Manifolds and the Shape of the Universe Stacy Hoehn Vanderbilt University [email protected] July 13, 2010 Stacy Hoehn Manifolds and the Shape of the Universe Stretching, shrinking, bending, and twisting are allowed. Examples of Homeomorphic Objects: What Does \Shape" Mean to a Topologist? In topology, two objects have the same shape if one can be deformed into the other without cutting or gluing. Objects with the same shape are called homeomorphic. Stacy Hoehn Manifolds and the Shape of the Universe Examples of Homeomorphic Objects: What Does \Shape" Mean to a Topologist? In topology, two objects have the same shape if one can be deformed into the other without cutting or gluing. Objects with the same shape are called homeomorphic. Stretching, shrinking, bending, and twisting are allowed. Stacy Hoehn Manifolds and the Shape of the Universe What Does \Shape" Mean to a Topologist? In topology, two objects have the same shape if one can be deformed into the other without cutting or gluing. Objects with the same shape are called homeomorphic. Stretching, shrinking, bending, and twisting are allowed. Examples of Homeomorphic Objects: Stacy Hoehn Manifolds and the Shape of the Universe A torus and a sphere are not homeomorphic. Homeomorphic or Not? A doughnut and a coffee cup are homeomorphic. Stacy Hoehn Manifolds and the Shape of the Universe Homeomorphic or Not? A doughnut and a coffee cup are homeomorphic. A torus and a sphere are not homeomorphic. Stacy Hoehn Manifolds and the Shape of the Universe 2 Locally, they both look two-dimensional Euclidean space R . The torus and the sphere are both called 2-manifolds because they share this property. 2-Manifolds Even though the torus and sphere are not homeomorphic, they do have something in common. Stacy Hoehn Manifolds and the Shape of the Universe The torus and the sphere are both called 2-manifolds because they share this property. 2-Manifolds Even though the torus and sphere are not homeomorphic, they do have something in common. 2 Locally, they both look two-dimensional Euclidean space R . Stacy Hoehn Manifolds and the Shape of the Universe 2-Manifolds Even though the torus and sphere are not homeomorphic, they do have something in common. 2 Locally, they both look two-dimensional Euclidean space R . The torus and the sphere are both called 2-manifolds because they share this property. Stacy Hoehn Manifolds and the Shape of the Universe Examples: 1-manifolds: 2-manifolds: 3 3-manifolds: R and the universe n-Manifolds Definition An n-manifold is a topological space that locally looks like n n-dimensional Euclidean space R . Stacy Hoehn Manifolds and the Shape of the Universe n-Manifolds Definition An n-manifold is a topological space that locally looks like n n-dimensional Euclidean space R . Examples: 1-manifolds: 2-manifolds: 3 3-manifolds: R and the universe Stacy Hoehn Manifolds and the Shape of the Universe The Surface of the Earth Locally, the surface of the Earth looks like a 2-dimensional plane, so it is a 2-manifold. If we only saw this local picture, it would be reasonable to believe that the Earth is an infinite plane, a sphere, a torus, or any other 2-manifold. What other 2-manifold possibilities are there, and how can we eliminate the other possibilities? Stacy Hoehn Manifolds and the Shape of the Universe Yes! There are actually infinitely many compact 2-manifolds. But fortunately they are all made out of simple building blocks. Compact 2-Manifolds We will restrict our attention to 2-manifolds that are finite. These are called compact 2-manifolds. The sphere and torus are both compact 2-manifolds. Are there any others? Stacy Hoehn Manifolds and the Shape of the Universe Compact 2-Manifolds We will restrict our attention to 2-manifolds that are finite. These are called compact 2-manifolds. The sphere and torus are both compact 2-manifolds. Are there any others? Yes! There are actually infinitely many compact 2-manifolds. But fortunately they are all made out of simple building blocks. Stacy Hoehn Manifolds and the Shape of the Universe This square, with its opposite sides identified, helps us depict the torus in the plane. The Torus To help us visualize the other compact 2-manifolds, we will first view the torus a little bit differently. We will construct a torus by gluing together opposite edges of a square. Stacy Hoehn Manifolds and the Shape of the Universe The Torus To help us visualize the other compact 2-manifolds, we will first view the torus a little bit differently. We will construct a torus by gluing together opposite edges of a square. This square, with its opposite sides identified, helps us depict the torus in the plane. Stacy