Exploring the Topology of the Projective Plane

Stephen Barr's fun book "Experiments in Topology" helps us begin to think topologically by building and considering paper models. In this session, we will explore the projective plane. In Barr's book the projective plane is introduced in Chapter 2 on pages 34 and 38 and more extensively in Chapter 6 on pages 78-107.

The introductory section of Chapter 1 on pages 1-9 (especially the account on homeomorphism) and Chapter 2 on pages 20-39 provide important background information for this session. Since the text on the Möbius strip and Klein bottle introduce experiments that may help in understanding the even more subtle projective plane, briefly skimming Chapters 3, 4, and 5 might be helpful.

The hope and intention for this session is to build our intutions about topology and thinking topologically about surfaces through exploring paper models of the projective plane.

It is recommended that participants build as many of the paper models discussed in the book as they have time for. Building the models and thinking about them ahead of time will help with our task of exploring the subject during our discussion.

To guide your exploration of the text and to guide our exploration during the meetup, I have organized this list of 22 questions: http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.06.pdf

Here are some selected questions for the discussion from the full list (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.06.pdf):

● Inspired by the book, the questions that follow or your own initiative, what additional experiments did you attempt? What did you learn from these supplemental experiments?

● According to chapter 1 of Barr's book, what is the rule for homeomorphically distorting one surface into another? What caveats does Barr mention for using his rule? How does Barr's definition compare to Wikipedia's definition of homeomorphism?

● Why would a projective plane with a hole in it be deformable to a Möbius strip? Is the projective plane with a hole in it, therefore, homeomorphic to the Moebius strip?

● Why is a sphere with a hole in it deformable to a plane? Is a sphere with a hole in it, therefore, homeomorphic to a plane?

● What is a cross-cap? How would you describe it? How would you describe it to a child?

● On pages 82-€“85, Barr describes the Martin Gardner model of a projective plane (Figures 8-11). Do you understand the model? How does it work? How does your imagination interpret the model to see the inherent self-intersection that is required?

● In several of the experiments discussed in Chapter 6 on "€œThe Projective Plane"€ subtleties associated with connectivity when cutting and re-joining models are explored. What are these subtleties? What is the topological point of exploring them in our paper models? What caveats must we keep in mind when interpreting a model with cuts in it?

● The point of the latter part of Chapter 6 is to determine if the projective plane (and the Möbius strip) is symmetric. What does it mean for a topological surface to be symmetrical? How can you explain the symmetry of the projective plane with (paper) models? Can you imagine an explanation that is clear enough to explain the symmetry of the projective plane to a child?

This meetup will be part of a series exploring the content of Stephen Barr's fun book "Experiments in Topology".

"Experiments in Topology" is available from Dover (http://store.doverpublications.com/0486259331.html)

"Experiments in Topology" at Google Books (https://books.google.com/books?id=9TMx6ABV-98C)

Each event will be made as accessible to newcomers as possible. Key concepts will be reviewed and an effort to explain technical terms will be made. If anything is unfamiliar to you, please ask and we will try to clarify.