advertisement

Classification of 2n-Manifolds Using Algebraic Methods David Zhang Advisor: Dr. James Turner Manifolds Classification Using Loops Gluing Spheres Higher Dimensions The earth is a sphere, but it looks like a plane when you stand on it. This is because the earth is so large and our field of vision is too small to capture the curvature of the earth. This phenomena is essential to the idea of a manifold: a space in which if you “zoom in” at any point, it will look like a point, a line, a plane, or an Euclidean space of dimension three or higher. Although we can study the topological structure of simple manifolds such as balls and cubes by using our imagination and common sense, the classification of higher dimension manifolds requires more systematic methods. One way of doing this is by using a “contracting loop”. If you draw a loop on a sphere or a box, no matter what the loop looks like, it can always contract to a point. On the other hand, on a torus at least two kinds of loops cannot contract to a point without leaving the surface. Another important fact about a highly connected 2n-manifolds is that we can obtain any of them by gluing a (2n-1)sphere to a bouquet of n-spheres. For the two-dimensional case, a torus can be obtained by gluing the boundary of a disk to a skeleton made of circles. It turns out that the structure of a 2nmanifold is determined by how we glue it, which is described by its attaching map. The attaching map can be considered as a member of the (2n-1)-th homotopy group of the bouquet of nsphere. Thus we can compare two 2nmanifolds by studying the associated homotopy group elements. The goal of our research is to develop an algebraic method to classify 2nmanifolds. By using tools such as Pialgebra and Hilton Basis, we convert a 2n-manifold into an integer vector. The overall stragety is shown in the following flowchart: For example, a circle is an 1-manifold and a torus is a 2-manifold. Our universe is a 3-manifold, but physicists have not figured out whether it is a 3-Euclidean space, a 3-sphere, or a 3-torus. Structure of Manifolds Manifolds come in all shapes and sizes. However, in topology, the study of geometric properties unaffected by continuous changes, we are only concerned with the structure of a manifold. A ball and a cube look quite different, but if the ball is made of PlayDoh, you can reshape the ball into a cube without tearing or connecting any parts. However, there is no way you can reshape a ball into a doughnut without poking a whole or connecting two ends. This means that a ball is topologically equivalent to a cube, while topologically different from a torus. The contractibility of loops on a surface gives rise to an algebraic structure called homotopy groups. Each manifold has a unique homotopy group. It turns out that two manifolds have the same structure (of the same homotopy type) if and only if their homotopy groups are the same. Manifold Cell Complex Attaching Map Homotopy Group Element Vector What we discovered is that two 2nmanifolds are of the same homotopy type if and only if their associated vectors can be mapped to each other by a change-of-basis matrix that satisfies certain properties. This necessary and sufficient condition holds for manifolds of any even dimensions. Hopf fibration, a 3-dimensional representation of a 3-sphere, which resides in a 4-demsnional space. References A mug can be continuously transformed into a doughnut and therefore the two objects are topologically equivalent Any loop on a sphere can contract to a point, but this is not true for loops on a torus By gluing the edges of a square in different ways, we can form a torus or a sphere. Photo credits: Intelligent Perception (mug), Scientific American (loops on torus), Wikipedia (loop on sphere, Hopf fibration), Annenberg Learner (gluing a square).