18.906
Homotopical Methods in Algebraic Topology
Continues the development of algebraic topology, with a focus on homotopical questions and computational tools. Topics include basic homotopy theory, classifying spaces, spectral sequences, and cohomology operations. Usually, only one of 18.906 or 18.916 is offered in a given academic year.
Spring 2020 · Algebraic Topology II
Topics
- Limits, Colimits, and Adjunctions
- Cartesian Closure and Compactly Generated Spaces
- Basepoints and the Homotopy Category
- Fiber Bundles
- Fibrations, Fundamental Groupoid
- Cofibrations
- Cofibration Sequences and Co-exactness
- Weak Equivalences and Whitehead’s Theorems
- Homotopy Long Exact Sequence and Homotopy Fibers
- Serre Fibrations and Relative Lifting
- Connectivity and Approximation
- Cellular Approximation, Obstruction Theory
- Hurewicz, Moore, Eilenberg, Mac Lane, and Whitehead
- Representability of Cohomology
- Obstruction Theory
- Vector Bundles
- Principal Bundles, Associated Bundles
- I -invariance of Bun G , and G -CW Complexes
- The Classifying Space of a Group
- Simplicial Sets and Classifying Spaces
- The Čech Category and Classifying Maps
- Why Spectral Sequences?
- The Spectral Sequence of a Filtered Complex
- Serre Spectral Sequence
- Exact Couples
- The Gysin Sequence, Edge Homomorphisms, and the Transgression
- The Serre Exact Sequence and the Hurewicz Theorem
- Double Complexes and the Dress Spectral Sequence
- Cohomological Spectral Sequences
- Serre Classes
- Mod C Hurewicz and Whitehead Theorems
- Freudenthal, James, and Bousfield
- Chern Classes, Stiefel-Whitney Classes, and the Leray-Hirsch Theorem
- H *( BU ( n )) and the Splitting Principle
- The Thom Class and Whitney Sum Formula
- Closing the Chern Circle, and Pontryagin Classes
- Steenrod Operations
- Cobordism
- Hopf Algebras
- Applications of Cobordism
Readings
Prerequisites & course information
Prereq: 18.905 G (Fall) 3-0-9 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Spring 2020
- MIT OCW calendar ↗
- MIT OCW syllabus ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.