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

Learning Resource Types

assignment Problem Sets
notes Lecture Notes