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Browse materialsMIT 18.906 · Spring 2020

Algebraic Topology II

Subjects: Geometry · Mathematics

Art: Max Brückner (1860 – 1934), Brückner Vielflache Tafel 07 · Public domain

Instructor: Prof. Haynes Miller
Term: Spring 2020
Level: Graduate
What's inside: 6 problem sets · 2 lecture notes · 8 other files · 5 web pages
Materials open: MIT OCW

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About this course

Source: MIT OpenCourseWare 18.906 Algebraic Topology II, Spring 2020, Prof. Haynes Miller — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Lecture Notes

Assignments

9 files

MIT OpenCourseWare, Prof. Haynes Miller, 18.906 Algebraic Topology II, Spring 2020.

License: CC BY-NC-SA 4.0 · Edition: Spring 2020 · Open the original course on OCW ↗

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Algebraic Topology II

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Lecture 1: Limits, Colimits, and Adjunctions

Lecture 2: Cartesian Closure and Compactly Generated Spaces

Lecture 3: Basepoints and the Homotopy Category

Lecture 4: Fiber Bundles

Lecture 5: Fibrations, Fundamental Groupoid

Lecture 6: Cofibrations

Lecture 7: Cofibration Sequences and Co-exactness

Lecture 8: Weak Equivalences and Whitehead’s Theorems

Lecture 9: Homotopy Long Exact Sequence and Homotopy Fibers

Lecture 10: Serre Fibrations and Relative Lifting

Lecture 11: Connectivity and Approximation

Lecture 12: Cellular Approximation, Obstruction Theory

Lecture 13: Hurewicz, Moore, Eilenberg, Mac Lane, and Whitehead

Lecture 14: Representability of Cohomology

Lecture 15: Obstruction Theory

Lecture 16: Vector Bundles

Lecture 17: Principal Bundles, Associated Bundles

Lecture 18: I -invariance of Bun G , and G -CW Complexes

Lecture 19: The Classifying Space of a Group

Lecture 20: Simplicial Sets and Classifying Spaces

Lecture 21: The Čech Category and Classifying Maps

Lecture 22: Why Spectral Sequences?

Lecture 23: The Spectral Sequence of a Filtered Complex

Lecture 24: Serre Spectral Sequence

Lecture 25: Exact Couples

Lecture 26: The Gysin Sequence, Edge Homomorphisms, and the Transgression

Lecture 27: The Serre Exact Sequence and the Hurewicz Theorem

Lecture 28: Double Complexes and the Dress Spectral Sequence

Lecture 29: Cohomological Spectral Sequences

Lecture 30: Serre Classes

Lecture 31: Mod C Hurewicz and Whitehead Theorems

Lecture 32: Freudenthal, James, and Bousfield

Lecture 33: Chern Classes, Stiefel-Whitney Classes, and the Leray-Hirsch Theorem

Lecture 34: H *( BU ( n )) and the Splitting Principle

Lecture 35: The Thom Class and Whitney Sum Formula

Lecture 36: Closing the Chern Circle, and Pontryagin Classes

Lecture 37: Steenrod Operations

Lecture 38: Cobordism

Lecture 39: Hopf Algebras

Lecture 40: Applications of Cobordism