This is an archived course. A more recent version may be available at ocw.mit.edu.

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Readings

The primary book for the course is:

Amazon logo Massey, W. S. "Algebraic Topology: An Introduction." Graduate Texts in Mathematics. Vol. 56. New York, NY: Springer-Verlag, 1977. ISBN: 0387902716.
Students should note that there are two other books in the GTM series (GTM 70 and GTM 127) by Massey, which are different books.

The material covered in this course is also contained in the beginning of:

Amazon logo Hatcher, Allen. Algebraic Topology. Cambridge, UK: Cambridge University Press, 2001. ISBN: 0521795400 (paperback.)
This is an excellent text with many examples and pictures, and it can by found online at Allen Hatcher's Homepage.

The text by Massey is our primary source because it spends more time on the material we plan on covering, and gives a very careful exposition.

Finally, for instructions on how to present a lecture for this class, please read Guidelines for Lectures (PDF)

SES # TOPICS READINGS
1 Organizational Meeting  
2 n-manifolds and Orientability, Compact, Connected 2-manifolds Lecture 1

Massey: Chapter 1, Sec. 2-3

Lecture 2

Massey: Chapter 1, Sec. 4
3 Classification Theorem for Compact Surfaces, Triangulation Lecture 3

Massey: Chapter 1, Sec. 5

Lecture 4

Massey: Chapter 1, Sec. 6, and "Step 1" of the proof in Sec. 7
4 Classification Theorem for Compact Surfaces (cont.), Euler Characteristic Lecture 5

Massey: Chapter 1, the rest of section Sec. 7

Lecture 6

Massey: Chapter 1, Sec. 8
5 Review of Group Theory, Homotopy and the Fundamental Group Lecture 7

Review group theory using the notes on Basic Group Theory (PDF), or the online Group Theory notes by J. S. Miline

Lecture 8

Massey: Chapter 2, Sec. 2, 3
6 The Fundamental Group (cont.), Homotopy Equivalence and Homotopy Type Lecture 9

Massey: Chapter 2, end of Sec. 3, and Sec. 4

Lecture 10

Massey: Chapter 2, Sec. 8
7 The Fundamental Group of a Circle, Retracts, Brower Fixed-Point Theorem Lecture 11

Massey: Chapter 2, Sec. 5

Lecture 12

Massey: Chapter 2, part of Sec. 4 after thm 4.1, Sec. 6
8 Weak Product of Groups, The Fundamental Group of a Torus, Free Abelian Groups Lecture 13

Massey: Chapter 3, Sec. 2, and Chapter 2, Sec. 7

Lecture 14

Massey: Chapter 3, Sec. 3
9 Free Products, Free Groups, Presentations of Groups Lecture 15

Massey: Chapter 3, Sec. 4, Sec. 5

Lecture 16

Massey: Chapter 3, Sec. 6, maybe a bit from Sec. 5
10 Siefert-Van Kampen Theorem and its Generalization Lecture 17

Massey: Chapter 4, first half of Sec. 2

Lecture 18

Massey: Chapter 4, second part of Sec. 2
11 Applications of the Siefert-Van Kampen Theorem, Structure of the Fundamental Group of a Compact Surface Lecture 19

Massey: Chapter 4, Sec. 3

Lecture 20

Massey: Chapter 4, Sec. 4, beginning of Sec. 5
12 Fundamental Groups on Closed Surfaces, Application to Knot Theory Lecture 21

Massey: Chapter 4, second part of Sec. 5

Lecture 22

Massey: Chapter 4, Sec. 6
13 Covering Spaces, Path Lifting Lemma, Homotopy Lifting Lemma Lecture 23

Massey: Chapter 5, Sec. 2

Lecture 24

Massey: Chapter 5, Sec. 3
14 Fundamental Group of a Covering Space, Lifting of Arbitrary Maps to a Covering Space Lecture 25

Massey: Chapter 5, Sec. 4

Lecture 26

Massey: Chapter 5, Sec. 5
15 Homomorphisms and Isomorphisms of Covering Spaces, Action of the Fundamental Group on Fibers of Covering Spaces Lecture 27

Massey: Chapter 5, Sec. 6

Lecture 28

Massey: Chapter 5, Sec. 7
16 Regular Covering Spaces and Quotient Spaces, Borsuk-Ulam Theorem for the 2-sphere Lecture 29

Massey: Chapter 5, Sec. 8

Lecture 30

Massey: Chapter 5, Sec. 9
17 The Existence Theorem for Covering Spaces, Induced Covering Space over a Subspace Lecture 31

Massey: Chapter 5, Sec. 10

Lecture 32

Massey: Chapter 5, Sec. 11
18 Graphs, Trees, Fundamental Group of a Graph Lecture 33

Massey: Chapter 6, Sec. 2-3

Lecture 34

Massey: Chapter 6, Sec. 4-5
19 Euler Characteristic and Coverings of Graphs, Generators of Subgroups of Free Groups Lecture 35

Massey: Chapter 6, Sec. 6-7

Lecture 36

Massey: Chapter 6, Sec. 8
20 Delta Complex, Singular Chains, Homology Lecture 37

Hatcher: Chapter 2, Sec. 1 (pp. 102-105)

Lecture 38

Hatcher: Chapter 2, Sec. 1 (pp. 105-107)
21 Singular Homology, The Homomorphism pi_1(X) -> H_1(X) Lecture 39

Hatcher: Chapter 2, Sec. 1 (p. 108 - top of p. 110)

Lecture 40

Hatcher: Chapter 2, appendix A (pp. 166-168)
22 Degree of a Map and its Applications, Higher Homotopy Groups Lecture 41

Hatcher: Chapter 2, Sec. 2 (pp. 134-135)

Lecture 42

Hatcher: Chapter 4, Sec. 1 (pp. 340-343)
23 Cell Complex, Whitehead's Theorem Lecture 43

Hatcher: Chapter 0, Sec. 1 (pp. 5-6), bit from Chapter 4
24 Presentations of Final Projects  
25 Presentations of Final Projects (cont.)