Algebraic Topology II

Calendar

LEC #	TOPICS	KEY DATES
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 Transgression
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