Getting to know a beetle s world classification of closed 3 dimensional manifolds

Transcription

1 Getting to know a beetle s world classification of closed 3 dimensional manifolds Sophia Jahns Tübingen University December 31, 2017 Sophia Jahns A beetle s world December 31, / 34

2 A beetle s world introducing 2-dimensional manifolds Design a computer game with a beetle crawling on an infinite plane R 2. No bumping into walls, no being forced to turn around. But: the beetle should stay on a little quadratic screen. Icon: ERGOSIGN, CC BY-NC-ND 3.0. Sophia Jahns A beetle s world December 31, / 34

3 A beetle s world introducing 2-dimensional manifolds A solution: Beetle crawls across the top edge of the screen: make him enter through the bottom edge. Beetle crawls across the left edge of the screen: make him enter through the right edge. He does not change his speed or direction when crossing edges. Icon: ERGOSIGN, CC BY-NC-ND 3.0. Sophia Jahns A beetle s world December 31, / 34

4 A beetle s world introducing 2-dimensional manifolds What happened to the beetle s plane? By deciding that the top edge of the screen is the same as the bottom edge of the screen, we have identified a strip of the plane R 2 with the strip lying above it, and the one above that, and so on. Then, we have identified a strip of the plane with the one lying next to it, etc. We say that by identifying parts of the plane R 2, we have made a quotient. What we obtain is called a torus. Sophia Jahns A beetle s world December 31, / 34

5 A beetle s world introducing 2-dimensional manifolds A torus: important example of a 2-dimensional manifold. A 2-dimensional manifold: an object that looks around each point like the plane R 2, but maybe deformed. Another important example: 2-dimensional sphere S 2. The surface of the Earth is a sphere. From our perspective, it looks like we re living on a plane R 2, possibly deformed by hills and valleys. Sophia Jahns A beetle s world December 31, / 34

6 How to annoy the beetle geometry vs topology Describing the new habitat of beetle: What are the shortest paths between two points? How to measure angles? How to measure distances? In other words: what is the geometry of the quotient torus? Angles, shortest paths, lengths are the same as in the flat plane. Hence: flat torus The beetle needs to know the geometry of his habitat to find good paths! Mathematical structure describing the geometric properties: a metric. Sophia Jahns A beetle s world December 31, / 34

7 How to annoy the beetle geometry vs topology Another way to make a torus: Crafting a torus out of a rubber sheet. By Gimps.de-team at the German language Wikipedia, CC BY-SA 3.0, The above torus is not flat! Sophia Jahns A beetle s world December 31, / 34

8 How to annoy the beetle geometry vs topology We can deform any manifold (bumps and dents, squeezing): we can change its metric. Two tori with different metrics. Left: by Philip Rideout - Own work, CC BY-SA 3.0, Right: by Ag2gaeh - Own work, CC BY-SA 4.0, The metric (and hence the geometric properties) changes, the topology (the general shape, disregarding deformations) stays the same, as long as we don t cut or glue. Sophia Jahns A beetle s world December 31, / 34

9 How to annoy the beetle geometry vs topology But: the topology puts some restrictions on the geometry! Example: trying to flatten a sphere at the pole caps will result in even more curvature around the equator, and vice versa. It is impossible to make a flat sphere. By Tomruen - Own work, CC BY-SA 4.0, Sophia Jahns A beetle s world December 31, / 34

10 Letting the beetle fly Manifolds of any dimension Letting the beetle fly into the third dimension: Instead of a flat plane R 2, start with the flat space R 3. The beetle should always be visible in a cube. (Imagine a cubic hologram wherein the beetle is displayed.) Beetle leaves through one side of the cube: make him reappear through the opposing side. The beetle keeps his direction when crossing sides. This means: we identify a cubic area in space with the cubic area on top of it, behind it, left to it, and so on. Thus, we create a 3-dimensional flat torus. Sophia Jahns A beetle s world December 31, / 34

11 Letting the beetle fly Manifolds of any dimension The 3-dimensional torus is an example of a 3-dimensional manifold. In general: an n-dimensional manifold is an object that looks around each point like the space R n, but maybe somewhat deformed. A 1-dimensional manifold. Also a 1-dimensional manifold. Not a manifold. Sophia Jahns A beetle s world December 31, / 34

12 More habitats for the beetle standard constructions What other habitats (3-dimensional manifolds) could we design for the beetle? Important example: the 3-dimensional sphere S 3 = {(x, y, z, w) R 4 : x 2 + y 2 + z 2 + w 2 = 1}. We can deform the sphere S 3 (change its metric). Sophia Jahns A beetle s world December 31, / 34

