Download Pizzas, Bagels, Pretzels, and Euler`s Magical χ

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. 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"
≅
≅
It’s MY geometry!
≇
No tearing
≇
No glueing
More examples
≅
:
≅
≇
vs.
≅
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
Same
(“homeomorphic”)
in general topology
≅
Different in
knot theory
≇knot
Trefoil knot
Unknot
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
(β = number of boundaries)
● attached to two
boundaries
● decreases β by 1
3. Adding a lid
+
Lid
=
A lid can be attached to any boundary
+
Decreases β by 1
Lid
=
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
=
Neither can embed into the 3-D space!
The Klein Bottle
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
How does χ change when we add a polygon?
● ΔV=4
● ΔE=5
● ΔF=1
Δχ := ΔV - ΔE + ΔF = 0
Theorem: χ = 1 for all planar
complexes with no "holes".
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?
Theorem: χ is a topological invariant!
χ=2
Proof:
Trivial.
Left as an exercise.
χ=1
Δχ
Adding an...
●
●
●
●
●
-1
-1
-1
-1
+1
Ear
Bridge
Twisted ear
Twisted bridge
Lid
χ=0
χ=0
Δβ
+1
-1
0
-1
-1
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.
2.
3.
4.
Regular polyhedra
Critical points
Poincaré–Hopf theorem
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: Any "hairstyle" on a sphere has at least two
vortices (cowlicks).
Corollary: At any time, there are at least two places on
the earth with no winds.
References:
1. Wikipedia!
2. Topology of Surfaces by L. Christine Kinsey
Thank you!