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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!