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Point-set and Algebraic Topology: Spring.




Instructor: Dr. Stefan Forcey Textbook: We have 2, but they are both free! Download these as soon as you can.
[PS] A. Hatcher, Notes on Introductory Point-Set Topology

[AT] A. Hatcher, Algebraic Topology

Supplementary material:

Prasolov, V.V., Intuitive Topology A.M.S.

Weeks, Jeffrey. The Shape of Space.

Phylogenetic trees.


Course Syllabus.
The syllabus will include information about grading policies, schedules and outlines. You should definitely read it.

Homework due dates:
Each assignment will be a printed handout, but the questions are already listed on the syllabus. I'll pass out the printed version on the week of the topic and we will do some portion in class, leaving the remainder for you to finish and turn in. In addition, each student will get the chance to present those remaining solutions to one assignment for one part of the final grade, upon the next class period after the work is checked. For that purpose, keep a copy when it is your turn (or I can make a copy for you). You'll pick any four topics, and I'll try to give everyone a first or second choice.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             

  • Category of topological spaces and continuous maps.
    Reading [PS] Chapter 1,2 pgs 1-20.
    Answer the questions in number (1) on the course outline, in the Syllabus. Quiz handout with spacing provided;
    we'll do some work in class but you can think ahead!
    Take home quiz 1 due Jan. 23.

  • Connectivity, path connectivity. Reading [PS] Chapter 2 pgs 20-28.
    Take home quiz 2 due Jan. 30

    Check out this Mandelbrot gallery.
    Here's an excellent tutorial on the relationship between Mandelbrot and Julia sets, and on how the Fibonacci numbers appear.
    Here's the Wolfram Mathworld article stating that it is unknown whether the Mandlebrot set is path connected.

  • Metric spaces
    Take home quiz 3 due Feb. 4

  • Cut points
    Take home quiz 4 due Feb. 11

  • Surfaces
    Take home quiz 5 due Feb. 20

    • Example of cutting and gluing

  • Genus and Euler characteristic
    Take home quiz 6 due March 3

  • Homotopy and Fundamental Groups
    Take home quiz 7.
    • You may turn in when complete by email.

    Note that all you need to do is find a group presentation, not necessary to show what well-known group it is isomorphic to. (However, the latter is easy to look up for some of the examples, to check your work.)
    For surfaces, the method is to start just like when finding the Euler characteristic: chop it up into disks. Now, however, the generators are the loops that make up the boundaries of the disks. So write down a generator for each loop around each disk, that is, line segments that start and end at the same point. Most examples can be done with a single disk: you can use the results of the previous quizzes, especially the gluing diagrams like hexagons and octagons. The relations correspond to the disks, because now there will be two ways to "walk" around a disk by following generators; and these will be homotopic by filling in the disk with a family of loops. 
     Also, it is nice to know what happens when you start with just a plain disk: the fundamental group pi_1(D^2) = {e}; just the identity element, since every loop is homotopic to the single point. Some places also call this group "0." Recall that a sphere S^2 is found by starting with a disk and identifying all the points on its boundary to a single point. 

    Examples of calculating some fundamental groups.

    Finding a homotopy.

  • More Homotopy and Fundamental Groups
    Take home quiz 8.
    • You may turn in when complete by email.


    Top menu.
    Here is the lecture from April 1.



  • Even More Homotopy and Fundamental Groups
    Take home quiz 9. Due on April 7. Hint: look at this example again, in reverse order: Example of cutting and gluing

    Examples: answers to quiz 5

    • Top menu.
    Here is the lecture from April 2.

    This is a 3-part explanation of how the two answers for the fundamental group found in quiz 8 can be shown to be isomorphic.








    Top menu.
  • Knots and homotopy Knot notes Part 1

    • Take home quiz 10.

    Here is the discussion of the knot notes part 1.



  • Knot notes Part 2

    • Here is the discussion of the knot notes part 2.



    Here is a demonstration of the knot relation.



  • Knot quiz 1
    Take home quiz 10.

    • Required watching: the geometry...what it would feel like to live inside a space
    homeomorphic to the knot complement, imbued with hyperbolic geometry!




    Top menu.
  • Homology Download notes from April 9: Homology notes Part 1


    • Then watch the explanations!


    ***Error fixed in the pdf above: the Euler characteristic is 1.






    ***One error fixed in the pdf: 1-cycles are comprised of loops, not only single loops.


    ***Note: in the pdf download above, the correction to subscript "n+1" has already been made
    (hit reload on the pdf to see revisions if you already downloaded.)

    Top menu.
  • Homology Download notes from April 14: Homology notes Part 2


    • Then watch the explanations!










    Top menu.
  • Homology Download notes from April 16: Homology notes Part 3


    • Then watch the explanations!














    Top menu.

    Take home quiz 11. due April 28










Top menu.
Project.

Instructions and due dates.
To get a color pdf of your personalized graph,
replace the filename in the url of this site with (your) Lastname_project_blank.pdf

Sample project: now showing parts 1-6

Another sample project: showing part 1

Another sample project: showing parts 2-5

News about topology!
Shape of the universe: recent excitement
Puzzle: what is wrong about the first sentence in the caption of the picture at the top of this article?
Hint here.

NPR.org Scientists have twisted molecules into the tightest knot ever

Resources


CREDIT:
The text of this page was adapted, with permission, from an original course site by J.P. Cossey.