Fantastic Topological Surfaces and How to Classify Them Khorben Boyer Western Oregon University, Kboyer12@Mail.Wou.Edu

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2018-06-01 Fantastic Topological Surfaces and How to Classify Them Khorben Boyer Western Oregon University, [email protected]

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Recommended Citation Boyer, Khorben, "Fantastic Topological Surfaces and How to Classify Them" (2018). Academic Excellence Showcase Proceedings. 101. https://digitalcommons.wou.edu/aes/101

This Presentation is brought to you for free and open access by the Student Scholarship at Digital Commons@WOU. It has been accepted for inclusion in Academic Excellence Showcase Proceedings by an authorized administrator of Digital Commons@WOU. For more information, please contact [email protected], [email protected], [email protected]. Introduction Overview of Topological Surfaces How to work with Surfaces

Fantastic Topological Surfaces and How to Classify Them

Khorben Boyer Western Oregon University

May 31, 2018

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them The ﬁeld of Topology is concerned with the properties of space that are preserved under continuous deformations such as stretching, crumpling, and bending, but not tearing new holes or sewing up old ones.

Think of it as the ﬁeld of “rubber-sheet” geometry since the rigid shapes of the objects are not relevant.

In particular, mathematicians often study mathematical structures in topology called surfaces.

These surfaces come in a diverse array of variations and combinations.

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Our context:

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Think of it as the ﬁeld of “rubber-sheet” geometry since the rigid shapes of the objects are not relevant.

In particular, mathematicians often study mathematical structures in topology called surfaces.

These surfaces come in a diverse array of variations and combinations.

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Our context: The ﬁeld of Topology is concerned with the properties of space that are preserved under continuous deformations such as stretching, crumpling, and bending, but not tearing new holes or sewing up old ones.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them In particular, mathematicians often study mathematical structures in topology called surfaces.

These surfaces come in a diverse array of variations and combinations.

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Think of it as the ﬁeld of “rubber-sheet” geometry since the rigid shapes of the objects are not relevant.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them These surfaces come in a diverse array of variations and combinations.

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Think of it as the ﬁeld of “rubber-sheet” geometry since the rigid shapes of the objects are not relevant.

In particular, mathematicians often study mathematical structures in topology called surfaces.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Think of it as the ﬁeld of “rubber-sheet” geometry since the rigid shapes of the objects are not relevant.

In particular, mathematicians often study mathematical structures in topology called surfaces.

These surfaces come in a diverse array of variations and combinations.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Sphere, S2 Torus, T2

Klein Bottle, K 2-holed Torus, T2#T2

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Examples of Topological Surfaces:

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Torus, T2

Klein Bottle, K 2-holed Torus, T2#T2

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Examples of Topological Surfaces:

Sphere, S2

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Klein Bottle, K 2-holed Torus, T2#T2

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Examples of Topological Surfaces:

Sphere, S2 Torus, T2

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them 2-holed Torus, T2#T2

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Introduction

Examples of Topological Surfaces:

Sphere, S2 Torus, T2

Klein Bottle, K

Examples of Topological Surfaces:

Sphere, S2 Torus, T2

Klein Bottle, K 2-holed Torus, T2#T2

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Question: If two surfaces are not the same, how can we tell them apart?

Question: How many categories of surfaces are there?

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Motivating Questions

Question: When are two surfaces the same?

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Question: How many categories of surfaces are there?

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Motivating Questions

Question: When are two surfaces the same?

Question: If two surfaces are not the same, how can we tell them apart?

Question: When are two surfaces the same?

Question: If two surfaces are not the same, how can we tell them apart?

Question: How many categories of surfaces are there?

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums

Compact How to work with surfaces

Locally Euclidean Polygonal Presentations

Homeomorphism Klein Lemma

Coﬀee and Donuts General Remarks

The Classiﬁcation Theorem Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums

How to work with surfaces

Locally Euclidean Polygonal Presentations

Homeomorphism Klein Lemma

Coﬀee and Donuts General Remarks

The Classiﬁcation Theorem Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces

Compact

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums

How to work with surfaces

Polygonal Presentations

Homeomorphism Klein Lemma

Coﬀee and Donuts General Remarks

The Classiﬁcation Theorem Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces

Compact

Locally Euclidean

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums

How to work with surfaces

Polygonal Presentations

Klein Lemma

Coﬀee and Donuts General Remarks

The Classiﬁcation Theorem Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces

