Dokuz Eylül University Information Package / Courses Catalog

Description of Individual Course Units

Course Unit Code	Course Unit Title	Type Of Course	D	U	L	ECTS
MAT 6064	ADVANCED TOPOLOGY AND GEOMETRY II	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

ASSISTANT PROFESSOR MURAT ALTUNBULAK

Offered to

Mathematics
Mathematics

Course Objective

The aim of this course is to give an interest of topological invariants of a space in algebraic topology with aid of homology theory. We intend to understand the geometric concepts by using algebraic tools.

Learning Outcomes of the Course Unit

1	will be able to know notions of singular homology, cohomology and homotopy groups
2	will be able to calculate the homology groups of naturally occurring topological spaces
3	will be able to use homology groups to say something about the homotopy type of a topological space
4	will be able to know duality theorems
5	will be able to know the relations between homotopy and homology groups

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week	Subject	Description
1	Standard simplex, an affine sigular simplex, The singular homology group
2	CW complexes
3	Singular homology, Subdivision
4	The Mayer-Vietoris sequence, Simplicial complexes, Simplicial maps
5	Multi Linear algebra, Differential forms, Integration of forms, Stokes theorem
6	Relationship to Singular homology, The de Rham s theorem
7	The cross product and the Künneth s theorem, A sign convention, A cup and cap products, The orientation bundle
8	Midterm Exam
9	Duality theorems, Duality and compact manifolds with boundary
10	Homotopy theory, Cofibrations
11	The compact-open topology, Homotopy groups
12	Fiber spaces
13	Free homotopy, Classical groups and Associated manifolds
14	The Hurzewicz theorem, The Whitehead theorem

Recomended or Required Reading

Textbook: Glen E. Bredon, Topology and Geometry, Springer, 1993, ISBN-13: 978-0387979267

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

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

*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

Office: B-220 (Math. Dept.)
Phone: (30)1 85 92

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	20	20
Preparation for final exam	1	30	30
Preparing assignments	10	5	50
Midterm	1	3	3
Final	1	3	3
TOTAL WORKLOAD (hours)			197

Contribution of Learning Outcomes to Programme Outcomes

PO/LO	PO.1	PO.2	PO.3	PO.4	PO.5	PO.8	PO.9	PO.10	PO.11
LO.1	4	3	3	4	3	5	3	4	3
LO.2	4	3	4	4	3	4	3	3	3
LO.3	4	4	4	4	3	5	3	4	3
LO.4	4	4	4	4	3	5	3	4	3
LO.5	4	4	4	4	3	5	3	4	3

COURSE UNIT TITLE

DEGREE PROGRAMMES

Description of Individual Course Units

Offered By

Level of Course Unit

Course Coordinator

Offered to

Course Objective

Learning Outcomes of the Course Unit

Mode of Delivery

Prerequisites and Co-requisites

Recomended Optional Programme Components

Course Contents

Recomended or Required Reading

Planned Learning Activities and Teaching Methods

Assessment Methods

Further Notes About Assessment Methods

Assessment Criteria

Language of Instruction

Course Policies and Rules

Contact Details for the Lecturer(s)

Office Hours

Work Placement(s)

Workload Calculation

Contribution of Learning Outcomes to Programme Outcomes

CONTACT INFORMATION