Betti Numbers and the Symmetry of the Projective Plane

We will explore several questions on Betti numbers and the nature and the symmetry of the projective plane from chapters 8 and 6 of Stephen Barr's book "Experiments in Topology". We will focus on pages 123-128 in Chapter 8 which discusses Betti numbers and pages 96-107 in Chapter 6 on the projective plane.

All participants might want to read pages 10-19 in Chapter 1 on Euler's theorem which is referenced in the material on Betti numbers. For completeness, you might want read all of Chapter 8, pages 120-135, as well.

Newcomers to the subject are advised to read pages 1-9 in Chapter 1, all of Chapter 2 (pages 20-39), and the beginning of Chapter 6 (pages 82-96) in order to follow the material on the projective plane.

Here is a set of 10 exercises which will guide our exploration during the event: http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.10.pdf

Here are the main questions for the discussion abstracted from the full list of exercises ( http://www.cjfearnley.com/MathCounts/TopologicalSurfaces.10.pdf ):

● What is the Betti number for a disk, an annulus, a torus, a Möbius strip, a Klein bottle, a sphere, and a projective plane? Do the different results reasonably characterize the connectivity of each surface? How? Why?

● What is the pattern for the number of twists that result after cutting along the axes of various variations on the Martin Gardner model of the projective plane discussed in Stephen Barr's text?

● In considering this rule for the number of twists and the full set of variations on the Martin Gardner model of the projective plane discussed in Stephen Barr's text, can you understand why the projective plane is symmetrical? Can you organize your understanding to be clear enough to explain the projective plane's symmetry to a child?

● How are the topological and geometrical projective planes related? Can you see the symmetry of the topological perspective in each of the geometrical models described at http://blog.cjfearnley.com/2012/07/24/models-of-projective-geometry

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)

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.