数学 409：拓扑学

Instructor (老师) : Yitzchak Elchanan Solomon (水神恩)

Fall Term 2, Session 2, 2021 / 秋天第二学期

TuTh 7:30AM – 9:30AM / 星期二+四上午 7：30 – 9： 30

Textbook (教科书）: “Introduction to Topology: Second Edition” by Gamelin and Greene.

**What is this course about?**

Math 409 is a first course in topology. Topology is a fundamental area of mathematics that studies continuity, connectivity, compactness, and shape. In addition to being an important field in its own right, topology is used throughout mathematics, in geometry, analysis, algebra, number theory, combinatorics, etc. Topology is also useful in many other scientific fields, like computer science, physics, and, as of late, machine learning. Studying topology develops abstract mathematical thinking and can benefit anyone looking to pursue advanced STEM studies.

This course is divided into several modules. The first module introduces topological concepts by studying metric spaces. The next module dives into abstract topological spaces: what they are, how they can be transformed, and different properties they can have. In the third module, we study the important concepts of compactness and connectedness in topological spaces, and how these relate to products and quotients of spaces. Finally, in the last module, we turn to algebraic topology by studying the fundamental group of a topological space and the elements of homotopy theory.

Topics include topology of metric spaces, abstract topological spaces, open and closed sets, connectedness, compactness, continuity, and completeness, subspaces, product and quotient spaces, separation axioms, homotopies of paths, the fundamental group, covering spaces, index theory, and applications (Borsuk-Ulam Theorem, Ham Sandwich Theorem, Fundamental Theorem of Algebra).

**How will this course be organized?**

The course will consist of daily lectures over Zoom. The lectures will be recorded and uploaded to YouTube for asynchronous students to access, as well as for synchronous students to review. The homeworks will be taken from the textbook, will be focused on proofs, and will be handed in weekly in LaTeX or neatly written up. Homework will be due to the Wednesday of the week after it is assigned. A midterm and a final will be used to assess your grade, the breakdown is as follows：Homework: 40%, Midterm: 30%, Final: 30%.

There will be one homework assignment each week, excluding the final week, with the lowest grade dropped. As the lowest homework grade will be dropped, no late homework will be accepted outside of emergency situations. The midterm will cover the first two modules, metric spaces and abstract topological spaces, and the final will cover the remaining two modules. Homework will be submitted on Gradescope.

There will also be a midterm and a final, both of which will be administered over Gradescope: Students will have 3 hours in a 72-hour time-window to work on problems. The exams will be open-book, but working with peers or using the internet will be prohibited. The exams will focus on proofs, and not on stating definitions/theorems.

LECTURE NOTES

LECTURE RECORDINGS

Weeks | Topics | Chapters | Assignments Due |

1-2 | Topology ofMetric Spaces (open and closed sets, completeness, products of metric spaces, compactness, continuity, contraction mappings) | 1 | Hwk 1 Hwk 1 Sols Hwk 2 Hwk 2 Sols |

3-4 | Abstract Topological Spaces (definition of topological spaces, subspaces, continuity, bases, separation axioms) | 2.1-2.5 | Homework 3 Hwk 3 Sols Midterm middle of week 4 Midterm Sols |

4-5 | Connectedness & Compactness (compactness and local compactness, connectedness and path connectedness, product and quotient spaces) | 2.6- | Homework 4 Hwk 4 Sols |

6-7 | Algebraic Topology (primer on group theory, path homotopy, the fundamental group, induced homomorphisms, covering spaces, index theory) | 3 | Homework 5 Hwk 5 Sols Final in week 8 |