Michael Albanese


Office: Pavillon Président Kennedy PK-5540 —— —— Email: michael.albanese[at]cirget.ca

This is a place for me to put notes that I have written up about various topics. If you have any questions, comments, suggestions, or corrections, I would love to hear them, just send me an email (or tell me in person).

Expository Notes

Steenrod Squares, Wu Classes, and Stiefel-Whitney Classes. There is a list of formulae for the first five Wu classes in terms of Stiefel-Whitney classes on nLab. There was no reference given for the necessary computations, so I tried to do them myself. In doing so, I realised I misunderstood what little I thought I knew about Steenrod squares and Wu classes. In this note I explain what is needed to compute, and hopefully understand, the formulae given on nLab.

Homotopy Groups of a Wedge Sum of Spheres. There is a trick for computing the first few homotopy groups of a wedge sum of spheres which uses cellular approximation. But how do you compute the remaining homotopy groups? The answer is given by Hilton's Theorem. After introducing the trick, I explain Hilton's theorem and how to implement it to calculate the homotopy groups of a wedge sum of spheres in terms of the homotopy groups of spheres.

The Hirzebruch χy Genus and a Theorem of Hirzebruch on Almost Complex Manifolds. The purpose of this note is to give an introduction to the Hirzebruch χy genus and to give a proof of a theorem of Hirzebruch which states that on a closed almost complex manifold M of dimension 4m we have χ(M) ≡ (-1)mσ(M) mod 4.

A modified version of this note appears as an appendix to a paper; see here.

Which Grassmannians are Spin/Spinc? The purpose of this note is to determine which (unoriented, oriented, and complex) grassmannians are spin, and which ones are spinc. In order to achieve this goal, formulae for the first and second Stiefel-Whitney classes of a tensor product are derived. The corresponding non-orientable analogues pin+, pin-, and pinc are also considered.

The Normal Bundle of a Sphere Bundle is Trivial. Given a Riemannian metric on a smooth vector bundle E → M, one can form its sphere bundle S(E). The purpose of this note is to show that the normal bundle of the inclusion of total spaces S(E) → E is trivial.

Notes for Talks and Lectures

I plan to improve all of these at some point.

Almost Complex Structures and Obstruction Theory. These are notes for a lecture I gave in John Morgan's Homotopy Theory course at Stony Brook in Fall 2018.

Enlargeable Manifolds. These are notes for a talk I gave in the Student Differential Geometry Seminar at Stony Brook in Spring 2019.

The Yamabe Invariant of Complex Surfaces. These are notes for a series of four online lectures delivered as part of the 6th Geometry-Topology Summer School hosted by the Feza Gürsey Institute. Videos of the lectures can be found on YouTube.

Translations

Wu - The i-Squares in a Grassmannian Variety. This is a translation of Wu's paper Les i-carrés dans une variété grassmannienne. The original can be found here.

Habegger - A manifold of dimension 4 with even intersection form and signature -8. This is a translation of Habegger's paper which was written in French. The original can be found here.