Introduction to Topology   Fall 2015

Homework Set 9.

To be handed in Tuesday, December 1st.
  • Download as pdf
  • Hints to problem 1 and problem 3a
  • Send me an email if you have any questions related to the hw. I will also have office hours on Monday 3:30-4:30pm (!!!)
  • Note: I will not have office ours on Thursday.
  • Homework Set 8.

    To be handed in Tuesday, Nov 24th.
  • 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.)
  • Mapping [0,1] onto the unit square by a continuous map: A Peano-curve. Namely, the Hilbert-curve. (An example of a "space filling curve".
  • Space filling curve art and art generator
  • Homework Set 7.

    To be handed in Friday, Nov 13th.
  • Download as pdf
  • Examples of deformation retracts for practice
  • Here are two nice animations: one for "nullhomotopic loops" i.e. loops that are homotopic to the constant loop. and another
  • illustrating that 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)
  • Summary of what we are/will be discussing: Retracts, deformation retracts, homotopy equivalence
  • Homework Set 6.

    To be handed in Friday, Nov 6th.
  • Download as pdf
  • sequence of diagrams for problem 1
  • Proof of the classification of compact, connected surfaces discussed in class
  • the surface with boundary for problem 2
  • Reading on generators, relations and the free group on two generators

  • In addition: FYI
  • Homotopy of paths from Allen Hatcher's book
  • The fundamental group definition, properties etc
  • Some homotopy practice problems
  • Two additional proofs of the Classification Theorem of compact, conected surfaces: based on "cutting away know parts of the surface and Conway's ZIP proof

      Date & place: Tuesday, October 20th, 8-10 am in 105
      Special office hours, review session: Monday, Oct 19th, Room 104, 8:30-10am  
  • Midterm info, sample problems

  • Answers:
    page 1, page 2, page 3, page 4, page 5
      Summary of new material from this week, for the midterm:
  • Triangulation of surfaces
  • Summary — the Euler number (these summaries are from here)  
  • Homework Set 5.

    To be handed in Friday, Oct 16th.
  • Download as pdf
  • Homework Set 4.

    To be handed in Friday, Oct 9th.
  • Download as pdf
  • Proof of the Heine-Borel theorem - missing part.
  • Exercises 3
    interior, closure, boundary, the Hausdorff property
  • Homework Set 3.

    To be handed in Friday, Oct 2nd.
  • Download as pdf
  • NOTES FOR WEEK 3: formal treatment of "pasting" ("gluing", "identifying") and cut and paste arguments from Munkres in case of pairwise identification of edges of polygons
  • Homework Set 2.

    To be handed in Friday, Sept 25th.
  • Download as pdf
  • Exercises 2. (For practice, no need to hand these in.)
  • Illustrations:
  • Homework Set 1.

    To be handed in Friday, Sept 18th.
  • 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: