I.The Basic Information of the Course(Time New Rome, 12 points, bold font)
Course Number:202012420034
The EnglishName of theCourse：Algebraic Topology
The ChineseName of theCourse：代数拓扑学
In-classHours andAllocation：total class hours:32, classroom teaching: 32 classhours
Credit(s):2
Semester:2
AppliedDiscipline(Professional Degree Category)：Mathematics
CourseObjectOriented：Post-Graduate,Academic Doctor
EvaluationMode:Closed-book Examination
TeachingMethod:Classroom teaching
CourseOpening Department:Department of Mathematics
Notes:Filling explanation to applied discipline: the general courses like foreign languages, ideological and political theory course may fill in “AllDisciplines”,Mathematics general course may fill inScience andEngineeringDiscipline, and the other courses fill in “Name ofApplicableDiscipline” according to actual teaching objects, and the number of applicable discipline may be more than one.
II.Prerequisite Course(Time New Rome 12 points bold font)
General Topology, Abstract Algebra
III. The Objectives and Requirements of the Course(Time New Rome 12 points, total words: about 200 words)
Topology, together with Algebra and Analysis, forms the foundation of modern mathematics. Algebraic topology provides algebraic methods in dealing with problems in topology.
This course is an elective course for graduate students majoring in pure mathematics. It is aimed at showing the application of algebraic methods in solving topological problems. During the course, the students will acquire the basic knowledge through the lectures and exercises, and obtain the calculation skills through all the examples. In a word, they will be prepared for the potential research on topics related to topology.
Having finished the course, the students are supposed to be familiar with the important definitions and theorems in algebraic topology, and be proficient in basic proofs.
IV. The Content of the Course(Time New Rome, 12 points, bold font, 1000-2000 words)
Description:Other course should reflect the content and cases of “Courses for Ideological and Political education”, besides the Course for Ideological and Political Education.
Chapter 0 Preparation (6 class hours)
0.1 Homotopy and homotopy type: definitions and examples of contractions, mapping cones, the definitions of homotopic maps and homotopic spaces, contractible spaces and null-homotopic maps
0.2 CW complexes: definition and examples of CW complexes, Euler characteristic, characteristic maps of cells
0.3 Actions on spaces: the product, quotient, suspension, join, wedge sum and smash product of CW complexes
0.4 Properties and examples of homotopical equivalence
Chapter 1 Fundamental groups and covering spaces (13 class hours)
1.1 The construction of the fundamental group: path and homotopy, including path, path homotopy, homotopy class, fundamental groups of based spaces, fundamental groups of path-connected spaces and simply-connected spaces; fundamental groups of spheres, including the calculation of the fundamental groups of spheres, and the application of these fundamental groups in proving certain classical theorems, e.g., the fundamental theorem of algebra, Brouwer fixed point theorem and Borsuk-Ulam Theorem etc.; the induced homomorphisms on fundamental groups and the properties of induced homomorphisms
1.2 Van Kampen Theorem: the free product of groups and examples; Van Kampen Theorem; CW complexes and fundamental groups
1.3 Covering spaces: path lifting property, including homotopy lifting and path lifting; definition and classification of covering spaces; the existence of covering spaces; universal covering spaces; covering transformations and group actions
Chapter 2 Homology (13 class hours)
2.1 simplicial homology and singular homology: Δ-complexes, Δ-complex structure on spaces; simplicial homology, calculation and examples of simplicial homology on Δ-complexes; singular homology, including singular simplexes, singular complexes and the construction of singular homology groups; homotopical invariance of cohomology groups; exact sequences and excision theorem, including the definitions of exact sequences and short exact sequences, relative homology groups, excision theorem; the relation between simplicial homology and singular homology
2.2 calculation and application: degree of self-maps of spheres, the definition and properties; homology of cellular complexes; Mayer-Victoris sequences and homology with coefficients
Chapter 1 Introduction (**classhours)
Courses for Ideological and Political education:Inspire the students to devote themselves into the career of pure math. A large number of qualified young mathematicians are necessary for theimprovement of our nation's position in pure math, and also for our national rejuvenation.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1.Allen Hatcher. Algebraic Topology.Cambridge University Press, 2002.
2.William S. Massey. Algebraic Topology: An Introduction.Springer-Verlag, 1990.
3.姜伯驹.同调论.北京大学出版社. 2006.
4. Merkurjev and Vishik, Operations in connective K-theory, arxiv: 2006.12193, 2020.
B. Learning Resources (Time New Rome 12 points)
1.MIT open courseware:
https://ocw.mit.edu/courses/mathematics/18-905-algebraic-topology-i-fall-2016/
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