Variations of the Möbius Band to Explore the Nature of Homeomorphism

Stephen Barr defines a homeomorphism (https://en.wikipedia.org/wiki/Homeomorphism) with a "no-cutting-or-joining rule": "distortions are only allowed if one does not disconnect what was connected (like making a cut or a hole), nor connect what was not (like joining the ends of the previously unjoined string, or filling in the hole)". It is an intuitively conceptual definition instead of a mathematically rigorous one. Its subtleties require a deeper exploration.

This session will explore the nature of homeomorphism by considering several variations on a Möbius strip made from a strip of paper.

In particular, we will consider the minimal length Möbius band. How much distortion is permitted in a homeomorphism of a Möbius strip? What does connectedness mean in a highly folded model of a Möbius band?

In addition, we will consider the conical Möbius band. How much of the edges of a strip of paper must be joined to properly produce a Möbius band?

Our guide for exploring these mind-bending ways to imagine homeomorphism is chapters 3-4 of Stephen Barr's fun book Experiments in Topology. A review of chapters 1 and 2 is recommended for newcomers and for all participants to deepen our understanding of the basic concepts.

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 topologically will help with our task of exploring the subject during our discussion.

To further guide your thinking about the variations on a Möbius strip that we will discuss, I organized this list of 27 questions (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.02.pdf) to guide your preparations in thinking about the book.

Here are some selected questions for the discussion from the full list (http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.02.pdf):

• 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?

• According to chapter 1 of Barr's book, what is the definition for a simply connected surface? How does Barr's definition compare to Wikipedia's definition of a simply connected space?

• What is the property of orientability? Which of the sphere, disk (the 2D surface bounded by a circle), cylinder, torus, and Möbius band are orientable and which are nonorientable? How can you explain orientability with paper models of topological surfaces?

• In a paper model of a Möbius strip, what happens when you cut through the middle of the strip? What are the resulting surface(s)? How many sides, edges, and separate pieces are there? What logic explains this behavior?

• Can one build a paper model of a Möbius strip whose width is greater than its length (let us define the width to be the length of each of the two opposite edges of the rectangular strip of paper that are glued together when making the Möbius band)? That is, can the width to length ratio (width/length) ever exceed 1? Note: the length of a Möbius band might be said to be the length of its one and only edge.

• How can one build a paper model of a Möbius band from a square strip of paper?

• What is the maximum width to length ratio for a paper model of a Möbius strip?

• How can one build such a "minimum length'' Möbius band from paper?

• What is the width to length ratio of the minimal length paper Möbius strip which can still be cut along the centerline of the strip? Why does the property of cutting a Möbius strip through its centerline fail when the width to length ratio exceeds that amount?

• In considering the variations on a Möbius band in chapter 3, how should we think about the connectedness of the joined edge of a Möbius band? How is the homeomorphic property preserved in each of these variations?

• In considering the sequence of experiments in chapter 3 where Möbius strips of decreasing length to width ratios (or increasing width to length ratios) are considered, what did you learn about the nature of homeomorphism and paper representations of topological surfaces?

• In considering the variations on a Möbius band in chapter 4, how should we think about the distortion and connectedness in the joined edge of the strip of paper used to make a Möbius band? How is the homeomorphic property preserved in each of these variations?

• In considering the sequence of experiments discussed in chapter 4 where Möbius strips with various extents of connectedness are considered, what did you learn about the nature of homeomorphism and paper representations of topological surfaces?

• How do the properties of simply connected surfaces and orientability play a role in understanding the variations on a Möbius band considered in chapters 3 and 4?

• Given the considerations in chapters 3 and 4, what subtleties, limitations, and caveats must we heed about the nature of the distortions allowed and the requirements of connectedness and continuity in Stephen Barr's definition of a homeomorphism?

This meetup is 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)