The Hairy Ball Theorem

Jacob Mazor 2013 Mentor: Dr. Emily Landes – Mathematics Faculty, Technion Proving the Hairy Ball Theorem Generalization to n-holed donuts The Hairy Ball Theorem: Any head of hair with Vector Fields always has at least one bald spot. In order to prove the hairy ball theorem We can generalize the results for any 2 Mathematically, every smooth vector that we must also look at vector fields (i.e. dimensional orientable without exists on a sphere has at least one zero. hair on a head). We are primarily boundary, i.e. n-holed donut, which we also call

concerned with the points where the Tn. Starting with the most basic case, we look at vanishes, known as the figure 3, and see that for a 휒 = 0. We then

zeros (i.e. bald spots). Def: The index of look at T2. We can “cut” it along the blue outline a zero is the number of times the vector in figure 7 into two donuts with holes in them field causes a particle to rotate about which we call A and B. The simplicial itself as it moves clockwise once around decomposition is shown in figure 8. To

the zero; counted positively for determine χ(T2) we find χ(A) and χ(B). clockwise rotation and negatively for 휒 퐴 = 휒 퐵 = −1 and we find the total by counter-clockwise rotation. The indices adding them and subtracting away the common Spheres and Topological Surfaces of figures 4, 4, and 5 are 1, 1, and -1 elements, i.e. the hole. Since the hole contains respectively. three vertices and three edges, it cancels out, A sphere is a topological surface, and in order and 휒 푇2 = −2. to examine topological surfaces we can “break them down” into what are known as simplicial decompositions, which are essentially triangulations of the surface. Def: The for a simplicial decomposition S 휒 푆 = 푛푢푚푏푒푟 표푓 푣푒푟푡푖푐푒푠 − 푛푢푚푏푒푟 표푓 푒푑푔푒푠 + 푛푢푚푏푒푟 표푓 푓푎푐푒푠 Figure 3: Sink Figure 4: Source Figure 5: Saddle

Figures 1 and 2 are examples of simplicial zero with index 1 zero with index 1 zero with index -1 Figure 7: Cutting a 2-holed decompositions of a sphere and a torus Def: The Lefschetz number, L(v), to be donut. Manifold A is shown below. (donut). the sum of the indices of the vector Figure 8: Simplicial field v on that surface. Amazingly, the decomposition of 2-holed Figure 1: Simplicial Figure 2: Torus and Lefschetz number is independent of donut. The top is A, and the decomposition of a sphere simplicial decomposition bottom is B. the choice of vector field. Thus we We can similarly break up T into A, B and D, can calculate L from any vector field, 3 shown in figure 9, where D has two holes. We in particular one obtained from a count 휒 퐴 = 휒 퐵 = −1 and 휒 퐷 = −2 We simplicial decomposition. Figure 6 determine χ(T ) using a similar method to the shows how one can construct a vector 3 one for T , giving us 휒 푇 = −4 field from a simplicial decomposition, 2 3 by relating each of the triangles in the simplicial decomposition to zeros as shown. Figure 9: Simplicial

decomposition of T3. From top to bottom, Figure 6: Correlation the decompositions are between vector fields of A, D and B and simplicial We have a triangulation of a sphere, and after decomposition expansion, we can realize it as sitting on the sphere. We can then produce a stereographic N-holed donuts will have (n-2) center pieces, so projection which relates a sphere and a plane, As shown above, the L for a surface is the T can be realized as n-2 copies of D, as well as A thus mapping our simplicial decomposition to a same as the Euler characteristic. This is n and B. Since the intersection subtracted is zero, planar graph on ℝ2. because we have related zeros with 휒 푇 = −1 2 − 2 푛 − 2 = 2 − 2푛 Under this the face indices of -1 to edges which are counted 푛 closest to the pole becomes pushed outwards negatively, and zeros with indices 1 to towards infinity. In order to calculate the Euler vertices and faces which are counted Acknoledgements characteristic in the setting, we count the positively. This has two nice I would like to thank Dr. Emily Landes for her help guiding me through this project. outside as well as one face. Looking at this consequences. I would also like to thank Professor Yoav Moriah for supporting simplicial decomposition of the sphere, we find 1. A vector field on a sphere must always the project. that the Euler characteristic is 2. It can be have a L=2, meaning that a vector field shown by induction that it is always 2. If we on the surface must have at least one References 1. Mark Anthony Armstrong, Basic , consider adding another vertex, and connecting zero, proving the hairy ball theorem. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1983, Corrected reprint of the 1979 original. it with n edges, we add another n-1 faces, 2. Since L is invariant of the choice of MR 705632 (84f:55001) meaning that the Euler characteristic will vector field for any surface S, and any 2. Murray Eisenberg and Robert Guy, A proof of the hairy ball theorem, Amer. Math. Monthly 86 (1979), no. 7, change by 1-(n)+(n-1)=0. Therefore, the Euler simplicial decomposition of a S gives 572–574. MR 542769 (80i:57018) characteristic of a sphere is independent of the rise to a vector field such L(v) coincides 3. Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974. MR choice of simplicial decomposition, and we can with the χ(S), it follows that χ(S) in 0348781 (50 #1276) define the Euler characteristic of a sphere itself independent of the choice of simplicial 4. Marcus C. Werner, A Lefschetz fixed point theorem in gravitational lensing, J. Math. Phys. 48 (2007), no. 5, to be 2. decomposition. 052501, 9. MR 2329848 (2008e:83014)