Introduction to Topology   Spring 2016

  Date & place: Tuesday, May 24th, 8-10 am in 105
  Special office hours: Monday, May 23rd, 8:30-10:30am  
  • Sample problems for the final
  • Twisted prism for the last problem
  • Last two topics.

    Homework Set 7.

    To be handed in Friday, Apr 22th.
  • Download as pdf
    Two nice animations:
  • one for "nullhomotopic loops" i.e. loops that are homotopic to the constant loop. and another
  • 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)

  • In addition: FYI - READING
  • class notes on:Path homotopy and the definition of the Fundamental Group - Based on notes taken by Casey Shiring, University of Wisconsin at La Crosse and BSM Fall 2015.
  • Alternatively, Homotopy of paths from Allen Hatcher's book and The fundamental group definition, properties etc from Messer's booktwo nice animations:
  • Reading on generators, relations and the free group on two generators
  • Homework Set 6.

    To be handed in Friday, April 15th.
  • 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
  • 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, April 5th, 8-10 am in 105
      Special office hours: Monday, April 4th, 2-4pm  
  • Midterm info, sample problems
  • surface.pdf file for "long" sample problem 5
  • Some answers: True-false 9, long problems 1-4, long problems 5
  •   Summary of new material from this week, for the midterm:
  • Triangulation of surfaces
  • Summary — the Euler number (these summaries are from here)
  • The connected sum of surfaces and Euler characteristics. The classification theorem.
  • Illustration 1, Illustration 2, Illustration 3.

  • Homework Set 5.

    To be handed in Friday, March 18th.
  • Download as pdf
  • More info on Commutative diagrams
  • Illustrations: 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, March 11th.
  • Download as pdf
  • Proof of the Heine-Borel theorem - missing part.
  • notes on:compactness - Based on notes taken by Casey Shiring, University of Wisconsin at La Crosse and BSM Fall 2015.
  • notes on: interior, closure, boundary, the Hausdorff property
  • Homework Set 3.

    To be handed in Friday, March 4th.
  • Download as pdf
  • Related reading: formal treatment of "pasting" ("gluing", "identifying") and cut and paste arguments from Munkres in case of pairwise identification of edges of polygons
  • Week 2.

  • Homework 2 due Friday, February 26th
  • practice problems (no need to hand them in)
  • Here are summaries for the quotient topologies:
    1. from Munkres, discussing the torus case in detail
    2. from Messer and Straffin, discussing the Mobius case in detail
  • Illustrations: