Office: |
3-104, Mathematics | —— | —— | Email: |
michael.albanese[at]stonybrook.edu |

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.

**Wu - The Squares of Grassmannians**. This is a translation of Wu's paper which was written in
French. The original can be found here.

**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

**Which Grassmannians are Spin Manifolds?** The purpose of this note is to determine which
(unoriented, oriented, and complex) grassmannians are spin manifolds. In order to achieve this goal, formulae for the first and second
Stiefel-Whitney class of a tensor product are derived.

**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. *I plan to improve these at some point.*

**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.

**Enlargeable Manifolds**. These are notes for a talk I gave in
the Student Differential Geometry Seminar at Stony Brook in Spring 2019. *I plan to improve these at some point.*