Algebraic Topology II
(Math 612)
Spring 2016
Instructor:
Paul Aspinwall
Credits: 1.00, Hours: 03.0
Time: TuTh 1:25PM - 2:40PM.
Location: Physics 227
Requirements
Prerequisits
Math 611, or consent from me.
Homework
is here.
Exams
There will be a take-home final exam due in on Thursday, May 5 at 10:00 PM.
Synopsis
A rough outline is as follows
- Cohomology
- Some Homological Algebra
- Singular Cohomology
- Universal Coefficients Theorem (for both cohomology and
homology)
- De Rham Cohomology (fairly briefly)
- The cup and cap product
- The Künnuth Formula
- Poincaré duality
- Spectral sequences
- Presheaves and Čech cohomology
- Equivalence of Čech, singular, and De Rham cohomology
- The Leray-Serre spectral sequence for fibre bundles
- Group Cohomology and the Cartan-Leray spectral sequence
- Further homotopy
- Long exact sequence for fibre bundles
- The path fibration
- The Hurewicz Theorem
- Eilenberg MacLane spaces
- Computation of some homotopy groups for spheres
Textbooks
The course will be based on two texts:
- A. Hatcher, Algebraic Topology I, available over
the web (but
fairly inexpensive anyway).
- R. Bott and L.W. Tu, Differential Forms in Algebraic
Topology, Springer-Verlag 1982.
where Hatcher will be used for cohomology. From then on, we will
tend more to follow Bott and Tu.
It may also be useful to refer to
- Edwin H. Spanier, Algebraic topology,
Springer-Verlag 1966.
- Kenneth S. Brown, Cohomology of Groups,
Springer-Verlag 1982.
- Charles A. Weibel, An Introduction to Homological
Algebra, Cambridge 1994.
- Saunders MacLane, Homology, Springer 1995 (reprint of 1975).
- James R. Munkres, Topology: A First Course,
Prentice Hall, 1974.
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