Comments on material

I will be following some modified lecture notes of David Zureick-Brown. My general plan is to throw ourselves into the deep end of homotopy theory, build up lots of examples and dig our way out. This means we can't always prove everything but we will get to see how computations work more and get a better sense of how things fit together. The skeleton for the class is going to be the Geraschenko-Teichner notes but we are going skip around a little and fill things in (the notes themselves are incomplete, but the structure is more modern than the older treatments).

Below is a list of comments that will help guide you through the videos.

  1. The lectures/videos alone are not sufficient for this course.You will need to do a lot of nonlinear reading and find sources that you like.
  2. You shouldn't read all of the references. You need to find your own path through the material. This means discussing with others what you found useful and what you didn't find useful during weekly Zulip discussions.
  3. Collect your questions for Zulip discussions.
Category Theory
Read this as we go. Here is a bible for category theory: Awodey's Course Notes.
  1. May Chapter 2
  2. Geraschenko Teichner Lemma 15.1: smash products as left adjoints to hom
  3. Sato 2.2: product and quotient spaces
  4. Sato 2.3: quotients
  5. Sato 1.2: homotopy equivalence
  6. Sato 1.3: topological pairs
  7. Hatcher Chapter 0, "operations on spaces"
Homotopy and Topology Basics
Homotopy Groups of Spheres, Brauer Fixed Point, Borsuk-Ulam
Mapping Cylinder and Mapping Path Space (Fibrations and Cofibrations)
There are two parts of "Eckmann-Hilton Duality". Homotopy lifts and homotopy extension. Homotopy lifts are about fibrations (like vector bundles or covering spaces). Homotopy extensions are about cofibrations (like inclusion of one cell of a CW complex into another.) We are going to introduce this first BIG DEFINITIONS: Geraschenko-Teichner Definition 12.9. The main examples of a fibration is a well, a fiber bundle. The main example of a cofibration is an inclusion of a cell complex. These ideas are what fit into the modern treatment of infinity categories.
Higher Homotopy Groups
There are three ways that I know of to prove that higher homotopy groups are commutative. We are going to use that we have a cogroup object in the category hTop.
Manifolds and CW complexes
I'm not sure where this is going to fit in yet. I know that we need examples, and we need time to develop fun examples. I also know that to do Delta complexes like in Hatcher, we want to view them as realizations of dSets. Also, the given the modern point of view we are going to need to talk about sSets at some point and their realizations. (Let me say you need to set up the Complexes here carefully, or CW, Delta, and simplicial things don't look nice)
Fundamental Groups and Van Kampen
Fundamental Groups and Covering Spaces
Euler Characteristics, Chain Complexes and Homology
Tensor products of chain complexes are needed later for homotopy invariance.
Homology: Eilenberg-Steenrod
At the suggestion of Sean Tilson, I'm going to try setting up the axioms for homology first.
Cohomology
Last time I taught this course we ran out of time and I was rushing to do anything here. This time, I'm cutting out the construction of universal covers which should give us more time.