Introduction to Topology   Fall 2014


  • Summary of properly discontinuous group actions and fundamental groups.
  • Summary for the final
  • Practice/review problems for the final
  • Figure 1 for the last problem
  • Figure 2 for the last problem

    Homework Set 10.

    To be handed in Wednesday, Dec 10.
  • Download as pdf .
  • Summary of what we discussed about covering spaces so far (We talked about a different proof for the path lifting and homotopy lifting lemmas, which uses compactness and connectedness of the image of paths/homotopies. The one given here is from Munkres.)

    Homework Set 9.

    To be handed in Wednesday, Nov 26th.
  • Download as pdf .
  • Problem 3. contains some reading in group theory:
  • Here is an animation showing coverings of a circle as well.
    (The infinite spring of course is homeomorphic to R.)
  • Homework Set 8.

    To be handed in Friday, Nov 21th.
  • Download as pdf .
  • More examples of deformation retracts.
  • A nice animation for "nullhomotopic loops" i.e. loops that are homotopic to the constant loop.
  • Summary of what we are/will be discussing: Retracts, deformation retracts, homotopy equivalence
  • Homework Set 7.

    To be handed in Friday, Nov 14th.
  • Download as pdf .
  • Animation: the fundamental group of the torus is abelian(for the meridian "alpha" and longitudinal "beta" loops we have that alpha*beta is homotopic to beta*alpha)
  • The fundamental group - EXPANDED!!!
  • Homotopy of paths
  • Homework Set 6.

    To be handed in Friday, Nov 7th, in class, or Monday, Nov 10th in Anna's office.
  • Download as pdf .
  • Proof 2., cut-and-paste to normal form
  • Series of diagrams for 1.c.
  • Cube identification for problem 2.b
  • Enlargment, cube identification for problem 2.b
  • Gluing polyhedra to get 3-manifolds

  • See also,
  • Proof 1., reversing the connected sum operation
  • John Conways ZIP proof
  • Orientation of surfaces in terms of a triangulation.

  •   Midterm: Oct 22nd, 8:15-10 am in 104  
  • Midterm info and practice problems
  • Triangulation of surfaces
  • Summary — the Euler number (the summary is from here)  

  • Homework Set 5.

    To be handed in Friday, Oct 17th.
  • Download as pdf
  • A Peano-curve. Namely, the Hilbert-curve. (An example of a "space filling curve".
  • Space filling curve art and art generator
  • Homework Set 4.

    To be handed in Friday, Oct 10th.
  • Download as pdf
  • Proof of the Heine-Borel theorem (the missing part).
  • Commutative diagrams
  • Homework Set 3.

    To be handed in Friday, Oct 3rd.
  • Download as pdf
  • Exercises 3.
  • Summary on interior, closure, boundary, the Hausdorff property and convergence
  • Summary on the subspace, quotient topologies
  • Summary on the product topologies (still working on it...)
  • Homework Set 2.

    To be handed in Friday, Sept 26th.
  • Download as pdf
  • Exercises 2. (For practice, no need to hand these in.)
  • Illustrations: Stereographic projection - animation
  • Stereographic projection from a a complex analysis textbook (since they place the unit sphere differently in the coordinate system as we did in class, the formulae are different a bit)
  • The Klein Bottle (and Mobius strip) (animation)
  • If you go to Prague, check out this a Mobius mural.
  • M.C. Escher's famous Mobius strip
  • Week 1. summary

  • Practice problems and summary of what we did so far <
  • Homework Set 1.

    To be handed in Friday, Sept 19th.
  • Download as pdf
  • Exercises 1. (for practice, no need to hand these in)
  • Illustration: In topology the cube = the sphere
  • Literature used


    Special dates this semester: