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 of 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 of 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. Extensivity2. Idempotence3. Preservation of binary unions4. 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

≅ ≅

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≅ :

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

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≇≇

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

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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 BottleNeither 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) = 1χ := V - E + F = 1

● V = 9● E = 10● F = 2χ := V - E + F = 1

A polygon complex

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

● ΔV=4● ΔE=5● ΔF=1Δχ := 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 0-1 -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 orientable 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 polyhedra2. Critical points3. Poincaré–Hopf theorem4. 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