13 More habitats for the beetle quotients Remember the quotient torus: construction of a new manifold (the torus) from a given manifold (R 2 or R 3 ). Similar constructions to find more manifolds? Problem: not every quotient is a manifold (see picture). Identifying the North Pole and the South Pole of a sphere does not give you a manifold. By Ag2gaeh - Own work, CC BY-SA 4.0, Sophia Jahns A beetle s world December 31, / 34

14 More habitats for the beetle quotients Quotient construction of torus in terms of maps: f (x, y) = (x + 1, y) g(x, y) = (x, y + 1) Every point in R 2 was identified with its image under f and its image under g, and the images of these images, and so on. Important condition: no point should get mapped too close to itself under any of the maps. There are many quotient manifolds of S 3. Sophia Jahns A beetle s world December 31, / 34

15 More habitats for the beetle quotients Question: if the manifold we start with has a metric (we can measure lengths and angles), do we automatically get a metric on the quotient? Answer: only if the maps (in the torus example f and g) respect the metric: every triangle has to get mapped to a triangle with same angles and side lengths. Sophia Jahns A beetle s world December 31, / 34

16 More habitats for the beetle products Stacking tyres (circles S 1 ) on a pole (the real line R) is making a product of S 1 and R. The product is a cylinder R S 1. Photo: George P. Lewis. Sophia Jahns A beetle s world December 31, / 34

17 More habitats for the beetle products Threading the tyres on a rope and then tying both ends of the rope together is a product S 1 S 1. The product S 1 S 1 is (again!) a torus. The torus as a product S 1 S 1. By DaveBurke - Own work, CC BY 2.5, Sophia Jahns A beetle s world December 31, / 34

18 More habitats for the beetle products A product inherits a metric from the factors: The beetle on the stack of tyres sees no curvature if he looks straight up or down. (R not curved) The beetle sees curvature if he looks left or right. (Every tyre S 1 is curved.) On the torus of tyres, the beetle sees curvature in every direction. Hence, the torus or tyres S 1 S 1 is not a flat torus. Same product construction for higher dimensions, for example S 1 S 2, or S 1 S 1 S 1, or R 5 S 2, or... Sophia Jahns A beetle s world December 31, / 34

19 Walking in circles topological explorations Finding closed loops on a torus is like taking a walk with a ball of wool and tying the ends together upon return. Some closed loops can be deformed to a point. A closed loop on a torus which can be deformed to a point. CC BY-SA 3.0, Sophia Jahns A beetle s world December 31, / 34

20 Walking in circles topological explorations Some closed loops on the torus cannot be deformed to a point. In the plane R 2, all closed loops can be deformed to a point. Hence: closed loops are a tool to distinguish between the plane and the torus! In pink and red: two closed loops on a torus which cannot be deformed to a point or into each other. CC BY-SA 3.0, We do not distinguish between closed loops that can be deformed one into the other. Sophia Jahns A beetle s world December 31, / 34

21 Walking in circles topological explorations CC BY-SA 3.0, Use integers to count how many times a loops goes around a hole. The sign (+ or ) keeps track of direction. Use red integers Z for loops of the red type and pink integers Z for loops of the pink type. For example, (1, 3, 2 ) means: around the central hole once, circling the jam filling of the donut three times, again around the central hole but in opposite direction. Sophia Jahns A beetle s world December 31, / 34

22 Walking in circles topological explorations The closed loops correspond to the maps that we used to form the quotient. For instance, the pink loop that circles the central hole once clockwise can correspond to (x, y) (x + 1, y). Creating quotients more closed loops. Hence, good question: is this manifold a quotient of something simpler? A manifold where all loops can be contracted to a point are called simply connected. Sophia Jahns A beetle s world December 31, / 34

23 How many worlds for a beetle? posing the problem Habitats of the beetle we re interested in have to be: orientable (The beetle should be able to tell left from right not like on a Moebius strip.) A Moebius strip in non-orientable. closed (The beetle should not be able to walk into infitite distances like on R 2 or fall off an edge like on a disk.) Sophia Jahns A beetle s world December 31, / 34

24 How many worlds for a beetle? posing the problem Goal Get a good overview of all closed, orientable, 3 dimensional manifolds! Strategy: Cutting the manifold into simpler pieces. Classifying the simpler pieces as quotients of some special manifolds. Our manifolds do not come with prescribed geometric properties, but we can use geometry to classify them! Sophia Jahns A beetle s world December 31, / 34

25 Breaking the problem into pieces Back to 2 dimensional manifolds: Create all orientable, closed 2-dimensional manifolds from tori and spheres: Removing disks from the manifolds and gluing the manifolds together along the new boundary: Two tori were glued together, now another one gets glued in. CC BY-SA 3.0, Reversing this process: cutting along circles and gluing back in disks. Sophia Jahns A beetle s world December 31, / 34

