COURSE UNIT TITLE

: ADVANCED TOPOLOGY AND GEOMETRY I

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 6035 ADVANCED TOPOLOGY AND GEOMETRY I ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

Offered to

Mathematics (English)
Mathematics (English)

Course Objective

The aim of this course is to give an interest of smooth manifold theory in algebraic topology. We intend to understand the geometric concept by using algebraic tools. It aims to combine algebraic topology and differential topology.

Learning Outcomes of the Course Unit

1   will be able to recall the fundamental notions of topology
2   will be able to determine the homotopy type of spaces
3   will be able to use implicit and inverse function theorems
4   will be able to explain Sard s theorem
5   will be able to explain the action of first fundamental group on fibers

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Metric space, Topological space, Neighborhood, Continuity, Homeomorphism, Basis, Subbasis
2 Countability, Subspaces, Connectivity and components
3 Compactness, Paracompact spaces, Quotient spaces, Compactification
4 Homotopy, Homotopy inverse
5 Contraction, Contractable space, Deformation retract
6 Topological groups, Homomorphism between topological groups, Symmetric group
7 Group action, Some examples of Lie groups
8 Midterm
9 Differential manifolds, The mean value theorem, The Banach contraction principle
10 The implicit function theorem, examples
11 The inverse function theorem, Differentiable manifold, Local coordinates, Tangent vectors and differential
12 Sard s theorem and regular values, Vector fields and Flows, Tangent bundles
13 Fundamental groups, Homotopy groups
14 Bundle theory, Covering spaces, The lifting theorem, The action of first fundamental group on the fiber

Recomended or Required Reading

Textbook:
1. Glen E. Bredon, Topology and Geometry, Springer, 1993, ISBN-13: 978-0387979267
Supplementary Book:
References:
2. Saunders MacLane, Homology, Springer, 1995, ISBN-13: 978-3540586623
3. William S. Massey, Algebraic Topology: an introduction, Springer, 1977, ISBN-13: 978-0387902715
4. Marvin J. Greenberg and John R. Harper, Westview Press, 1981, ISBN-13: 978-0805335576
5. Edwin Henry Spanier, Algebraic Topology, Springer, 1994, ISBN-13: 978-0387944265
6. John Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974, ISBN-13: 978-0691081229
Materials:

Planned Learning Activities and Teaching Methods

Lecture notes, Presentation, Problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 MTE MIDTERM EXAM
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40


Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

Email: bedia.akyar@deu.edu.tr

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 13 3 39
Preparations before/after weekly lectures 13 4 52
Preparation for midterm exam 1 23 23
Preparation for final exam 1 30 30
Preparing assignments 10 5 50
Midterm 1 3 3
Final 1 3 3
TOTAL WORKLOAD (hours) 200

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.133544333
LO.2435434333
LO.3444434343
LO.44444334343
LO.54444334343

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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