COURSE UNIT TITLE

: ALGEBRAIC TOPOLOGY

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5032 ALGEBRAIC TOPOLOGY ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSISTANT PROFESSOR ASLI GÜÇLÜKAN ILHAN

Offered to

Mathematics
Mathematics

Course Objective

The aim of this course is to introduce students to the elementary concepts of homotopy, homology and cohomology theories.

Learning Outcomes of the Course Unit

1   will be able to explain the main concepts of this course such as homotopy, homology, cohomology, etc.
2   will be able to compute the fundamental groups of elementary spaces
3   will be able to compute the homology groups of elementary spaces
4   will be able to compute the cohomology groups of spaces
5   will be able to use fiber bundles to make computations
6   will be able to know the main theorems of this course such as the van Kampen theorem, Künneths formula, Hurwicz theorem, Poincaré duality, etc.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Introduction to Algebraic Topology: What is the role of Algebraic Topology Which kinds of problems can be solved with aid of it Categories and Functors, Homeomorphisms, Homotopy equivalences,Operations on Spaces(product space, quotient space, topological sum)
2 Operations on spaces (Suspension, Join, Wedge sum, Smash product, Collapsing spaces, Attaching spaces), Cell complexes, Two Criteria for Homotopy equivalence, The homotopy extension property
3 Homotopy of paths, Homotopy of maps, The fundamental group, The fundamental group of the circle, Induced homomorphisms
4 Free products of groups, The van Kampen theorem and applications to Cell complexes
5 Lifting properties, the classification of covering spaces, Deck transformations and group actions, Higher homotopy groups
6 The idea of homology, Simplicial complexes, Simplicial homology
7 Singular homology, Homotopy invariance,
8 Midterm
9 Exact sequences and excision, The equivalence of Simplicial and Singular homology
10 Cellular homology, Mayer-Vietoris sequences, Homology with coefficients
11 Homology and Fundamental group, Cohomology of groups, The universal coefficient theorem
12 Cohomology of spaces, Cup product, The Cohomology ring, A Künneth formula
13 Poincaré Duality, Fiber bundles and Vector bundles, Spectral Sequences
14 Higher homotopy groups of spheres, Homology groups of spheres, Hurewicz Theorem

Recomended or Required Reading

Textbook:
1. Allen Hatcher, Algebraic topology, Cambridge University Press, 2001, ISBN-13: 978-0521795401
2. Hajime Sato, Algebraic Topology: an intuitive approach, AMS, 1999, ISBN-13: 978-0821810460
Supplementary Book(s):
3. I. M. Singer and J. A. Thorpe, Lecture notes on elementary Topology and Geometry, Springer, 1976, ISBN-13: 978-0387902029
4. Marvin J. Greenberg and John R. Harper, Algebraic Toology: a first course, Westview Press, 1981, ISBN-13: 978-0805335576
References:
5. Glen E. Bredon, Topology and Geometry, Springer, 2010, ISBN-13: 978-1441931030
6. Edwin H. Spanier, Algebraic Topology, Springer, 1994, ISBN-13: 978-0387944265
7. Dale Husemöller, Fiber Bundles, Springer-Verlag, Berlin, 1993, ISBN-13: 978-3540940876
8. Saunders MacLane, Homology, Springer, 1995, ISBN-13: 978-3540586623

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)

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
Tutorials 0
Preparations before/after weekly lectures 13 3 39
Preparation for midterm exam 1 20 20
Preparation for final exam 1 30 30
Preparation for quiz etc. 0
Preparing assignments 4 5 20
Preparing Group Assignments 4 5 20
Preparing presentations 0
Final 1 3 3
Midterm 1 3 3
Quiz etc. 0
TOTAL WORKLOAD (hours) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.1334533343
LO.2334433333
LO.3333433333
LO.4333433333
LO.5333433343
LO.6333533343

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