18.906 | Spring 2020 | Graduate

Algebraic Topology II

Lecture Notes

Complete lecture notes (PDF - 1.4MB)

Basic Homotopy Theory (PDF) 
1 Limits, Colimits, and Adjunctions
2 Cartesian Closure and Compactly Generated Spaces
3 Basepoints and the Homotopy Category
4 Fiber Bundles
5 Fibrations, Fundamental Groupoid
6 Cofibrations
7 Cofibration Sequences and Co-exactness
8 Weak Equivalences and Whitehead’s Theorems
9 Homotopy Long Exact Sequence and Homotopy Fibers
The Homotopy Theory of CW Complexes (PDF)  
10 Serre Fibrations and Relative Lifting
11 Connectivity and Approximation
12 Cellular Approximation, Obstruction Theory
13 Hurewicz, Moore, Eilenberg, Mac Lane, and Whitehead
14 Representability of Cohomology
15 Obstruction Theory
Vector Bundles and Principal Bundles (PDF)  
16 Vector Bundles
17 Principal Bundles, Associated Bundles
18 I-invariance of BunG, and G-CW Complexes
19 The Classifying Space of a Group
20 Simplicial Sets and Classifying Spaces
21 The Čech Category and Classifying Maps
Spectral Sequences and Serre Classes (PDF) 
22 Why Spectral Sequences?
23 The Spectral Sequence of a Filtered Complex
24 Serre Spectral Sequence
25 Exact Couples
26 The Gysin Sequence, Edge Homomorphisms, and the
27 The Serre Exact Sequence and the Hurewicz Theorem
28 Double Complexes and the Dress Spectral Sequence
29 Cohomological Spectral Sequences
30 Serre Classes
31 Mod C Hurewicz and Whitehead Theorems
32 Freudenthal, James, and Bousfield
Characteristic Classes, Steenrod Operations, and Cobordism (PDF) 
33 Chern Classes, Stiefel-Whitney Classes, and the Leray-Hirsch Theorem
34 H*(BU(n)) and the Splitting Principle
35 The Thom Class and Whitney Sum Formula
36 Closing the Chern Circle, and Pontryagin Classes
37 Steenrod Operations
38 Cobordism
39 Hopf Algebras 
40  Applications of Cobordism 

Course Info

As Taught In
Spring 2020
Learning Resource Types
Lecture Notes
Problem Sets