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