Courses:

Seminar in Topology >> Content Detail



Study Materials



Readings

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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 #TOPICSREADINGS
1Organizational Meeting
2n-manifolds and Orientability, Compact, Connected 2-manifoldsLecture 1

Massey: Chapter 1, Sec. 2-3

Lecture 2

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

Massey: Chapter 1, Sec. 5

Lecture 4

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

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

Lecture 6

Massey: Chapter 1, Sec. 8
5Review of Group Theory, Homotopy and the Fundamental GroupLecture 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
6The Fundamental Group (cont.), Homotopy Equivalence and Homotopy TypeLecture 9

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

Lecture 10

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

Massey: Chapter 2, Sec. 5

Lecture 12

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

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

Lecture 14

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

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

Lecture 16

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

Massey: Chapter 4, first half of Sec. 2

Lecture 18

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

Massey: Chapter 4, Sec. 3

Lecture 20

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

Massey: Chapter 4, second part of Sec. 5

Lecture 22

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

Massey: Chapter 5, Sec. 2

Lecture 24

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

Massey: Chapter 5, Sec. 4

Lecture 26

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

Massey: Chapter 5, Sec. 6

Lecture 28

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

Massey: Chapter 5, Sec. 8

Lecture 30

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

Massey: Chapter 5, Sec. 10

Lecture 32

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

Massey: Chapter 6, Sec. 2-3

Lecture 34

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

Massey: Chapter 6, Sec. 6-7

Lecture 36

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

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

Lecture 38

Hatcher: Chapter 2, Sec. 1 (pp. 105-107)
21Singular 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)
22Degree of a Map and its Applications, Higher Homotopy GroupsLecture 41

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

Lecture 42

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

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

 








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