# Graduate Topology Seminar

The graduate topology seminar is a working seminar for Masters and Ph.D. students. Past topics have included: characteristic classes, Sullivan’s rational homotopy theory, and cohomology operations.

In 2023, the topic is an introduction to homotopical algebra in the form of model categories and infinity categories. We will start by following the paper “Homotopy Theory and Model Categories” by Dwyer and Spalinski. We will then progress to more advanced papers (to be determined by student interest). Seminar meets **Wednesdays at 11:00 am in Peter Hall 162**.

## Semester 2, 2023

♦ **August 9th:** Chandan Singh

Topic: Review of categories, functors, limits and colimits.

♦ **August 23rd:** Olivia Borghi

Topic: Introduction to model categories and the homotopy category

♦ **August 30th:** No Seminar

♦ **September 6th**: Kurt Stoeckl

Topic: The projective model structure on chain complexes

♦ **September 13th:** Marcy Robertson

Topic: The model structure on the category of topological spaces

♦ **September 20th:** Guillaume Laplante-Anfossi

Topic: The small object argument and postnikov towers

♦ **September 27th:** Jonah Nelson

Topic: Derived functors

♦ **October 4th:** TBD

Topic: Dwyer-Kan localisation

♦ **October 11th:** TBD

Topic: Relative categories as a model for infinity categories

♦ **October 18th:** Olivia Borghi

Topic: An introduction to decomposition spaces

**Semester 1, 2021**

GTS did not run.

Students were encouraged to participate in the Graduate Class on Cohomology Operations and Applications.

**Semester 2, 2020.**

**Semesters 1 & 2 2020 – Milnor’s lecture notes on the h-cobordism theorem**

♦ Monday December 7th: **Ethan Armitage**

**Title: Some applications of the h-cobordism theorem **(Sections 8 & 9)

**Abstract:** In this talk we finish the discussion of the removal of low (high) index handles and so complete the proof of the h-coboridsm theorem. We will then discuss some applications, including Smale’s proof of the Generalised Poincaré Conjecture.

♦ Monday November 30th: **Jayden Hammet**

**Title: Cancellation of general handles in general **(Sections 7 & 8)

**Abstract: **This talk continues the process of cancelling handles. The main line of argument reaches it conclusion with the removal of all handles from an h-coborism with no 0-, 1-, (n-1)- or n-handles and we summarise this. We then proceed to removing 0- and 1-handles (and dually n- and (n-1)-handles).

♦ November 3: **Ethan Armitage**

**Title: Cancellation of handles (critical points) in the middle dimension **(Sections 6 & 7)

**Abstract:** In this talk we will start the process of cancelling handles, or critical points, whose index is between 2 and n-2 in simply connected cobordisms. This involves first rephrasing the cancellation theorem in terms of intersection numbers for a simply connected manifold and then showing that every morse function with no critical points of index 0,1,n-1 or n can be deformed so that the left- and right-hand sphere of all critical points have intersection number +/-1.

♦ September 29: **Ethan Armitage**

**Title: Cancallation of handles **(Section 5)

**Abstract:** In this talk we prove that, given strong assumptions on a gradient like vector field, handle attachment of an index lambda+1 handle can cancel an index lambda handle. We will then state a simple topological condition that allows us to find a gradient like vector field satisfying those strong assumptions.

♦ September 1: **Ethan Armitage**

**Title: Rearrangement of cobordisms and an introduction to handle cancellation **(Sections 4 & 5)

**Abstract:** in this talk we will prove sufficient conditions for when a composition of elementary cobordisms can be ‘rearranged’ and as a corollary prove the existence of self-indexing morse functions. We will then discuss when the composition of elementary cobordisms is the trivial cobordism.

♦ August 18: **Jayden Hammet **

**Title: Elementary cobordisms (**Section 3)

**Abstract:** This talk analyses the effect of crossing a critical point on the topology of a cobordism and establishes the fundamental relationship between handle addition and isolated singularities of Morse functions.

♦ May 18: **Ethan Armitage **(Section 2)

**Title: Existence of a Morse function**

**Abstract:** This talk continues the reading seminar on Milnor’s classic monograph, “Lecture notes on the h-cobordism theorem,” covering the existence of Morse functions on bordisms.

♦ May 11: **Ethan Armitage**

**Title: Introduction and Morse functions**

**Abstract: **This talk starts a reading seminar on Milnor‘s classic monograph, “Lecture notes on the h-cobordism theorem”. We’ll begin with a quick introduction to the h-cobordism theorem and proceed to Section 2 on Morse functions.