Compact

Locally Euclidean

Homeomorphism

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums

How to work with surfaces

Polygonal Presentations

Klein Lemma

General Remarks

The Classiﬁcation Theorem Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces

Compact

Locally Euclidean

Homeomorphism

Coﬀee and Donuts

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Connected Sums

How to work with surfaces

Polygonal Presentations

Klein Lemma

General Remarks

Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces

Compact

Locally Euclidean

Homeomorphism

Coﬀee and Donuts

The Classiﬁcation Theorem

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them How to work with surfaces

Polygonal Presentations

Klein Lemma

General Remarks

Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces Connected Sums

Compact

Locally Euclidean

Homeomorphism

Coﬀee and Donuts

The Classiﬁcation Theorem

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Polygonal Presentations

Klein Lemma

General Remarks

Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces Connected Sums

Compact How to work with surfaces

Locally Euclidean

Homeomorphism

Coﬀee and Donuts

The Classiﬁcation Theorem

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Klein Lemma

General Remarks

Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces Connected Sums

Compact How to work with surfaces

Locally Euclidean Polygonal Presentations

Homeomorphism

Coﬀee and Donuts

The Classiﬁcation Theorem

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them General Remarks

Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces Connected Sums

Compact How to work with surfaces

Locally Euclidean Polygonal Presentations

Homeomorphism Klein Lemma

Coﬀee and Donuts

The Classiﬁcation Theorem

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Conclusion

Introduction Context Overview of Topological Surfaces Important Questions How to work with Surfaces Important Concepts Outline of Key Concepts

Surfaces Connected Sums

Compact How to work with surfaces

Locally Euclidean Polygonal Presentations

Homeomorphism Klein Lemma

Coﬀee and Donuts General Remarks

The Classiﬁcation Theorem

Surfaces Connected Sums

Compact How to work with surfaces

Locally Euclidean Polygonal Presentations

Homeomorphism Klein Lemma

Coﬀee and Donuts General Remarks

The Classiﬁcation Theorem Conclusion

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them We shall now proceed to deﬁne compact and locally Euclidean.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Surfaces

Deﬁnition A surface is a 2-dimensional compact locally Euclidean shape.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Surfaces

Deﬁnition A surface is a 2-dimensional compact locally Euclidean shape.

We shall now proceed to deﬁne compact and locally Euclidean.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Deﬁnition Let X be a surface. X is compact if it is closed and bounded.

3 In R :

Torus, T2 Inﬁnite Cylinder

Compact Not Compact

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Compact Space

First, we will deﬁne the topological concept of a compact space.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them 3 In R :

Torus, T2 Inﬁnite Cylinder

Compact Not Compact

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Compact Space

First, we will deﬁne the topological concept of a compact space. Deﬁnition Let X be a surface. X is compact if it is closed and bounded.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Torus, T2 Inﬁnite Cylinder

Compact Not Compact

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Compact Space

First, we will deﬁne the topological concept of a compact space. Deﬁnition Let X be a surface. X is compact if it is closed and bounded.

3 In R :

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Inﬁnite Cylinder

Compact Not Compact

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Compact Space

First, we will deﬁne the topological concept of a compact space. Deﬁnition Let X be a surface. X is compact if it is closed and bounded.

3 In R :

Torus, T2

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Inﬁnite Cylinder

Not Compact

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Compact Space

First, we will deﬁne the topological concept of a compact space. Deﬁnition Let X be a surface. X is compact if it is closed and bounded.

3 In R :

Torus, T2

Compact

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Not Compact

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Compact Space

First, we will deﬁne the topological concept of a compact space. Deﬁnition Let X be a surface. X is compact if it is closed and bounded.

3 In R :

Torus, T2 Inﬁnite Cylinder

Compact

First, we will deﬁne the topological concept of a compact space. Deﬁnition Let X be a surface. X is compact if it is closed and bounded.

