**Math 121b (Algebraic Topology)**
**Kiyoshi Igusa**
**Spring 2006/ Spring 2009**

Updated: 07/14/11

What is new: added notes from 2009.

I taught Math 121b in Spring 2006 and Spring 2009 using Hatcher's book. These are partial lecture notes from those two courses. Here is the **Syllabus**.

**Notes**

- Notes from first three weeks (These are notes from 2006.) Sections 1-5:

1. Introduction and examples

1.1. ∆-complex

1.2. ∆-cohomology

1.3. Cohomology of a sphere

2. Universal coefficient theorem

2.1. Reduction to the free resolution case

2.2. Free resolutions and Ext

2.3. UCT

2.4. Problems

3. Cohomology of spaces

3.1. Singular cohomology

3.2. Relative cohomology and long exact sequences

4. Cup product

4.1. Cup product at the cochain level

4.2. Cup product in cohomology

5. Ku ̈nneth formula

5.1. Logic

5.2. Tensor product of chain complexes

5.3. Skew-commutativity of cup product

5.4. Homotopy invariance

5.5. Eilenberg-Zilber

5.6. The Yoneda element in general - Poincare Duality (Incomplete notes from 3/23/09.)

6. Poincare Duality 1

6.1. Manifolds 2

6.2. Orientation 3

6.3. Orientation sheaf 9

6.4. Cap product 11

6.5. Proof for good coverings 15

6.6. Direct limit 18

6.7. Proof of Poincare duality 23

- Homotopy Theory (Notes from 4/20/09.)

7. Homotopy theory 1

7.1. Adjoint maps 1

7.2. Fibrations 5

7.3. Cofibrations 8

7.4. Cell complexes 11

7.5. Cellular approximation 14

7.6. Proof of Key Lemma 17

7.7. Compactly generated topology 20

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