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Assignment Instructions

Prove the Classification of Surfaces Theorem using the Conway ZIP proof (in the attached

PDF document). The proof should be around 5-7 pages, APA 6th edition format.

Due: April 19, 2018 8:00 am EST

Classification of Surfaces Theorem

There are 3 versions of the theorem but all 3 theorems are equivalent:

Theorem 1: Any closed surface is homeomorphic either to:

1.1 The Sphere

1.2 The sphere with a finite number of handles added

1.3 The sphere with a finite number of discs removed and replaced by Mobius strips.

Theorem 2: Any closed surface is homeomorphic either to:

2.1 The sphere

2.2 A connected sum of tori

2.3 A connected sum of projective planes

Theorem 3: Any closed surface is homeomorphic either to:

3.1 The sphere

3.2 A connected sum of tori

3.3 A connected sum of a tori and one projective plane

3.4 A connected sum of a tori and two projective planes

Conways ZIP Proof

George K. Francis

Jeffrey R. Weeks

December 28, 2000

Surfaces arise naturally in many different forms, in branches of mathematics

ranging from complex analysis to dynamical systems. The Classification Theorem, known since the 1860s, asserts that all closed surfaces, despite their diverse

origins and seemingly diverse forms, are topologically equivalent to spheres with

some number of handles or crosscaps (to be defined below). The proofs found

in most modern textbooks follow that of Seifert and Threlfall [5]. Seifert and

Threlfalls proof, while satisfyingly constructive, requires that a given surface

be brought into a somewhat artificial standard form. Here we present a completely new proof, discovered by John H. Conway in about 1992, which retains

the constructive nature of [5] while eliminating the irrelevancies of the standard

form. Conway calls it his Zero Irrelevancy Proof, or ZIP proof, and asks that

it always be called by this name, remarking that otherwise theres a real danger

that its origin would be lost, since everyone who hears it immediately regards

it as the obvious proof. We trust that Conways ingenious proof will replace

the customary textbook repetition of Seifert-Threlfall in favor of a lighter, fatfree nouvelle cuisine approach that retains all the classical flavor of elementary

topology.

We work in the realm of topology, where surfaces may be freely stretched and

deformed. For example, a sphere and an ellipsoid are topologically equivalent,

because one may be smoothly deformed into the other. But a sphere and a

doughnut surface are topologically different, because no such deformation is

possible. All the figures in the present article depict deformations of surfaces.

For example, the square with two holes in Figure 1A is topologically equivalent

to the square with two tubes (1B), because one may be deformed to the other.

More generally, two surfaces are considered equivalent, or homeomorphic, if and

only if one may be mapped onto the other in a continuous, one-to-one fashion.

That is, its the final equivalence that counts, whether or not it was obtained

via a deformation.

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Figure 2. Crosshandle

Figure 1. Handle

Let us introduce the primitive topological features in terms of zippers or

zip-pairs, a zip being half a zipper. Figure 1A shows a surface with two boundary circles, each with a zip. Zip the zips, and the surface acquires a handle (1D).

If we reverse the direction of one of the zips (2A), then one of the tubes must

pass through itself (2B) to get the zip orientations to match. Figure 2B shows

the self-intersecting tube with a vertical slice temporarily removed, so the reader

may see its structure more easily. Zipping the zips (2C) yields a crosshandle

(2D). This picture of a crosshandle contains a line of self-intersection. The selfintersection is an interesting feature of the surfaces placement in 3-dimensional

space, but has no effect on the intrinsic topology of the surface itself.

Figure 3. Cap

Figure 4. Crosscap

If the zips occupy two halves of a single boundary circle (Figure 3A), and

their orientations are consistent, then we get a cap (3C), which is topologically

trivial (3D) and wont be considered further. If the zip orientations are inconsistent (4A), the result is more interesting. We deform the surface so that

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corresponding points on the two zips lie opposite one another (4B), and begin

zipping. At first the zipper head moves uneventfully upward (4C), but upon

reaching the top it starts downward, zipping together the other two sheets

and creating a line of self-intersection. As before, the self-intersection is merely

an artifact of the model, and has no effect on the intrinsic topology of the surface. The result is a crosscap (4D), shown here with a cut-away view to make

the self-intersections clearer.