3 In R :

Torus, T2 Inﬁnite Cylinder

Compact Not Compact

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them The triangle on a local patch of Earth will add up to approximately 180◦as seen here with 40◦+ 50◦+ 90◦= 180◦. But the triangle at a much larger hemispheric scale will not add up to 180◦as seen here with 50◦+ 90◦+ 90◦= 230◦. Thus, the Earth’s exterior is a 2-dimensional shape i.e. a surface.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Locally Euclidean Spaces

Next, a surface must be locally Euclidean, meaning that near each point, the surface “looks ﬂat”, but may not be ﬂat overall.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Locally Euclidean Spaces

Next, a surface must be locally Euclidean, meaning that near each point, the surface “looks ﬂat”, but may not be ﬂat overall.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them But the triangle at a much larger hemispheric scale will not add up to 180◦as seen here with 50◦+ 90◦+ 90◦= 230◦. Thus, the Earth’s exterior is a 2-dimensional shape i.e. a surface.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Locally Euclidean Spaces

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Thus, the Earth’s exterior is a 2-dimensional shape i.e. a surface.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Locally Euclidean Spaces

Next, a surface must be locally Euclidean, meaning that near each point, the surface “looks ﬂat”, but may not be ﬂat overall. The triangle on a local patch of Earth will add up to approximately 180◦as seen here with 40◦+ 50◦+ 90◦= 180◦. But the triangle at a much larger hemispheric scale will not add up to 180◦as seen here with 50◦+ 90◦+ 90◦= 230◦.

Deﬁnition A homeomorphism is a way to bend or stretch a surface into another without creating or destroying “holes”.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Theorem (Classiﬁcation Theorem)

Let M be a compact 2-dimensional surface. Then M is homeomorphic to exactly one of the following: 2 S , the sphere. 2 2 T #...#T , a connected sum of tori. 2 2 P #...#P , a connected sum of projective planes.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights How to Classify

The Classiﬁcation Theorem of Compact Surfaces answers our questions.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Then M is homeomorphic to exactly one of the following: 2 S , the sphere. 2 2 T #...#T , a connected sum of tori. 2 2 P #...#P , a connected sum of projective planes.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights How to Classify

The Classiﬁcation Theorem of Compact Surfaces answers our questions.

Theorem (Classiﬁcation Theorem)

Let M be a compact 2-dimensional surface.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them 2 S , the sphere. 2 2 T #...#T , a connected sum of tori. 2 2 P #...#P , a connected sum of projective planes.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights How to Classify

The Classiﬁcation Theorem of Compact Surfaces answers our questions.

Theorem (Classiﬁcation Theorem)

Let M be a compact 2-dimensional surface. Then M is homeomorphic to exactly one of the following:

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them 2 2 T #...#T , a connected sum of tori. 2 2 P #...#P , a connected sum of projective planes.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights How to Classify

The Classiﬁcation Theorem of Compact Surfaces answers our questions.

Theorem (Classiﬁcation Theorem)

Let M be a compact 2-dimensional surface. Then M is homeomorphic to exactly one of the following: 2 S , the sphere.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them 2 2 P #...#P , a connected sum of projective planes.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights How to Classify

The Classiﬁcation Theorem of Compact Surfaces answers our questions.

Theorem (Classiﬁcation Theorem)

Let M be a compact 2-dimensional surface. Then M is homeomorphic to exactly one of the following: 2 S , the sphere. 2 2 T #...#T , a connected sum of tori.

The Classiﬁcation Theorem of Compact Surfaces answers our questions.

Theorem (Classiﬁcation Theorem)

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them This is an example of a Connected Sum. One can think of this as eﬀectively “suturing” or “stitching” one surface onto another. Theorem The connected sum of surfaces is a surface.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Connected Sums

Surfaces can be combined or “glued” together:

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Connected Sums

Surfaces can be combined or “glued” together:

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them One can think of this as eﬀectively “suturing” or “stitching” one surface onto another. Theorem The connected sum of surfaces is a surface.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Connected Sums

Surfaces can be combined or “glued” together:

This is an example of a Connected Sum.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Theorem The connected sum of surfaces is a surface.

Introduction Deﬁnitions and Terminology Overview of Topological Surfaces The Classiﬁcation Theorem How to work with Surfaces Topological Insights Connected Sums

Surfaces can be combined or “glued” together:

This is an example of a Connected Sum. One can think of this as eﬀectively “suturing” or “stitching” one surface onto another.

Surfaces can be combined or “glued” together:

This is an example of a Connected Sum. One can think of this as eﬀectively “suturing” or “stitching” one surface onto another. Theorem The connected sum of surfaces is a surface.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them It turns out that every surface has a polygonal presentation. Theorem (Presentation Theorem) Every compact surface admits what is called a polygonal presentation.

