# 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.

**GTS will not run 2021 Semester 1**

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

**Due to the pandemic, the graduate seminar ran online in 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.