18.905 | Fall 2016 | Graduate

Algebraic Topology I

Calendar

LEC # TOPICS KEY DATES
I. Singular Homology
1 Introduction: Singular Simplices and Chains  
2 Homology  
3 Categories, Functors, Natural Transformations  
4 Categorical Language  
5 Homotopy, Star-shaped Regions  
6 Homotopy Invariance of Homology  
7 Homology Cross Product Problem set 1 due
8 Relative Homology  
9 The Homology Long Exact Sequence  
10 Excision and Applications  
11 The Eilenberg Steenrod Axioms and the Locality Principle  
12 Subdivision Problem set 2 due
13 Proof of the Locality Principle  
II. Computational Methods
14 CW-Complexes  
15 CW-Complexes II  
16 Homology of CW-Complexes Problem set 3 due
17 Real Projective Space  
18 Euler Characteristic and Homology Approximation  
19 Coefficients  
20 Tensor Product  
21 Tensor and Tor  
22 The Fundamental Theorem of Homological Algebra Problem set 4 due
23 Hom and Lim  
24 Universal Coefficient Theorem  
25 Künneth and Eilenberg-Zilber  
III. Cohomology and Duality
26 Coproducts, Cohomology  
27 Ext and UCT  
28 Products in Cohomology Problem set 5 due
29 Cup Product (cont.)  
30 Surfaces and Nondegenerate Symmetric Bilinear Forms  
31 Local Coefficients and Orientations  
32 Proof of the Orientation Theorem  
33 A Plethora of Products  
34 Cap Product and “Cech” Cohomology  
35 Cech Cohomology as a Cohomology Theory Problem set 6 due
36 The Fully Relative Cap Product  
37 Poincaré Duality  
38 Applications  
Oral Exam during Final Exam Week

Course Info

Instructor
Departments
As Taught In
Fall 2016
Level
Learning Resource Types
Problem Sets
Lecture Notes