Math 422 Elementary Geometric Topology
 
 Schedule:
   Mondays   13:40-15:30 M-102
  Thursdays  10:40-11:30 M-102

Course Content

Topology of subsets of Euclidean space. Topological surfaces. Surfaces in R^n. 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. .

   Syllabus:
Week 1-2 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.
Week 3-4-5-6 Topological surfaces; Examples of complexes, cell complexes, surfaces, triangulations, classification of surfaces, surfaces with boundary.
Week 7-8-9-10 The Euler characteristic; Topological invariants, graphs and trees, the Euler characteristic and the sphere, the Euler characteristic and surfaces, Map-coloring problems, graphs revisited.
Week 11-12-13-14 Homology; the algebra of chains, simplicial complexes, homology, more computations, Betti numbers and the Euler characteristic.


Course Objective

To introduce and illustrate the main ideas of geometric and algebraic topology (construction of spaces, connected sums and quotients of surfaces) and to provide a foundation for further study in geometric and algebraic topology.

Prerequisite Course: Math 252

Textbook: ''Topology of Surfaces''  by L. Christine Kinsey.
Supplementary Resources:''A first course in Geometric Topology and Differential Geometry''  by Ethan D. Bloch

Exams:  The assesment is based upon two midterm exams (30 % each) and a final exam (40 %) and an oral exam (20% )
Midterm 1   March 09, 2020, Monday, at 17:40, Exam Place is    U-3
Midterm 2   April 13, 2020, Monday, at 17:40, Exam Place is    
Final            

Suggested Exercises:
Chapter 2, exercises in section 2.1:  2.1,2.2,2.5,2.9,2.12
                exercises in section 2.2:  2.19
                exercises in section 2.3:  2.25
                exercises in section 2.4:   2.26, 2.28
                    exercises in section 2.5:  2.31
Chapter 3,exercises after section 3.1: 3.4
               exercises after section 3.2:  3.8, 3.9, 3.10, 3.13, 3.14, 3.15, 3.19
                   exercises after section 3.3:  3.23, 3.24, 3.25, 3.26, 3.27
             exercises after section 3.4:  3.28
             exercises after section 3.5:   3.31, 3.32, 3.33, 3.34, 3.35, 3.36
Chapter 4, exercises after section 4.1:  4.2, 4.3,
                    exercises after section 4.2:  4.4, 4.5, 4.6, 4.7, 4.8
                    exercises after section 4.3:  4.9, 4.10, 4.11, 4.12
                    exercises after section 4.4:  4.13, 4.14, 4.15, 4.17, 4.18, 4.19, 4.20
Chapter 5, exercises after section 5.2:  5.2, 5.3, 5.4
                    exercises after section 5.3:  5.5
                    exercises after section 5.4:  5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, 5.15, 5.16, 5.17
Chapter 6, exercises after section 6.1:  6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8
                    exercises after section 6.2:  6.9, 6.10, 
                    exercises after section 6.3:  6.11, 6.12, 6.13, 6.14, 6.15, 6.16, 6.17, 6.18
Chapter 7, exercises after section 7.1:  7.1, 7.2,
                    exercises after section 7.2:  7.3
                    exercises after section 7.3:  7.4, 7.5, 7.6, 7.7
                exercises after section 7.4:  7.8, 7.9, 7.10, 7.11, 7.12, 7.13, 7.14
Chapter 9, exercises after section 9.1:  9.1, 9.2, 9.3, 9.4, 9.5, 9.6
                    exercises after section 9.2:  9.7, 9.8, 9.9, 9.10, 9.12, 9.13
Chapter 10, exercises :  10.1, 10.2, 10.3, 10.4, 10.5,