26 Breaking the problem into pieces Decomposing 3-dimensional manifolds: Cut the manifold open along 2-dimensional spheres, glue in 3-dimensional balls to fill the gaps. Only finitely many spheres which do not overlap. It does not matter where exactly and in which order we cut. Result is much more complicated than in the 2-dimensional case! Some components might be S 1 S 2, the others are irreducible: every sphere S 2 that lies in one of them encloses no gap in the manifold. Some irreducible components might need to be cut even more! Sophia Jahns A beetle s world December 31, / 34

27 Breaking the problem into pieces Cut some components (that are not S 1 S 2 ) in a different way: Cut along tori S 1 S 1. Only finitely many tori which do not overlap. The tori have to enclose a gap in the manifold. Sophia Jahns A beetle s world December 31, / 34

28 The beetle with ruler and protractor model geometries On S 2 and on R 2, the beetle cannot know on which point he sits these manifolds looks the same everywhere, they are homogeneous. He cannot tell in which direction he looks, S 2 and R 2 are isotropic. But he knows if he is on S 2 or on R 2 : On R 2, he measures an angle sum of 180 in every triangle. On S 2, he measures an angle sum larger than 180 in every triangle. Sophia Jahns A beetle s world December 31, / 34

29 The beetle with ruler and protractor model geometries A homogeneous, isotropic habitat where the beetle measures an angle sum of less than 180 in every triangle: the hyperbolic plane H 2. A model of the hyperbolic plane H 2 ; the metric is such that all depicted polygons have the same size. By Tomruen - tiling 73-t0.png, CC BY-SA 3.0, Like S 2 and R 2 have 3 dimensional siblings S 3 and R 3, there is also a hyperbolic space H 3. Sophia Jahns A beetle s world December 31, / 34

30 The beetle with ruler and protractor model geometries Other 3 dimensional homogeneous habitats that are simply connected (where every loop can be contracted to a point): Products R S 2 and R H 2. Harder to describe: SL(2, R), the Nil geometry, and the Solv geometry. These 8 habitats are the only ones that are homogeneous, simply connected, complete (you cannot walk to an edge and fall off), and have at least one closed quotient. (Remember: the closed, orientable manifolds are the worlds we re after!) These 8 manifolds with their metrics are called model geometries. Sophia Jahns A beetle s world December 31, / 34

31 Classifying the beetle s worlds geometrization Thurstons s geometrization conjecture Every closed, orientable 3 dimensional manifold can in a unique way be cut into pieces such that each of the pieces admits a metric such that it is a quotient of one of the eight model geometries. For each piece, the model geometry is unique. This gives a nice overview of closed, orientable, 3 dimensional manifolds! For a special class of manifolds this was proven by William Thurston (1980). Photo: William Thurston By Bergman, George M. - i d = 6119, CCBY SA3.0, https : //commons.wikimedia.org/w/index.php?curid = Sophia Jahns A beetle s world December 31, / 34

32 Classifying the beetle s worlds geometrization Hamilton s idea to use Ricci flow: Ricci curvature is some specific kind of curvature. Ricci flow contracts the manifold where the Ricci curvature is positive and enlarges it where the Ricci curvature is negative. This makes the manifolds looks simpler. Problem: in this process, the manifolds can develop singularities (like sharp ridges). So, Hamilton s proof does not work in all scenarios! Photo: Richard Hamilton 1982 By George M. Bergman - i d = 5223, GFDL, https : //commons.wikimedia.org/w/index.php?curid = Sophia Jahns A beetle s world December 31, / 34

33 Classifying the beetle s worlds geometrization Perelman s proof: Applies to all closed, orientable 3 dimensional manifolds! Uses Hamilton s Ricci flow ideas. But: Perelman cuts out the singularities! Has not yet appeared in a peer-reviewed journal, but many mathematicians have checked it. Perelman was offered the Fields medal for this work but declined. Photo: Grigori Perelman 1993 By George M. Bergman - Mathematisches Institut Oberwolfach (MFO), GFDL, Sophia Jahns A beetle s world December 31, / 34

34 Classifying the beetle s worlds geometrization What all this means for the beetle: He knows that his world can be cut up into pieces that resemble one of eight special models. He can investigate the closed loops. (Remember: closed loops give hints how the quotient was made.) But: even knowing for each piece the model geometry and all the closed loops does not give full information about the beetle s world. To find out more, the beetle has to start doing research. Mathematicians worldwide try to help him. Icon: ERGOSIGN, CC BY-NC-ND 3.0. Sophia Jahns A beetle s world December 31, / 34