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.
Semester 2, 2021
TopFest – Back to School
A mini-conference for graduate students of Christian Haesemeyer, Marcy Robertson and Diarmuid Crowley:
♦ Dates and times tba, pending COVID lockdowns.
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.