Ori Livson

Construction of the Real Projective Plane from a Capped Cylinder

I.e., RP² via identification of the two caps on cylinder with an orientation-reversing twist.

Introduction

The Klein bottle is a non-orientable surface, meaning it impossible to distinguish between its inside or outside (see Figure 1). It is closely related to the well-known Möbius strip in that a Klein bottle can be produced by gluing two Möbius strips together along their boundaries. However, a Klein bottle is a 2D surface that can only be embedded 4D space, lest the surface self-intersects (see this plus maths article for further illustrations and explanations).

A well-known construction of the Klein bottle involves taking a cylinder / annulus (i.e., S1 x [0,1] ) and gluing its openings together with an orientation reversing twist (see Figure 2)

However, for a paper I recently coauthored¹, we were confronted with the same construction but with a capped cylinder is, i.e., a cylinder with a disk joined to each of its openings.²

To reiterate, the construction we’re talking about is taking a capped cylinder, and identifying (i.e., gluing) its caps such that the caps are assigned opposite orientations (along their boundaries) in that identification, just like in Figure 2.

Interestingly, the capped cylinder version yields another well-known non-orientable surface known as the real projective plane (RP²) - see Figure 3 for a visualisation.

Moreover, I was especially surprised that the answer was not readily available like the Klein bottle construction of Figure 2. The purpose of this post is to briefly outline why the above construction on the capped cylinder produces the real projective plane.

The Real Projective Plane

The real projective plane can be constructed in many different ways. For instance, by the fundamental polygon (i.e., gluing diagram) of Figure 4, or by gluing a Möbius Strip to a disk along their boundaries (see this article for an example). The construction of RP² that we’re interested is via the identification of antipodal points on a sphere (see Figure 5).

The Construction via a Capped Cylinder

We begin with an intuitive version of the argument, then outline the formal version.

Intuitive version: The first fact to observe is that a capped cylinder C is homomorphic to the 2D sphere S², the way to see this is to split the two surfaces into the following parts:

• C into its top and bottom caps D_top, D_bot, and an (uncapped) cylinder C closed by the caps.

• S² into a Northern hemisphere, Southern hemisphere and an equatorial band.

Then we have a homeomorphism by mapping the caps to the hemispheres and the (uncapped) cylinder to the equatorial band (see Figure 6).

Thus, we can intuit that our construction is equivalent to RP² if we can show that the construction (i.e., oppositely identifying the caps of the cylinder with an orientation reversing twist), passed through a homeomorphism to the sphere S², identifies all the sphere’s antipodal points.

Indeed, we find that identifying the caps of the cylinder with an orientation reversing twist amounts to identification of antipodal points across the hemispheres. Then, it remains to show that this identification drags the equatorial band into antipodal identifications as well. Intuitively one can see this by examining how neighbourhood identifications translate across the homeomorphism (see Figure 7), were one to sufficiently shrink down the equatorial band.

Formal version: formally, we prove that our construction is equivalent to RP² by showing the construction itself (i.e., the mapping of open neighbourhoods in Figure 7) comrpises a 2-sheeted covering map from the sphere (see Footnote 3 for a summary of 2-sheeted covering maps).

The result then follows because:

1. It is well known that if there exists a 2-sheeted covering map from a surface X to Y, then Y is non-orientable if and only if X is connected. Hence, our construction is firstly non-orientable because the sphere is obviously connectied .

2. If our construction starts with a sphere (which has Euler characteristic 2) and then collapses it with a 2-to-1 identification, the resulting surface must then have half the Euler characteristic (i.e., 1). So, by the classification theorem on closed surfaces, the only non-orientable surface with an Euler characteristic of 1 is RP².