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 |
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