This theorem enables us to convert any surface to its polygonal presentation provided it is compact. Working with the polygonal presentations is simpler and easier than dealing with surfaces directly.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Surface Mechanics

How do we work with surfaces?

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Theorem (Presentation Theorem) Every compact surface admits what is called a polygonal presentation.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Surface Mechanics

How do we work with surfaces? It turns out that every surface has a polygonal presentation.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them This theorem enables us to convert any surface to its polygonal presentation provided it is compact. Working with the polygonal presentations is simpler and easier than dealing with surfaces directly.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Surface Mechanics

How do we work with surfaces? It turns out that every surface has a polygonal presentation. Theorem (Presentation Theorem) Every compact surface admits what is called a polygonal presentation.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Working with the polygonal presentations is simpler and easier than dealing with surfaces directly.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Surface Mechanics

How do we work with surfaces? It turns out that every surface has a polygonal presentation. Theorem (Presentation Theorem) Every compact surface admits what is called a polygonal presentation.

This theorem enables us to convert any surface to its polygonal presentation provided it is compact.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Surface Mechanics

How do we work with surfaces? It turns out that every surface has a polygonal presentation. Theorem (Presentation Theorem) Every compact surface admits what is called a polygonal presentation.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them a a a b

b b b a

S2 T2 polygonal presentation: polygonal presentation: ha, b|abb−1a−1i ha, b|aba−1b−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 2 Polygonal Presentations of the Sphere, S , and the Torus, T .

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them a b

b a

T2 polygonal presentation: polygonal presentation: ha, b|abb−1a−1i ha, b|aba−1b−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 2 Polygonal Presentations of the Sphere, S , and the Torus, T .

a a

b b

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them a b

b a

T2 polygonal presentation: ha, b|abb−1a−1i ha, b|aba−1b−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 2 Polygonal Presentations of the Sphere, S , and the Torus, T .

a a

b b

S2 polygonal presentation:

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them a b

b a

T2 polygonal presentation: ha, b|aba−1b−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 2 Polygonal Presentations of the Sphere, S , and the Torus, T .

a a

b b

S2 polygonal presentation: ha, b|abb−1a−1i

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them polygonal presentation: ha, b|aba−1b−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 2 Polygonal Presentations of the Sphere, S , and the Torus, T .

a a a b

b b b a

S2 T2 polygonal presentation: ha, b|abb−1a−1i

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ha, b|aba−1b−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 2 Polygonal Presentations of the Sphere, S , and the Torus, T .

a a a b

b b b a

S2 T2 polygonal presentation: polygonal presentation: ha, b|abb−1a−1i

2 2 Polygonal Presentations of the Sphere, S , and the Torus, T .

a a a b

b b b a

S2 T2 polygonal presentation: polygonal presentation: ha, b|abb−1a−1i ha, b|aba−1b−1i

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them a b

b a

P2 P2 polygonal presentation: polygonal presentation: ha, b|ababi ha|aai

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 Polygonal Presentations of the Projective Plane, P .

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them P2 polygonal presentation: polygonal presentation: ha, b|ababi ha|aai

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 Polygonal Presentations of the Projective Plane, P .

a b

b a

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them P2 polygonal presentation: ha, b|ababi ha|aai

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 Polygonal Presentations of the Projective Plane, P .

a b

b a

P2 polygonal presentation:

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them P2 polygonal presentation: ha|aai

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 Polygonal Presentations of the Projective Plane, P .

a b

b a

P2 polygonal presentation: ha, b|ababi

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them polygonal presentation: ha|aai

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 Polygonal Presentations of the Projective Plane, P .

a b

b a

P2 P2 polygonal presentation: ha, b|ababi

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ha|aai

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

2 Polygonal Presentations of the Projective Plane, P .

a b

b a

P2 P2 polygonal presentation: polygonal presentation: ha, b|ababi

2 Polygonal Presentations of the Projective Plane, P .

a b

b a

P2 P2 polygonal presentation: polygonal presentation: ha, b|ababi ha|aai

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them a b

b a

K polygonal presentation: ha, b|abab−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

Polygonal Presentation of the Klein Bottle, K.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them polygonal presentation: ha, b|abab−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

Polygonal Presentation of the Klein Bottle, K.

a b

b a

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ha, b|abab−1i

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Presentation Examples

Polygonal Presentation of the Klein Bottle, K.

a b

b a

K polygonal presentation:

Polygonal Presentation of the Klein Bottle, K.

