Topology

Homework

• Homework 1
• Homework 2
• Homework 3

Syllabus

Course description

Topology comes from the Greek, τόπος, which means “place”. So from a purely linguistic point of view topology should be the study of places! In reality, topology is a subject much akin to geometry, but it is much more relaxed in many respects. For example, in geometry one is often concerned with the isometries (distance-preserving transformations) of a geometric space, whereas in topology there is not necessarily a notion of distance. Nevertheless, we can think of topology as a tool to encode the notion of “nearness.” As such, continuous maps between topological spaces are the fundamental kinds of objects in topology (just consider: we think of a function as continuous if points which are near in the domain are near in the codomain; this is much easier to intuit than the $\epsilon-\delta$ definition).

We will use the text Topology Now! by Robert Messer and Phil Straffin.

Course objectives

By the end of this course, students should be able to:

• Recognize surfaces and three-dimensional manifolds in Euclidean space
• Give a plethora of examples of such objects
• Determine isotopy of simple knots using knot diagrams and Reidemeister moves
• Give examples of metric and topological spaces
• Determine when certain topological spaces are homeomorphic (or not!)

Course Outline And Topics

More than likely, we will not cover topics in this order. Moreover, some material will be emphasized less than others (e.g., knot theory). Decisions will be somewhat dependent on the interests of the students.

1. Deformations

   • Equivalence
   • Bijections
   • Continuous Functions
   • Topological Equivalence
   • Topological Invariants
   • Isotopy

2. Knots and Links

   • Knots, Links and Equivalences
   • Knot Diagrams
   • Reidemeister Moves
   • Colorings
   • The Alexander Polynomial
   • Skein Relations
   • The Jones Polynomial

3. Surfaces

   • Definitions and Examples of Surfaces
   • Cut-and-Paste Techniques
   • The Euler Characteristic, Orientability
   • Classification of Surfaces
   • Surfaces Bounded by Knots

4. Three-Dimensional Manifolds

   • Examples of Three-Dimensional Manifolds
   • The Euler Characteristic
   • Gluing Polyhedral Solids

5. The Fundamental Group

   • Algebraic Properties
   • Invariance of the Fundamental Group
   • The sphere and the Circle

6. Metric and Topological Spaces

   • Metric Spaces
   • Topological Spaces
   • Connectedness
   • Compactness
   • Quotient Spaces

Grading procedure

• Three take home exams (25% each)
• Homework (20% total)
• Presentations (5% total)

Take-home exams are expected to be completed individually without the aid of your classmates; you are welcome to use the book and your notes, but you may not consult resources online. Homework will be assigned weekly. You are highly encouraged to work on the homework problems together with your classmates as this promotes learning, however, you should each write your final solutions independently. You will also expected to present some of your solutions to homework problems in front of the rest of the class. Your grade for the presentations will be determined based on the clarity of your communication, not so much on the validity of your results. The presentations are primarily an opportunity for learning how to communicate, which is why they account for a smaller portion of the overall grade.

Important dates

Makeups

If an emergency arises which requires you to miss an exam, I must be made aware at least two hours prior to the start time of your exam. Note: proper documentation will be required before a makeup arrangement is considered.

Academic dishonesty/misconduct

Academic misconduct includes, but is not limited to, cheating, plagiarism and forgery, and soliciting, aiding, abetting, concealing, or attempting such acts. Plagiarism may consist of copying, paraphrasing, or otherwise using written or oral work of another without proper acknowledgment of the source or presenting oral or written material prepared by another as ones own. At minimum, cheating will result in that assignment receiving a grade of zero.

Official communication

Your SIUE student e-mail account is the official method to communicate between you and your instructor. Official communication will not be sent to your personal e-mail (yahoo, wildblue, gmail etc.).

Accessible Campus Community & Equitable Student Support

Students needing accommodations because of medical diagnosis or major life impairment will need to register with Accessible Campus Community & Equitable Student Support (ACCESS) and complete an intake process before accommodations will be given. Students who believe they have a diagnosis but do not have documentation should contact ACCESS for assistance and/or appropriate referral. The ACCESS office is located in the Student Success Center, Room 1270. You can also reach the office by e-mail at myaccess@siue.edu or by calling (618)~650-3726. For more information on policies, procedures, or necessary forms, please visit the ACCESS website at www.siue.edu/access.

Disclaimer

This syllabus is subject to change by the instructor if deemed necessary for the benefit of student learning or to correct errors and omissions.