Special Mathematics Lecture (Topology and homotopy)

Goals of the Course

【Standardized across all programs】 In a global and interconnected society, mathematics play a central role. They are at the root of all communications tools, and serve as a universal language for natural and social sciences. All current technological developments are based on them. Mathematics can also be learned, enjoyed, and shared by everybody. Studying mathematics is also learning humility, tolerance, and diversity.

The aim of these Special Mathematics Lectures is to provide a solid knowledge on various mathematical topics. Concomitantly, students will develop a rigorous argumentation and a critical thinking. Each semester, a new subject is proposed, and these lectures are accessible to all students of Nagoya University, independently of their major and of their academic age. These special lectures also offer a unique opportunity to study in an international, transdisciplinary, and friendly environment.

Objectives of the Course

To acquire basic knowledge of topological spaces and their properties by studying a variety of examples. Students will also obtain an elementary understanding of homotopy theory as a gateway to further studies in algebraic topology.

Course Content or Plan

The topic will be an introduction to topology, which is a mathematical framework to understand continuity and equivalence in much more general settings. We will also give a basic overview of homotopy theory, which is one example of the interaction of topology with abstract algebra.

Planned content:

Metric spaces

• open/closed sets
• continuity and completeness
• compactness

Topological spaces

• open/closed sets, bases, separation axioms
• continuous maps and homeomorphisms
• subspaces and quotient spaces
• compactness and connectedness

Homotopy

• homotopic paths and the fundamental group
• covering spaces
• higher dimensional homotopy

Course Evaluation Method and Criteria

The grade will be determined by exercises and homework questions submitted by students throughout the semester. Achievement of the objectives stated in the [Objectives of the Course] section will be considered the criterion for passing the course.

Students who wish to withdraw from the topic are asked to inform the instructor.

Study Load (Self-directed Learning Outside Course Hours)

Exercises and homework questions will be given to students. Students are expected to read their notes, and to be familiar with the content of the previous lectures before each new lecture.

Notice for Students

It is expected that the students will show a certain maturity in studying independently and in choosing some exercises and problems to solve. Study sessions might be organized on a semi-regular basis.

Textbook

Theodore W. Gamelin and Robert E. Greene, Introduction to Topology (2nd edition), Dover Publications, 1999 It is not necessary to purchase this textbook.

Lecture summary notes and other references will also be distributed.

Reference Book

• Martin D. Crossley, Essential Topology, Springer, 2005
• James R. Munkres, Topology, Prentice Hall, 2000

Lecture Materials

• SML (Topology and Homotopy), Summary Notes
• SML (Topology and Homotopy), Problem Sheet

Reference Website

Special Mathematics Lecture (SML, Spring 2026) Topology and Homotopy

This lecture is provided under Creative Commons Attribution-Non Commercial-ShareAlike 4.0 International.

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