a b

b a

K polygonal presentation: ha, b|abab−1i

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them d 2 −1 −1 One, T , has ha, b|aba b i, c c 2 −1 −1 One, T , has hc, d|cdc d i

d b

a a 2 2 2 2 Thus, T #T gives T #T −1 −1 −1 −1 b ha, b, c, d|aba b cdc d i ha, b|aba−1b−1i polygonal presentation:

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Important Implication

To ﬁnd the connected sum of two polygonal presentations, 2 2 just concatenate them such as, T #T , the 2-holed torus.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ha, b|aba−1b−1i polygonal presentation:

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Important Implication

To ﬁnd the connected sum of two polygonal presentations, 2 2 just concatenate them such as, T #T , the 2-holed torus. d 2 −1 −1 One, T , has ha, b|aba b i, c c 2 −1 −1 One, T , has hc, d|cdc d i

d b

a a 2 2 2 2 Thus, T #T gives T #T −1 −1 −1 −1 b ha, b, c, d|aba b cdc d i Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Important Implication

d b

a a 2 2 2 2 Thus, T #T gives T #T −1 −1 −1 −1 b ha, b, c, d|aba b cdc d i Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ha, b|aba−1b−1i polygonal presentation: Proof: By the following sequence of elementary transformations, we ﬁnd that the Klein bottle has this series of presentations:

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Klein Lemma

Lemma (Klein Lemma) 2 2 The Klein bottle is homeomorphic to P #P .

Lemma (Klein Lemma) 2 2 The Klein bottle is homeomorphic to P #P . Proof: By the following sequence of elementary transformations, we ﬁnd that the Klein bottle has this series of presentations:

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ha, b|abab−1i ≈ ha, b, c|abc, c−1ab−1i (cut along c) ≈ ha, b, c|bca, a−1cbi (rotate and reﬂect) ≈ hb, c|bbcci (paste along a and rotate)

The presentation in the last line is a standard presentation of a

connected sum of two projective planes.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Klein Lemma

Transforming the Klein bottle K to P2#P2.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ≈ ha, b, c|abc, c−1ab−1i (cut along c) ≈ ha, b, c|bca, a−1cbi (rotate and reﬂect) ≈ hb, c|bbcci (paste along a and rotate)

The presentation in the last line is a standard presentation of a

connected sum of two projective planes.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Klein Lemma

Transforming the Klein bottle K to P2#P2. ha, b|abab−1i

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ≈ ha, b, c|bca, a−1cbi (rotate and reﬂect) ≈ hb, c|bbcci (paste along a and rotate)

The presentation in the last line is a standard presentation of a

connected sum of two projective planes.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Klein Lemma

Transforming the Klein bottle K to P2#P2. ha, b|abab−1i ≈ ha, b, c|abc, c−1ab−1i (cut along c)

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them ≈ hb, c|bbcci (paste along a and rotate)

The presentation in the last line is a standard presentation of a

connected sum of two projective planes.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Klein Lemma

Transforming the Klein bottle K to P2#P2. ha, b|abab−1i ≈ ha, b, c|abc, c−1ab−1i (cut along c) ≈ ha, b, c|bca, a−1cbi (rotate and reﬂect)

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them The presentation in the last line is a standard presentation of a

connected sum of two projective planes.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up Klein Lemma

Transforming the Klein bottle K to P2#P2. ha, b|abab−1i ≈ ha, b, c|abc, c−1ab−1i (cut along c) ≈ ha, b, c|bca, a−1cbi (rotate and reﬂect) ≈ hb, c|bbcci (paste along a and rotate)

The presentation in the last line is a standard presentation of a

connected sum of two projective planes.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them In particular, the elementary transformations allow us to reduce all possible polygonal presentations of surfaces to just a few general types.

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up General Remarks

For the sake of brevity, the Classiﬁcation Theorem itself is systematically proved through various applications of these elementary transformations as demonstrated in our proof of the Klein Lemma.

In particular, the elementary transformations allow us to reduce all possible polygonal presentations of surfaces to just a few general types.

Khorben Boyer Western Oregon University Fantastic Topological Surfaces and How to Classify Them Theorem (Classiﬁcation Theorem)

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up General Remarks

In summary, these concepts and tools enable:

Introduction Polygonal Presentations Overview of Topological Surfaces Important Example Lemma How to work with Surfaces Wrapping Up General Remarks

In summary, these concepts and tools enable: Theorem (Classiﬁcation Theorem)