Pizzas, Bagels, Pretzels, and Euler's Magical ---- an informal - - PowerPoint PPT Presentation

Pizzas, Bagels, Pretzels, and Euler's Magical χ

• --- an informal introduction to topology

What is topology?

Given a set X , a topology on X is a collection T of subsets

• f X, satisfying the following axioms:
• 1. The empty set and X are in T.
• 2. T is closed under arbitrary union.
• 3. T is closed under finite intersection.

Equivalent definition: Given a set X , a topology on X is a collection S of subsets

• f X, satisfying the following axioms:
• 1. The empty set and X are in S.
• 2. S is closed under finite union.
• 3. S is closed under arbitrary intersection.

...Another equivalent definition:

Given a set X , a topology on X is an operator cl on P(X) (the power set of X) called the closure operator, satisfying the following properties for all subsets A of X:

• 1. Extensivity
• 2. Idempotence
• 3. Preservation of binary unions
• 4. Preservation of nullary unions

If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure operator.

What really is topology?

Topology is : "Gummy geometry"

≅ ≇ ≅ ≅ ≇ No tearing No glueing

It’s MY geometry!

vs.

More examples

≅ ≅ ≇ ≅

:

Topological invariants:

• holes & cavities
• boundaries & endpoints
• connectedness ("in one piece")
• etc...

Not topological invariants:

• size
• angle
• curvature
• etc...

Classify boldface capital letters up to “topological sameness”:

• G,I,J,L,M,N,S,U,V,W,Z,C,E,F,T,Y,H,K,X
• R,A,D,O,P,Q
• B

≇ ≇ ≇

Question:

They cannot deform into each other in the 3-D space, but they can if you put them in a 4-D space.

≅ ?

Hence they should be considered as the same topological object (“homeomorphic” objects). They just sit ("embed") differently in the 3-D space. Mathematical rigor is needed at some point to help our intuition!

Knot theory

≅ Trefoil knot Unknot Same (“homeomorphic”) in general topology Different in knot theory ≇knot

Applications of knot theory

Surfaces: compact 2-dimensional manifolds with boundaries

These are not surfaces in our sense:

Operations on surfaces:

• 1. Adding an ear
• 2. Adding a bridge

Bridge

• attached to one

boundary

• increases β by 1
• attached to two

boundaries

• decreases β by 1

(β = number of boundaries)

• 3. Adding a lid

Decreases β by 1

+ = Lid + = Lid A lid can be attached to any boundary

How to make a torus (the surface of a bagel)? Torus = disk + ear + bridge + lid

Annulus

• 4. Twisted ear

The Möbius strip Adding a twisted ear does not change β

The Möbius strip is unorientable: no up and down! Möbius Strip by Escher

Escher's paintings

The Möbius strip

The Möbius Resistor

Other unorientable surfaces

+ Lid = (The real projective plane) + Lid = The Klein Bottle Neither can embed into the 3-D space!

The Klein bottle in real life

Twisted bridge & more complicated surfaces

Fact: All surfaces can be built this way.

Topological invariants for surfaces:

• number of boundaries β
• orientability: can we distinguish between

inside and outside (or up and down)?

• the Euler characteristic

The Euler characteristic χ

• V(ertices) = 5
• E(dges) = 5
• F(aces) = 1

χ := V - E + F = 1

• V = 9
• E = 10
• F = 2

χ := V - E + F = 1

A polygon complex

Theorem: χ = 1 for all planar complexes with no "holes".

• ΔV=4
• ΔE=5
• ΔF=1

Δχ := ΔV - ΔE + ΔF = 0 How does χ change when we add a polygon?

χ = 0 In general, a planar complex with n holes has χ = 1 - n .

We may also define χ for other (not necessarily planar) complexes:

χ = 2 χ = 2 χ = 1

That’s all very nice, but what’s so magical about χ anyway?

Proof:

χ = 2 χ = 1

Trivial. Left as an exercise. Theorem: χ is a topological invariant!

Δχ Δβ

• Ear
• Bridge
• Twisted ear
• Twisted bridge
• Lid
• 1

+1

• 1
• 1
• 1
• 1
• 1

+1

• 1

Adding an...

χ = 0 χ = 0 χ = - 4

Question: What values can χ take?

Answer: χ ≤ 2 In fact, β+χ ≤ 2

Theorem: The pair (β,χ) classifies all

• rientable surfaces.

Non-orientable surfaces?

They are classified by (β,χ,q) where q=0,1,2 measures non-orientability.

Some mathematical applications of χ

• 1. Regular polyhedra
• 2. Critical points
• 3. Poincaré–Hopf theorem
• 4. Gauss-Bonnet formula

Regular polyhedra

Theorem: These are the only five regular polyhedra.

A soccer ball needs 12 pentagons

Vector fields

Index of singularities

Poincaré–Hopf theorem: Σ(indices of singularities) = χ

Corollary: Any vector field on a sphere has at least two vortices. Corollary: At any time, there are at least two places on the earth with no winds. Corollary: Any "hairstyle" on a sphere has at least two vortices (cowlicks).

Thank you!

References:

• 1. Wikipedia!
• 2. Topology of Surfaces by L. Christine Kinsey