The preceding constructions should make the concept of a surface clear to

non-specialists. Specialists may note that our surfaces are compact, and may

have boundary.

Comment. A surface is not assumed to be connected.

Comment. Figure 5 shows an example of a triangulated surface. All surfaces

may be triangulated, but the proof [4] is difficult. Instead we may consider the

Classification Theorem to be a statement about surfaces that have already been

triangulated.

Definition. A perforation is whats left when you remove an open disk from a

surface. For example, Figure 1A shows a portion of a surface with two perforations.

Definition. A surface is ordinary if it is homeomorphic to a finite collection

of spheres, each with a finite number of handles, crosshandles, crosscaps, and

perforations.

Classification Theorem (preliminary version) Every surface is ordinary.

Proof: Begin with an arbitrary triangulated surface. Imagine it as a patchwork

quilt, only instead of imagining traditional square patches of material held together with stitching, imagine triangular patches held together with zip-pairs

(Figure 5). Unzip all the zip-pairs, and the surface falls into a collection of

triangles with zips along their edges. This collection of triangles is an ordinary

surface, because each triangle is homeomorphic to a sphere with a single perforation. Now re-zip one zip to its original mate. Its not hard to show that the

resulting surface must again be ordinary, but for clarity we postpone the details

to Lemma 1. Continue re-zipping the zips to their original mates, one pair at

a time, noting that at each step Lemma 1 ensures that the surface remains ordinary. When the last zip-pair is zipped, the original surface is restored, and is

seen to be ordinary. 2

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Figure 5. Install a zip-pair along

each edge of the triangulation,

unzip them all, and then re-zip

them one at a time.

Figure 6. These zips only partially occupy the boundary circles, so zipping them yields a

handle with a puncture.

Lemma 1 Consider a surface with two zips attached to portions of its boundary. If the surface is ordinary before the zips are zipped together, it is ordinary

afterwards as well.

Proof: First consider the case that each of the two zips completely occupies a

boundary circle. If the two boundary circles lie on the same connected component of the surface, then the surface may be deformed so that the boundary

circles are adjacent to one another, and zipping them together converts them

into either a handle (Figure 1) or a crosshandle (Figure 2), according to their

relative orientation. If the two boundary circles lie on different connected components, then zipping them together joins the two components into one.

Next consider the case that the two zips share a single boundary circle, which

they occupy completely. Zipping them together creates either a cap (Figure 3)

or a crosscap (Figure 4), according to their relative orientation.

Finally, consider the various cases in which the zips neednt completely occupy their boundary circle(s), but may leave gaps. For example, zipping together

the zips in Figure 6A converts two perforations into a handle with a perforation on top (6B). The perforation may then be slid free of the handle (6C,6D).

Returning to the general case of two zips that neednt completely occupy their

boundary circle(s), imagine that those two zips retain their normal size, while

all other zips shrink to a size so small that we cant see them with our eyeglasses

off. This reduces us (with our eyeglasses still off!) to the case of two zips that

do completely occupy their boundary circle(s), so we zip them and obtain a

handle, crosshandle, cap, or crosscap, as illustrated in Figures 14. When we

put our eyeglasses back on, we notice that the surface has small perforations as

well, which we now restore to their original size. 2

The following two lemmas express the relationships among handles, crosshandles, and crosscaps.

4

Lemma 2 A crosshandle is homeomorphic to two crosscaps.

Proof: Consider a surface with a Klein perforation (Figure 7A). If the parallel

zips (shown with black arrows in 7A) are zipped first, the perforation splits in

two (7B). Zipping the remaining zips yields a crosshandle (7C).

If, on the other hand, the antiparallel zips (shown with white arrows in

Figure 7A) are zipped first, we get a perforation with a Mo¨bius bridge (7D).

Raising its boundary to a constant height, while letting the surface droop below

it, yields the bottom half of a crosscap (7E). Temporarily fill in the top half of

the crosscap with an invisible disk (7F), slide the disk free of the crosscaps

line of self-intersection (7G), and then remove the temporary disk. Slide the

perforation off the crosscap (7H) and zip the remaining zip-pair (shown with

black arrows) to create a second crosscap (7I).

