Tse Leung So University of Southampton 28th July 2017 Topology and Surfaces Tse Leung So University of Southampton 28th July 2017

What is Topology? Topology studies properties of objects under stretching, crumpling and bending, but not tearing or gluing. Geometry Vs Topology

Examples Fundamental Theorem of Algebra: Any polynomial p(z) = 𝑧 𝑛 + 𝑎 𝑛−1 𝑧 𝑛−1 +…+ 𝑎 1 𝑧+ 𝑎 0 has a root in complex numbers. For z far away from 0, p(z) = 𝑧 𝑛 + 𝑎 𝑛−1 𝑧 𝑛−1 +…+ 𝑎 1 𝑧+ 𝑎 0 = 𝑧 𝑛 1+ 𝑎 𝑛−1 𝑧 +…+ 𝑎 1 𝑧 𝑛−1 + 𝑎 0 𝑧 𝑛 ~ 𝑧 𝑛 The polynomial p(z) maps the interior to the interior. p(z)

Question: Can we spread the surface of a globe on a table without cutting it open?

Euler number Euler number is a topological invariant. χ = number of vertices – number of edges + number of faces = V – E + F A square has 4 vertices, 5 edges and 2 faces, so χ = 4 – 5 + 1 = 0. A ball can be formed by gluing the adjacent edges of a square. It has 3 vertices, 3 edges and 2 faces, so χ = 3 – 3 + 2 = 2.

Euler numbers of closed surfaces A surface is closed if it does not have a boundary. Eg. a sphere is closed but a square and a birthday hat are not. A torus (a doughnut) is formed by gluing the opposite edges of a square. It has 1 vertex, 3 edges and 2 faces, so χ = 1 – 3 + 2 = 0.

A 2-torus is formed by gluing two tori together. It has 1 vertex, 7 edges and 4 faces, so χ = 1 – 7 + 4 = 2.

In general, an n-torus is formed by gluing n tori together. Its Euler number is n – 2n (Exercise).

A projective plane is formed by identifying opposite points on a sphere. It is also formed by gluing the opposite edges of a square in a reverse direction. It has 2 vertices, 3 edges and 2 faces, so χ = 2 – 3 + 2 = 1. It is a non-orientable surface like a Mobius strip.

Surface V E F χ Sphere 3 2 Torus 1 2-torus 7 4 –2 3-torus 11 6 –4 Projective plane

Classification of orientable closed surfaces Any orientable closed surface can change to a sphere, a torus, a 2-torus, …, or an n-torus.

Sketch proof We give a label to a polygon as follows: d We give a label to a polygon as follows: Assign a letter and an arrow to each edge of a polygon. If the arrow of an edge is anti-clockwise, then its letter has exponent +1. Otherwise its letter has exponent – 1. Eg. the label of the polygon is ab-1cac-1d. Glue any two edges of the same letter according to their arrows. a b c

Each letter appears only twice. If a letter appears once, then If a letter appears more than twice, then a b

Otherwise it contains a projective plane and is not orientable. One copy of each letter has exponent +1 and the other copy has exponent –1. Otherwise it contains a projective plane and is not orientable. b a aba-1b-1 a w2 w1 b aw1aw2 bbw2-1w1

We can cancel any pair of aa-1 in the label. w1aa-1w2 w1 w2 b a b a w1 w2 b a w1 w2 w1w2

each letter appears twice; In summary, each letter appears twice; one copy has exponent +1 and the other has exponent –1; any pair of aa-1 is cancelled. If w is empty, then w = aa-1bb-1 a b

Lemma: If w is not empty, then we want to show that. w = aba-1b-1w’ Lemma: If w is not empty, then we want to show that w = aba-1b-1w’ where a and b are letters, and w’ is a shorter label without a and b. Take a letter a such that the distance between a and a-1 is the shortest. Pick a letter b between them. Then we have w = w1 a w2 b w3 a-1 w4 b-1 w5. b w4 w1 a w2 w3 w5 c

b w4 w1 a w2 w3 w5 c a c w4 w1 b w5 w3 w2 b c w5 w1 w4 d w3 w2 d b w3 w2 c w5 w1 w4

Therefore w = ede-1d-1w4w3w2w5w1 = aba-1b-1w’. c w5 w1 w4 Therefore w = ede-1d-1w4w3w2w5w1 = aba-1b-1w’. e d c w3 w2 w5 w1 w4 c d e w3 w4 w2 w5 w1

For w = (a1b1a1-1b1-1) … (anbnan-1bn-1) Repeat this step for several times until w is in the form w = a1b1a1-1b1-1w1 = … = (a1b1a1-1b1-1) … (anbnan-1bn-1) For w = (a1b1a1-1b1-1) … (anbnan-1bn-1) c a1 a2 b1 b2 a1b1a1-1b1-1 a2b2a2-1b2-1

My research area Homotopy theory is to study properties of objects under continuous deformations. Besides Euler number, there are other topological invariants, eg. homology, cohomology, homotopy. They remain unchanged when we deform the objects continuously.

Thank you for listening.