Introduction to Topology   Fall 2016

Homework Set 10.

To be handed in Tuesday, Dec 6th.
  • Download as pdf
  • Hint to problem 3
  • Homework Set 9.

    To be handed in Tuesday, Nov 29th.
  • Download as pdf
  • Practice problems
  • Summary of what we are/will be discussing: Covering spaces I. - Based on notes taken by Casey Shiring, University of Wisconsin at La Crosse and BSM Fall 2015.
  • Here is an animation showing coverings of a circle.
    (The infinite spring of course is homeomorphic to R.)
  • Homework Set 8.

    To be handed in Friday, Nov 18th.
  • Download as pdf
  • Examples of deformation retracts for practice
  • Summary of what we are/will be discussing: Retracts, deformation retracts, homotopy equivalence
  • Homework Set 7.

    To be handed in Friday, Nov 11th.
  • Download as pdf
  • Practice problems, to be posted later...

  • 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 (he uses "composition of paths" instead of "concatenation of paths") and The fundamental group definition, properties etc from Messer's book.
  • Reading on generators, relations and the free group on two generators
  • Homework Set 6.

    To be handed in Friday, November 4th.
  • Download as pdf
  • sequence of diagrams for problem 2
  • Proof of the classification of compact, connected surfaces (to be) discussed in class
  • 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
  • FYI: The 2016 Nobel prize in physics is "for theoretical discoveries of topological phase transitions and topological phases of matter". It uses the classiciation of compact sonnected surfaces, click to read.
  • Nobel member uses cinnamon bun, bagel and pretzel to explain video
  • This is the animation mentioned in class: The Klein can be cut along a circle so that the result is a Mobius strip

      Date & place: Friday, October 21st, 8-10 am in 105
      Reminder: office hours: Thursday, 2-3:30pm  
  • Midterm info, sample problems
  • surface.pdf file for the last "long" sample problem
  • Answers to long 1, long 3, long4
  • Week 6.

  • Summary of what we covered on Friday (Oct 14th) The connected sum of surfaces and Euler characteristics. The classification theorem. and Illustration 1, Illustration 2, Illustration 3.
  • In addition, here is reading on the triangulation of surfaces and the Euler number (these summaries are from here)
  • Illustrations: A Peano-curve. Namely, the Hilbert-curve. (An example of a "space filling curve").
  • FYI: Space filling curve art and art generator
  • More info on Commutative diagrams
  • Week 5.

  • Homework 4: download as pdf . Due Friday, Oct 14th.
  • Proof of the missing part of the Heine-Borel thm
  • Notes on: compactness
  • Week 4.

  • Homework 4: download as pdf . Due Friday, Oct 7th.
  • Notes on: interior, closure, boundary, the Hausdorff property
  • Week 3.

  • Homework 3 due September 30th
  • Summary for subspace, quotient and product topologies as we discussed Week 2 and 3. Summary for "cut-and-paste arguments" is included as well.
  • 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 September 23rd
  • 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: