Introduction to Topology
Spring 2024
INFO FOR THE FINAL.
- The final will be on Friday, May 17th 8am-10am in 007.
- General rules: the final will be comprehensive with an emphasis on the second half of the course.
It will be closed books and notes, but an A4 or "letter" size "cheat sheet" (carefully prepared before the final) can be used.
You can write on both sides of the cheat sheet.
- Here is a summary for the final
- and sample/wrap-up problems for the final
- Solution to problem 4
Set 9-10.
- Here is hw9/10 due May 10 (Friday) (soft deadline, submit by Monday, May 13th the latest).
- Here is the latex version
- Solutions: problems 1, problem 2, part 1,
problem 2, part 2, problem 3,
problem 4, second pair,
problem 5
- for the extra credit: Podcast: finding a homotopy to show that the class of the constant loop x_0 is a left unit in the fundamental group; mp4 file;
- Here is an animation 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)
- Notes on retracts and deformation retracts.
- We will define covering maps/spaces on Tuesday.Notes on covering spaces.
-
Here is an animation showing coverings of a circle.
The infinite spring of course is homeomorphic to R and illustrates R covering S^1. In addition,
S^1 covering S^1 and "lifts of paths" are also shown.
Set 8.
- Here is hw8 due Monday, 29th April.
- Here is the latex version and figure 1,
figure 2, figure 3, figure 4
- An example of a simple flowchart - for the extra credit
- Solutions: Problem 1,
Problem 2,
Problem 3, part a. b.,
Problem 3, part c.,
- Notes on THE FUNDAMENTAL GROUP:
class notes on:
Path homotopy and the definition of the Fundamental Group. Dependence on the basepoint.
The fundamental group is a topological invariant.
- Additional reading:
Homotopy of paths from Allen Hatcher's book (he uses the phrase "composition of paths" instead of "concatenation of paths")
and The fundamental group definition, properties etc from Messer's book.
SET 7.
- HOMEWORK: Homework 7, due Friday, Apr 19th
note: some techniques needed for it will be discussed in class
- Here is the source file
- Solutions: problem 1, problem 2,
Problem 3. a-c,,
Problem 3. d,,
Problem 4
- Illustration for the homework: The platonic solids
- Notes on the connected sum of surfaces and
the classification theorem and the Euler characteristic (or Euler number).
- Alternatively, here is additional reading on triangulation of surfaces
and
the Euler characteristic (these summaries are
from here)
- Here is the proof of the classification of compact, connected surfaces as
will be discussed in class. The proof is from
this book *
- The 2016 Nobel prize in physics was given "for theoretical discoveries of topological phase transitions and topological phases of matter".
It uses the classiciation of
compact sonnected surfaces — click to read (especially the "answered by topology" part :) ) .
- Nobel prize winner physicist
explains the relation of the above physics result to topology – video
SET 4.
- homework 4, due March 12th, Tuesday
- source file in case you use latex
- Solutions problem 1, problem 2,
Problem 3, false part,
Problem 3, method 1,
Problem 3, method 2,
Problem 4
- Notes on interior, closure, boundary, the Hausdorff property and compactness
- Illustration:Stereographic projection - the ball is placed
on the plane - I used the unit sphere centered at the origin in class, so that the formulae for the projection work out to be simple.
- FYI:
The S^1xS^1xS^1 space that we looked at last class is such that "locally" (ie at every point of that space) you cannot distinguish
it from R^3. That is: microbes living in that space do not know "how that space folds into itself globally".
Similarly, people have the perception that they live in R^3, but we have no means (yet?) to decide what the Universe is like,
globally. Here is an article that discusses possible "shapes" of the Universe:
The Shape of the Universe: Ten Possibilities
from the Scientific American
- Practice questions in study cards mode.
SET 3.
- homework 3, due March 1st, Friday
- source file in case you use latex
- Solutions: problem 1,
problem 3, problem 4,
problem 5a and 5b, the case of vertices,
problem 5b, the case of "inner points" and "edge points",
problem 5c
- Reading for problem 2.
- illustration to problem 5:
gluing a doughnut surface from a hexagon
- Additional reading: formal treatment of
"pasting" ("gluing", "identifying") edges of polygons
and cut and paste arguments from Munkres.
- Still working on notes for week 2-3 (subspace, quotient, product topologies)
- ANIMATION: cutting the Klein bottle along a closed curve (circle) to get a Mobius strip
Find such a cut on the Klein diagram!
