A TOPIC NOT DISCUSSED IN THE COURSE AND AGREED WITH THE TEACHER. THE HOMEWORKS AIM AT TESTING THE STUDENT ABILITY TO APPLY THE THEORETICAL KNOWLEDGE WHICH THEY ACQUIRED DURING THE COURSE AND THE INDEPENDENT STUDY. THE SEMINAR AIMS AT TESTING THE DEGREE OF UNDERSTANDING OF THE THEORY DEVELOPED DURING THE LECTURES, INCLUDED ITS APPLICATIONS IN SETTINGS DIFFERENT FROM THOSE DISCUSSING DURING THE COURSE. THE SEMINARS ALSO TESTS THE COMMUNICATION SKILLS OF THE STUDENT CONCERNING THE SUBJECT OF THE COURSE. 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 (DURATION 45 MINUTES) ON THE TOPICS DISCUSSED IN THE COURSE. THE ORAL TEST AIMS AT TESTING THE DEGREE OF UNDERSTANDING OF THE THEORY DEVELOPED DURING THE LECTURES, THE ABILITY OF THE STUDENT TO APPLY THEIR THEORETICAL KNOWLEDGE AND THEIR COMMUNICATION SKILLS CONCERNING THE SUBJECTS OF THE COURSE. THE ORAL TEST WILL BE ON THE WHOLE PROGRAM AND WILL CONSIST OF: A FIRST "FREE QUESTION" WHICH WILL GIVE TO THE STUDENT THE OPPORTUNITY TO ILLUSTRATE A TOPIC OF THE PROGRAM OF THEIR CHOICE, AND TWO MORE QUESTIONS ON THE PARTS OF THE PROGRAM THAT THE STUDENT HS NOT YET DISCUSSED. IN THIS RESPECT THE PROGRAM SHOULD BE CONSIDERED AS DIVIDED INTO THREE MAIN PARTS: 1) MULTILINEAR ALGEBRA AND (CO)CHAIN COMPLEXES, 2) ALGEBRAIC APPLICATIONS, 3) GEOMETRIC APPLICATIONS. THE MINIMUM MARKS ARE GRANTED TO THE STUDENT WITH A BASIC KNOWLEDGE OF THE FOLLOWING TOPICS: MODULES OVER A RING, FREE MODULES, MULTILINEAR MAPS AND TENSOR PRODUCTS, (CO)CHAIN COMPLEXES, ALGEBRAIC OMOTOPIES, SHORT EXACT SEQUENCES OF COMPLEXES AND CONNECTING HOMOMORPHISM. GROUP, LIE ALGEBRA AND ASSOCIATIVE ALGEBRA (CO)HOMOLOGIES. SINGULAR HOMOLOGIES, GEOMETRIC AND ALGEBRAIC HOMOTOPIES BETWEEN SINGULAR CHAINS, MAYER-VIETORIS SEQUENCE. 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 | |
|---|---|
| L. VITAGLIANO, HOMOLOGY & COHOMOLOGY, A PRIMER FOR UNDERGRADUATES THROUGH APPLICATIONS, WORLD SCIENTIFIC. 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. |
| More Information | |
|---|---|
| WEB PAGE OF THE COURSE: HTTP://WWW.DIPMAT2.UNISA.IT/PEOPLE/VITAGLIANO/WWW/OMOLOGIA.HTML EMAIL: LVITAGLIANO@UNISA.IT |
BETA VERSION Data source ESSE3 [Ultima Sincronizzazione: 2025-10-07]
