Student topology seminar

The goal of the talk is to construct the classifying space of a group $G$ using Milnor's construction and prove that it is a universal principal $G$-bundle. We will start by introducing the notion of principal $G$-bundle and discussing some of their properties. No previous familiarity with the topic will be assumed.

Reference: Dale Husemöller, Fibre bundles.

We construct the classical Cech cohomology with coefficient in a sheaf and compute first Chern classes of homolomorphic line bundles over a complex space. An attempt at effective algebraic geometry computations in topology. We assume only that "variety" means "compact complex manifold."

References: Hartshorne, Bott/Tu, Serre's "GAGA" and "FAC".

Building on the classification of vector bundles, we will discuss how to assign and construct "universal" cohomological invariants called characteristic classes to vector bundles.

References: May, A Concise Course in Algebraic Topology; Milnor and Stasheff, Characteristic Classes.

I will define the homotopy groups and relative homotopy groups, discuss the long exact sequence associated with a fibration and other basic tools for computation, and calculuate some homotopy groups of spheres, including $\pi_3(S^2)$.

References: Hatcher, Algebraic Topology; Gray, Homotopy Theory.

Lecture notes

I will define the K-groups in the complex case and talk about some basic properties. If time allows, I will talk about the Product Theorem.

References: Hatcher, Vector Bundles and K-Theory.

Lecture notes

When a manifold $M$ has a group action, there is a refined cohomology theory on $M$, namely equivariant cohomology. In this talk, I will introduce equivariant cohomology (Borel construction) and apply it to enumerative geometry.

Steenrod squares are cohomology operations useful in many parts of Algebraic Topology. I will give an elementary construction of these operations and prove some of its properties. Time permitting I will talk about some of their applications.

References: Mosher and Tangora, Cohomology Operations and Applications in Homotopy Theory; Bredon, Topology and Geometry.

We present some basic facts on sphere bundles in preparation for Thom's results on Steenrod's problem.

Lecture notes

We will consider the cohomology of the Thom complexes $MO(k)$, $MSO(k)$ and their applications to Thom's investigations into Steenrod's problem: which homology cycles can be represented as embedded subvarieties?

Lecture notes

The goal of this talk is to prove Adams' theorem on the Hopf invariant. Along the way we will review computational tools in K-theory such as Adams operations, splitting principle, Thom isomorphism in K-theory and Atiyah–Hirzebruch spectral sequence. Time permitting we will also talk about vector fields on spheres.

Lecture notes

We will continue with the outline proposed for my last talk.

Lecture notes

I will give an introductory overview of the mathematical theory of knots.

I will define spectral sequences, describe the spectral sequence associated to a filtered chain complex, and give a simple example involving cellular homology.

References: McCleary, A User's Guide to Spectral Sequences; Spanier, Algebraic Topology.

I will talk about the Serre spectral sequence and some of its applications.

References: McCleary, A User's Guide to Spectral Sequences; Spanier, Algebraic Topology; Hatcher, Spectral Sequences in Algebraic Topology.

We will discuss the question of how many linearly independent vector fields exist on the sphere $S^{n-1}$. The upper bound answer to this question is one of the celebrated results of K-theory.

Lecture notes

I will deduce Bott periodicity from the Product Theorem and sketch an alternative proof from Quasifibrations and Bott periodicity.

References: A. Hatcher, Vector Bundles and K-Theory; M.A. Aguilar and C. Prieto, Quasifibrations and Bott periodicity.

I will discuss the braid groups from several points of view, each one yielding specific information about properties of that important family of groups.

I will discuss ring homomorphisms from the cobordism ring to a ring (multiplicative sequences). According to R. Thom, the dual ring of characteristic classes and the cobordism ring are "dual." Signature of manifolds gives us a ring homomorphism from the cobordism ring to $\mathbb{Z}$ and thus it can be expressed by characteristic classes.

Reference: Milnor and Stasheff, Characteristic Classes section 19.

Lecture notes

In this seminar we talk about braid groups on surfaces. We'll see the definiton, describe elements in these groups and find presentations for them.

I will discuss further aspects of the braid groups. However, students who missed my first lecture will be able to follow this discussion.