COURSE UNIT TITLE

: ELEMENTARY TOPOLOGY AND GEOMETRY

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 4012 ELEMENTARY TOPOLOGY AND GEOMETRY ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ASLI GÜÇLÜKAN ILHAN

Offered to

Mathematics (Evening)
Mathematics

Course Objective

The main goal of this course is to introduce surfaces from topological point of view and teach them to compute the homology and cohomology groups of surfaces.

Learning Outcomes of the Course Unit

1   will be able to explain what is homology, cohomology and Euler characteristic
2   will be able to express some complicated surfaces as a connected sum of well known surfaces
3   will be able to compute the Euler characteristic of surfaces
4   will be able to compute the homology and cohomology groups of some surfaces such as sphere, torus, cylinder, Möbius strip, etc.
5   will be able to compute Betti numbers of surfaces

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Review for point-set topology: Topological spaces, Continuous functions, Homeomorphisms
2 Review for point-set topology: Product space, Quotient space, Connectedness, Compactness
3 Topological surfaces, Surfaces in R^n, Surfaces via gluing
4 Surfaces via gluing, Connected sum
5 Classification of compact connected surfaces
6 Simplices, Simplicial complexes, Barycentric subdivisions
7 Simplicial approximation theorem
8 Fundamental group of a simplicial complex
9 Simplicial surfaces, Euler characteristics, Differentiable manifolds
10 Differentiable manifolds, Differential forms
11 Differential forms, Simplicial homology
12 Simplicial homology, Homology group of some surfaces
13 Homology group of some surfaces, De Rham s theorem
14 De Rham's cohomology

Recomended or Required Reading

Textbooks:
1. Bloch, E. D., A first course in Geometric Topology and Differential Geometry, Birkhauser, 1996, ISBN 978-0817638405
2. Singer, I. M., Thorpe, J.A., Lecture Notes on Elementary Topology and Geometry, Springer, 1976, ISBN 978-0387902029
Supplementary Books:
3. Armstrong, M. A., Basic Topology, Springer, 2010, ISBN 978-1441928191
4. Giblin, P.J., Graphs, Surfaces and Homology, 3rd ed, Cambridge University Press, 2010, ISBN 978-0521154055
5. Kinsey, L. C., Topology of Surfaces, Springer, 1993, ISBN 978-0387941028
References:
6. Hatcher, A., Algebraic Topology, Cambridge University Press, 2001, ISBN 978-0521795401
7. Bredon, G. E., Topology and Geometry, corrected ed., Springer, 1993, ISBN 978-0387979267

Planned Learning Activities and Teaching Methods

Lectures, Lecture notes and problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 ASG ASSIGNMENT
3 FINS FINAL EXAM
4 FCG FINAL COURSE GRADE MTE * 0.30 + ASG * 0.20 + FIN * 0.50
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + ASG * 0.20 + RST * 0.50


Further Notes About Assessment Methods

None

Assessment Criteria

Students' learning outcomes will be evaluated through written exams and homework assignments.

Language of Instruction

English

Course Policies and Rules

Students are required to have over 70% attendance in order to take the final exam.

Contact Details for the Lecturer(s)

e-mail: asli.ilhan@deu.edu.tr
Phone : 0 232 3018597

Office Hours

Monday: 14:40-16:40

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 4 56
Preparations before/after weekly lectures 14 3 42
Preparation for midterm exam 1 20 20
Preparation for final exam 1 35 35
Preparing assignments 2 10 20
Final 1 2 2
Midterm 1 2 2
TOTAL WORKLOAD (hours) 177

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.1543433333
LO.2544343343343
LO.35443433432343
LO.45434333333
LO.5543433333