Covering spaces and fibrations. Simplicial and CW complexes, Homology and cohomology, universal coefficients and Künneth formulas. Hurewicz theorem. Manifolds and Poincaré duality.

Hana Jia Kong

HW: Choose 6 questions from the questions after Chap 1 or Chap 2 in the textbook. There are 7 in total.
The last question asks you to prove that categories with coproducts and coequalizers have all colimits. If you pick this question, here is some hint.
1. Intuitively, a pushout (i.e. colimit of a diagram of the shape that we used in class) in spaces is like taking the sum of two things and gluing them by identifying points along a ``common sub''; taking coproducts does the taking the sum part, and taking coequalizers does the gluing. Try to write a pushout as a coequalizer where things in this coequalizer are expressed as coproducts of (one of multiple) things in the pushout.&nbsp;
2. Generalize 1. to arbitrary diagrams.

Homework 1

1. Choose six questions from
Chap 3 Question 1-6, Chap 4 Question 1-2, Chap 6 Question 1-2.
2. Read Chap 4 and 5.
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Homework 2

1. Choose 6 questions from Chapters 6-9 except Question (4) from Chap 9.
2. Read Chap 8 and convince yourself that the dual story with based fibrations works the same.
3. Read Chap 9, especially the part on homotopy groups of pairs. We will use pairs when we talk about homology in later classes.
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Homework 3

1. Choose 6 questions from Chapters 10-13.
2. Read Chap 10-13.

Homework 4

1. Choose 6 problems from 14 -- 16.
2. Read Chapter 14 -- 16.

Algebraic Topology (MATH 231A)

Lecture Notes

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Chapters 1-2

Chapters 3-4

Chapters 6-9

Chapters 10-12

Chapters 13-15

Chapter 17

Problem Sets