Michael Albanese

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).

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 χ_y Genus and a Theorem of Hirzebruch on Almost Complex Manifolds. The purpose of this note is to give an introduction to the Hirzebruch χ_y 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/Spin^c? The purpose of this note is to determine which (unoriented, oriented, and complex) grassmannians are spin, and which ones are spin^c. 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 pin^c 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.

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.

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.