Hoehn Manifolds and the Shape of the Universe You would see infinitely many copies of yourself in every direction! The Torus (continued) What would you see if you were a two-dimensional being living in a torus? Stacy Hoehn Manifolds and the Shape of the Universe The Torus (continued) What would you see if you were a two-dimensional being living in a torus? You would see infinitely many copies of yourself in every direction! Stacy Hoehn Manifolds and the Shape of the Universe The M¨obiusBand A M¨obiusband is constructed from a square by gluing the left side to the right side of the square after performing a half-twist. Stacy Hoehn Manifolds and the Shape of the Universe Note: The M¨obiusband is not a manifold because it has an edge. (It is called a manifold-with-boundary.) The M¨obiusBand (continued) A M¨obiusband contains an orientation-reversing curve. Clockwise becomes counterclockwise along this curve! Stacy Hoehn Manifolds and the Shape of the Universe The M¨obiusBand (continued) A M¨obiusband contains an orientation-reversing curve. Clockwise becomes counterclockwise along this curve! Note: The M¨obiusband is not a manifold because it has an edge. (It is called a manifold-with-boundary.) Stacy Hoehn Manifolds and the Shape of the Universe The Klein bottle is a 2-manifold. The Klein Bottle A Klein bottle is constructed from a square by gluing together the left and right edges the same way as for a torus, but now the top edge is flipped before being glued to the bottom edge. Stacy Hoehn Manifolds and the Shape of the Universe The Klein Bottle A Klein bottle is constructed from a square by gluing together the left and right edges the same way as for a torus, but now the top edge is flipped before being glued to the bottom edge. The Klein bottle is a 2-manifold. Stacy Hoehn Manifolds and the Shape of the Universe You would see infinitely many copies of yourself in every direction, but sometimes you would be flipped! The Klein Bottle (continued) What would you see if you were a two-dimensional being living in a Klein bottle? Stacy Hoehn Manifolds and the Shape of the Universe The Klein Bottle (continued) What would you see if you were a two-dimensional being living in a Klein bottle? You would see infinitely many copies of yourself in every direction, but sometimes you would be flipped! Stacy Hoehn Manifolds and the Shape of the Universe The Klein Bottle (continued) The Klein bottle contains an orientation-reversing curve since it contains a M¨obiusband. Manifolds that contain an orientation-reversing curve are called nonorientable. Manifolds that do not contain an orientation-reversing curve are called orientable. Stacy Hoehn Manifolds and the Shape of the Universe The projective plane is a nonorientable 2-manifold that can be obtained from a disk by making different identifications. The Sphere and Projective Plane The sphere is an orientable 2-manifold that can be obtained from a disk as shown below. Stacy Hoehn Manifolds and the Shape of the Universe The Sphere and Projective Plane The sphere is an orientable 2-manifold that can be obtained from a disk as shown below. The projective plane is a nonorientable 2-manifold that can be obtained from a disk by making different identifications. Stacy Hoehn Manifolds and the Shape of the Universe Amazingly, every compact 2-manifold is homeomorphic to either a sphere (orientable), a connected sum of tori (orientable), or a connected sum of projective planes (nonorientable). Connected Sum Given two 2-manifolds, we can create a new 2-manifold by taking their connected sum. To take the connected sum of two 2-manifolds, remove the inside of a small disk from each of them and then glue the two boundary circles of these disks together. Stacy Hoehn Manifolds and the Shape of the Universe Connected Sum Given two 2-manifolds, we can create a new 2-manifold by taking their connected sum. To take the connected sum of two 2-manifolds, remove the inside of a small disk from each of them and then glue the two boundary circles of these disks together. Amazingly, every compact 2-manifold is homeomorphic to either a sphere (orientable), a connected sum of tori (orientable), or a connected sum of projective planes (nonorientable). Stacy Hoehn Manifolds and the Shape of the Universe There are infinitely many 3-manifolds. A priori, any one of these 3-manifolds could be the shape of the universe. The Shape of the Universe No matter where we have been in the universe so far, if we choose a spot and travel out from it a short distance in all directions, we enclose a space that resembles a ball in 3-dimensional Euclidean space. Thus, the universe appears to be some 3-manifold.