The intrinsic topology of the surface does not depend on which zip-pair is

zipped first, so we conclude that the crosshandle (7C) is homeomorphic to two

crosscaps (7I). 2

Figure 7. A crosshandle is homeomorphic to two crosscaps.

Lemma 3 (Dycks Theorem [1]) Handles and crosshandles are equivalent

in the presence of a crosscap.

Proof: Consider a pair of perforations with zips installed as in Figure 8A. If,

on the one hand, the black arrows are zipped first (8B), we get a handle along

with instructions for a crosscap. If, on the other hand, one tube crosses though

5

itself (8C, recall also Figure 2B) and the white arrows are zipped first, we get a

crosshandle with instructions for a crosscap (8D). In both cases, of course, the

crosscap may be slid free of the handle or crosshandle, just as the perforation

was slid free of the handle in Figure 6BCD. Thus a handle-with-crosscap is

homeomorphic to a crosshandle-with-crosscap. 2

Figure 8. The presence of a crosscap makes a handle cross.

Classification Theorem Every connected closed surface is homeomorphic to

either a sphere with crosscaps or a sphere with handles. Proof: By the preliminary version of the Classification Theorem, a connected closed surface is

homeomorphic to a sphere with handles, crosshandles, and crosscaps.

Case 1: At least one crosshandle or crosscap is present. Each crosshandle is

homeomorphic to two crosscaps (Lemma 2), so the surface as a whole is homeomorphic to a sphere with crosscaps and handles only. At least one crosscap

is present, so each handle is equivalent to a crosshandle (Lemma 3), which is

in turn homeomorphic to two crosscaps (Lemma 2), resulting in a sphere with

crosscaps only.

Case 2: No crosshandle or crosscap is present. The surface is homeomorphic

is a sphere with handles only.

We have shown that every connected closed surface is homeomorphic to

either a sphere with crosscaps or a sphere with handles. 2

Comment. The surfaces named in the Classification Theorem are all topologically distinct, and may be recognized by their orientability and Euler number.

A sphere with n handles is orientable with Euler number 2 – 2n, while a sphere

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with n crosscaps is nonorientable with Euler number 2-n. Most topology books

provide details; elementary introductions appear in [6] and [2].

Nomenclature. A sphere with one handle is a torus, a sphere with two handles is a double torus, with three handles a triple torus, and so on. A sphere

with one crosscap has traditionally been called a real projective plane. That

name is appropriate in the study of projective geometry, when an affine structure is present, but is inappropriate for a purely topological object. Instead,

Conway proposes that a sphere with one crosscap be called a cross surface. The

name cross surface evokes not only the crosscap, but also the surfaces elegant

alternative construction as a sphere with antipodal points identified. A sphere

with two crosscaps then becomes a double cross surface, with three crosscaps

a triple cross surface, and so on. As special cases, the double cross surface is

often called a Klein bottle, and the triple cross surface Dycks surface [3].

References

[1] W. Dyck. Beitra¨ge zur Analysis situs I. Math. Ann., 32:457512, 1888.

[2] D. Farmer and T. Stanford. Knots and Surfaces. American Mathematical

Society, 1996.

[3] G. Francis and B. Collins. On knot-spanning surfaces: An illustrated essay

on topological art. In Michele Emmer, editor, The Visual Mind: Art and

Mathematics, chapter 11. MIT Press, 1993.

[4] T. Rado´. U¨ber den Begriff der Riemannschen Fla¨che. Acta Litt. Sci. Szeged,

2:101121, 1925.

[5] H. Seifert and W. Threlfall. Lehrbuch der Topologie. Teubner, Leipzig, 1934.

Translated into English as A Textbook of Topology, Academic Press, 1980.

[6] J. Weeks. The Shape of Space. Marcel Dekker, 1985.

University of Illinois at Urbana-Champaign, Urbana, IL 61801

gfrancis@math.uiuc.edu

88 State Street, Canton, NY 13617

weeks@geom.umn.edu

7

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