Differential Topology (ΓΕ4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ4
Semester	2
Course Title	Differential Topology
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in Greek)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	In this lecture we present applications of Algebraic and Differential Topology in the study of topological invariants of smooth manifolds.
General Competences	Work autonomously Work in teams Develop critical thinking skills

Syllabus

• Manifolds.
• Immersions, embeddings and submersions.
• Milnor’s proof of the fundamental theorem of algebra.
• Sard’s theorem and Morse functions.
• Partition of unity and Whitney’s embedding theorem.
• Homotopy and isotopy.
• Brouwer’s degree.
• Whitney’s approximation theorem.
• Differential forms and integration.
• Hopf's invariant.
• Hopf's degree theorem.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 78 Solution of Exercises-Homeworks 70.5 Course total 187.5
Student Performance Evaluation	Weakly HomeWorks, presentations in the blackboard of HomeWorks, written final examination.

Attached Bibliography

• T. Bröcker, K. Jänich, Introduction to differential topology, Cambridge Univ. Press, 1982.
• V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974.
• J. Milnor, Morse Theory, Annals of Mathematical Studies, 51. Princeton University Press, Princeton, N.J. 1963.
• J. Milnor, Topology from a differentiable viewpoint, The University Press of Virginia, Charlottesville, Va. 1965.