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An Introduction to Topology Linda Green - Marin Math Circle

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					An Introduction to Topology
          Linda Green
      Nueva Math Circle
     September 30, 2011




                        Images from virtualmathmuseum.org
                     Topics
•   The universe
•   Definitions
•   Surfaces and gluing diagrams
•   The universe
                             The Universe
  • Is the universe finite or infinite?
  • If we could step outside of it, what would it
    look like? What is its shape?




Many of the ideas in this talk are explored in more detail in The Shape of Space by Jeff Weeks
                       Dimension
                    Informal Definitions
•  1-dimensional: Only one number is required to specify a
  location; has length but no area. Each small piece looks like a
  piece of a line.
• 2-dimensional: Two numbers are required to specify a
  location; has area but no volume. Each small piece looks like a
  piece of a plane.
• 3-dimensional: Three numbers are required to specify a
  location; has volume. Each small piece looks like a piece of
  ordinary space.
         Topology vs. Geometry
• The properties of an object that stay the same
  when you bend, stretch or twist it are called the
  topology of the object. Two objects are
  considered the same topologically if you can
  deform one into the other without tearing,
  cutting, pinching, gluing, or other violent actions.
• The properties of an object that change when
  you bend, stretch, or twist are the geometry of
  the object. For example, distances, angles, and
  curvature are parts of geometry but not topology.
  Deforming an object doesn’t change
             it’s topology




A topologist is someone who can’t tell the difference between a coffee cup and a doughnut.
Which surfaces are topologically the
              same?
            Gluing diagrams
• What topological surface do you get when you
  glue (or tape) the edges of the triangle
  together as shown?
            Gluing diagrams
• What do you get when you glue the edges of
  the square together like this?
              Gluing diagrams
• What surface is this?



                                S2 (a sphere)

• And this?



                                T2 (a torus)
  Life inside the surface of a torus
• What happens as this 2-dimensional creature
  travels through its tiny universe?




What does it see when it looks forward? Backward?
Left? Right?
         Tic-Tac-Toe on a Torus
• Where should X go to win?




  What if it is 0’s turn?
              Tic-Tac-Toe on the Torus
• Which of the following positions are equivalent in torus tic-tac-
  toe?




• How many essentially different first moves are there in torus
  tic-tac-toe?
• Is there a winning strategy for the first player?
• Is it possible to get a Cat’s Game?
            Another surface
• What surface do you get when you glue
  together the sides of the square as shown?




                             K2 (a Klein bottle)
      Life in a Klein bottle surface
  What happens as this creature travels through its
  Klein bottle universe?




• A path that brings a traveler back to his starting point
  mirror-reversed is called an orientation-reversing path.
  How many orientation-reversing paths can you find?
• A surface that contains an orientation-reversing path is
  called non-orientable.
       Tic-Tac-Toe on a Klein bottle
• How many essentially different first moves are
  there in Klein bottle tic-tac-toe?




• Is there a winning strategy for the first player?
 What happens when you cut a Klein
          bottle in half?
• It depends on how you cut it.




      Cutting a Klein bottle


            Another Klein bottle video
      Three dimensional spaces
• How can you make a 3-dimensional universe
  that is analogous to the 2-dimensional torus?




• Is there a 3-dimensional analog to the Klein
  bottle?
          Name that Surface
• What two surfaces do these two gluing
  diagrams represent?
What topological surface is this?

				
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posted:3/30/2014
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