Algebraic Topology I (ΓΕ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ5
Semester	2
Course Title	Algebraic Topology I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special background. Specialized general knowledge. Skills development.
Prerequisite Courses	General Topology, Algebraic Structures I
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Algebraic topology is a twentieth century field of mathematics that can trace its origins and connections back to the ancient beginnings of mathematics. One of the strengths of algebraic topology has always been its wide degree of applicability to other fields. Nowadays that includes fields like physics, differential geometry, algebraic geometry, and number theory, biology, financial sciences, computer sciences. We expect familiarity with basic notions from point set topology. We study the compact open topology and function spaces as an introduction to homotopy between maps. The main Learning Outcomes can be described as the application of Cell complexes and the category of CW spaces in connection between homotopy-homology and important problems in geometry. Moreover, how do we compute using homotopy? How can we distinguish between topological spaces? We compute the fundamental groups of basic topological spaces and classify covering spaces. Singular homology is introduced along with the main technics of computations.

General Competences

Search for analysis and synthesis of data and information related with topological and geometrical problems. Working independently and in a Team work. Working in an interdisciplinary environment aiming at production of new research ideas related to the syllabus of the course.

Syllabus

Compact open topology. Homotopy, fundamental group. Homotopy of the circle. Cell complexes. Real and Complex projective spaces. Covering spaces. Deformations. Classification of covering spaces. Applications. Scheifert-Van Kampen Theorem. Fundamental groups of surfaces.

Singular homology. Homotopic maps and homology. The long exact sequence of a pair. Homology of the sphere. Relative homology, excision. The degree of maps of spheres. Fixed point Theorems.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working hours in class 28 Project 30 Assignments 40.5 Final 50 Course total 187.5
Student Performance Evaluation	Written Examination, Oral Presentation, tests, written assignments.

Attached Bibliography

• A first course in topology, j. Muncres, Prentice Hall.
• Algebraic Topology, A. Hatcher, https://www.math.cornell.edu/~hatcher/AT/
• A Concise Course in Algebraic Topology, J. P. May, https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf