Math 422 ~ Elementary Geometric Topology   (SPRING 2022)

Schedule:

Mondays    13:40-15:30, room: M-102
Wednesdays  12:40-13:30, room: M-102

Instructor: Fırat Arıkan,    Office: M-130,    Email: farikan(at)metu.edu.tr,   

Office hours: Monday 11-11:40, Wednesday 10-10:40 (or by appointment)

Zoom Link for Official Lecture Meetings and Office Hours:

https://zoom.us/j/6558344427?pwd=Y2ZBNFpGbkl4ZkFSNTB1KzRETzgzQT09

Announcements:  All course announcements will be posted here!

Exams and Grading:
 
There will be two Midterm Exams (%30 each) and a Final Exam (%40)

Midterm 1: Time:   April 19, 2022 (Tuesday) at 17:40
                 Topics included:  to be announced
 
Midterm 2: Time:   May 26, 2022 (Thursday) at 17:40
                    Topics included:  to be announced
 
Final Exam: Time:  to be announced
                      Topics included: Everything seen during the semester.

Content: Topology of subsets of Euclidean space. Topological surfaces. Surfaces in Rⁿ. Surfaces via gluing, connected sum and the classification of compact connected surfaces. Simplicial complexes and simplicial surfaces (simplicial complexes with underlying spaces that are topological surfaces). Euler characteristic.

Goals: The aim of this course is to introduce the student to some algebraic and differential topological ideas at an early stage emphasizing unity with geometry and more generally introduce the student to the relation of the modern axiomatic approach in mathematics to geometric intuition.

Course textbook:  ''Topology of Surfaces''  by L. Christine Kinsey.

Course Outline:

(2-3 Weeks) Chapters 1, 2 and 3: An overview, point-set topology in Euclidean space; Open closed subset, relative neighborhoods, continuity, compact sets, connected sets and applications. Also definition of topological spaces and basic constructions such as product spaces and quotient spaces.

(3-4 Weeks) Chapter 4: Topological surfaces; Examples of complexes, cell complexes, surfaces, triangulations, classification of surfaces, surfaces with boundary.

(3-4 Weeks) Chapter 5: The Euler characteristic; Topological invariants, graphs and trees, the Euler characteristic and the sphere, the Euler characteristic and surfaces, Map-coloring problems, graphs revisited

(3 Weeks)  Chapter 6: Homology; The algebra of chains, simplicial complexes, homology, more computations, Betti numbers and the Euler characteristic.

Other books: ''A first course in Geometric Topology and Differential Geometry''  by Ethan D. Bloch