### Spring 2023

### Kristen Hendricks

### Course Description:

This is a second course in algebraic topology. We will cover homotopy theory, characteristic classes, and possibly spectral sequences if time allows.

### Text:

Hatcher, Algebraic Topology

### Prerequisites:

Math 540 or equivalent

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### Spring 2022

### Hongbin Sun

### Course Description:

This is a continuation course of Math 540 Introduction to Algebraic Topology I. Continuing the topic's in the previous semester, the following topics will be covered.

1. Homotopy theory. Homotopy theory studies homotopy classes of maps from spheres to topological spaces, which is a natural generalization of the fundamental group, while their behaviors are very different from each other. The topics we will cover include: Whitehead's theorem, cellular approximation, excision theorem, Hurewicz theorem, long exact sequence of fiber bundles, connections with cohomology, Leray-Hirsch theorem and Gysin sequence.

2. Characteristic classes. For any vector bundle over a topological space, a characteristic class associate this vector bundle with a cohomology class of the base space. Characteristic classes are very useful in differential topology and differential geometry. The topics we will cover include: basics on vector bundles, Grassman manifolds, universal bundles, Stiefel-Whitney classes, Euler classes, Chern classes, Pontrjagin classes, the oriented cobordism ring. We will finish the course by talking about Milnor's construction of 7-dimensional exotic spheres.

### Text:

Allen Hatcher, Algebraic TopologyJohn Milnor & James Stasheff, Characteristic Classes

### Prerequisites:

Math 540 Introduction to Algebraic Topology I or equivalent

### *************************************

### Spring 2021 - Hongbin Sun

### Course Description:

This is a continuation course of Math 540 Introduction to Algebraic Topology I. Continuing the topic's in the previous semester, the following topics will be covered.

1. Homotopy theory. Homotopy theory studies homotopy classes of maps from spheres to topological spaces, which is a natural generalization of the fundamental group, while their behaviors are very different from each other. The topics we will cover include: Whitehead's theorem, cellular approximation, excision theorem, Hurewicz theorem, long exact sequence of fiber bundles, connections with cohomology, obstruction theory, Leray-Hirsch theorem and Gysin sequence.

2. Characteristic classes. For any vector bundle over a topological space, a characteristic class associate this vector bundle with a cohomology class of the base space. Characteristic classes are very useful in differential topology and differential geometry. The topics we will cover include: basics on vector bundles, Grassman manifolds, universal bundles, Stiefel-Whitney classes, Euler classes, Chern classes, Pontrjagin classes, Chern numbers, Pontrjagin numbers, the oriented cobordism ring.

### Text:

Allen Hatcher, Algebraic TopologyJohn Milnor & James Stasheff, Characteristic Classes

### Prerequisites:

Math 540 Introduction to Algebraic Topology I or equivalent

### Schedule of Sections:

### Previous Semesters

- Spring 2021 Prof. H.B. Sun
- Fall 2019 Prof. Weibel
- Spring 2017 Prof. Weibel