18.906 | Spring 2020 | Graduate

Algebraic Topology II


Basic Homotopy Theory
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 Problem set 1 due
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  
10 Serre Fibrations and Relative Lifting  
11 Connectivity and Approximation  
12 Cellular Approximation, Obstruction Theory Problem set 2 due
13 Hurewicz, Moore, Eilenberg, Mac Lane, and Whitehead  
14 Representability of Cohomology  
15 Obstruction Theory  
Vector Bundles and Principal Bundles  
16 Vector Bundles  
17 Principal Bundles, Associated Bundles  
18 I-invariance of BunG, and G-CW Complexes Problem set 3 due
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 
22 Why Spectral Sequences?  
23 The Spectral Sequence of a Filtered Complex  
24 Serre Spectral Sequence Problem set 4 due
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 Problem set 5 due
30 Serre Classes  
31 Mod C Hurewicz and Whitehead Theorems  
32 Freudenthal, James, and Bousfield  
Characteristic Classes, Steenrod Operations, and Cobordism 
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 Problem set 6 due
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