Math 843 | Algebraic Topology

Winter 2009
Mon: 12pm–1:30pm, Jeff 116
Wed: 9am–10:30am, Jeff 422

Algebraic topology is one of the major mathematical inventions of the twentieth century, and has had a profound influence on analysis, algebra, topology, number theory, and geometry.

This course is a reading course in algebraic topology. Its aim is to introduce the three main algebraic functors: the fundamental group, homology, and cohomology, and to see some of their applications.

The textbook for the course is the terrific book Algebraic Topology by Allen Hatcher. The book is available online from the author's homepage, or through Cambridge University Press. The lectures will be given by the participants, and attention to both the mathematical and expository details of the lectures is expected. The preliminary schedule below is likely to change as the course progresses.

1. The Fundamental Group

Homotopy of loops
Date Topic Speaker Chapter
Jan 19 Some underlying geometric notions. Jenny 0
21 Paths and homotopy, induced homomorphisms. Anika 1.1
26 The fundamental group of the circle and applications. Andrew 1.1
28 Van Kampen's theorem. Randy 1.2
Feb. 2 Applications to cell complexes. Jenny 1.2
4 Covering spaces, homotopy lifting. Nathan 1.3
9 The Classification of covering spaces. Emily 1.3
11 Deck transformations and group actions. Anika 1.3
23 Graphs and free groups. Emily 1.A

2. Homology

Barycentric subdivision
Date Topic Speaker Chapter
Feb. 25 Delta-complexes, simplicial and singular homology. Nathan 2.1
Mar. 2 Homotopy invarience, the fundamental group and H1, exact sequences and relative homology. Emily 2.1, 2.A
4 Excisions, equivalence of singular and simplicial homology. Andrew 2.1
9 Degrees of self-maps of spheres, CW-complexes, cellular homology. Randy 2.2
11 Homology of closed surfaces, projective spaces, lens spaces. Jenny 2.2
16 Euler Characteristic and Mayer-Vietoris sequence Anika 2.2
18 Coefficients in homology. Axiomatic homology, categories and functors, delta sets. Nathan 2.2–2.3
23 Classical applications of homology. Andrew 2.B
25 Simplicial approximation and the Lefschetz fixed-point theorem Randy 2.C

3. Cohomology

Poincare duality
Date Topic Speaker Chapter
Mar. 30 The universal coefficient theorem, cohomology, and the Ext functor. Randy 3.1
Apr. 1 Cup product and the cohomology ring Emily 3.2
6 The Künneth formula (restricted version) Nathan 3.2
8 Orientability and orientation of manifolds. Anika 3.3
13 Poincaré duality and cohomology with compact support. Jenny 3.3
15 Poincaré duality and the cup product. Andrew 3.3

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