Spring 2026
Zhenghao Rao
Course Description:
This is a second course in algebraic topology, after Math 540. We will cover cohomology, homotopy theory, and characteristic classes (if time permitted).
- Cohomology Theory: Cohomology is the algebraic counterpart to homology, with a natural graded ring structure: the cup product lets us multiply classes and reveal invariants that homology alone cannot see. For manifolds, Poincaré duality shows a clear matching between homology and cohomology. The topics we will cover include: cohomology groups, universal coefficient theorem, cup product and cohomology ring, Künneth formula, and Poincaré duality.
- Homotopy Theory: Homotopy groups study maps from spheres to topological spaces, which is a natural generalization of the fundamental group, but with very different behavior. Intuitively, homotopy groups record the shape of spaces. The topics we will cover include: Whitehead's theorem, cellular approximation, excision theorem, Hurewicz theorem, long exact sequence of fiber bundles, and connections with cohomology.
- Characteristic Classes: A characteristic class is a way of associating with each vector bundle over a space a cohomology class of the space. It is an invariant that measures how the global product structure is derived from the local product structure. The topics we will cover include: vector bundles, Grassmann manifolds, universal bundles, Stiefel-Whitney classes, Euler classes, and Chern classes.
Text:
Allen Hatcher, Algebraic Topology
Prerequisites:
Math 540 or equivalent
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Fall 2025
Feng Luo
Course Description:
This is a standard first course in algebraic topology. We will cover the following topics: fundamental group and covering spaces, singular, simplicial, and cellular homology.
Text:
Allen Hatcher, Algebraic Topology
Prerequisites:
Point-set topology and and abstract algebra (groups, rings, fields).
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Spring 2025
Kristen Hendricks
Course Description:
This is a second course in algebraic topology. We will cover cohomology (if not covered in the first semester), homotopy theory, characteristic classes, and possibly spectral sequences if time allows.
Text:
Cohomology and Homotopy Theory: A. Hatcher, Algebraic Topology and J. Milnor, Topology from the Differentiable Viewpoint. Characteristic Classes: J. Milnor and W. Stasheff, Characteristic Classes and A. Hatcher, Vector Bundles and K-Theory; Spectral Sequences: A. Hatcher, Spectral Sequences in Algebraic Topology and R. Bott and L. Tu, Differential Forms in Algebraic Topology
Prerequisites:
Math 540 or equivalent
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Spring 2024
Hongbin Sun
Course Description:
This course is a continuation of Math 540 Algebraic Topology I. The main topics of this course are cohomology theory and homotopy theory, with the following specific topics: cohomology groups, cup product, Poincare duality, homotopy groups, Whitehead theorem, Hurewicz theorem, fiber bundles, etc. Applications of algebraic topology to low-dimensional topology will also be given.
Text:
Allen Hatcher, Algebraic Topology
Prerequisites:
Math 540 Introduction to Algebraic Topology II
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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
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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