Math 225A: Differential Topology
Fall 2009
Time and Place: MWF 11-11:50 pm in MS 5148
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Instructor: Ciprian Manolescu
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E-mail: cm@math.ucla.edu
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Office: MS 6921
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Office Hours: Mon 10-11am, Tue 1:30-2:30pm, and by appointment
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- Section: Tue 11-11:50am in MS 5148
- Teaching
Assistant: Dustin Steinhauer
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E-mail: dsteinha@math.ucla.edu
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Office Hours: Mon 2pm in MS 3969
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Web page: http://www.math.ucla.edu/~cm/225a.html
Prerequisites: Real analysis in several variables (e.g.
the implicit function theorem) and point set topology.
Topics to be covered:
Manifolds, tangent vectors, smooth maps, tangent bundles and vector bundles
in general, Sard's theorem on the measure of
critical values, embedding theorems, vector fields and integral curves,
Ehresmann's fibration theorem,
transversality, degree theory, Lefshetz
fixed-point theorem, Euler characteristic.
Textbooks:
- Victor Guillemin and Alan Pollack,
Differential Topology, Prentice Hall, Inc., 1974.
- Michael Spivak, A Comprehensive Introduction to Differential
Geometry,
Vol. I, Third Edition, Publish or Perish, Inc., 2005.
We will cover roughly Chapters 1-3 from Guillemin and Pollack, and
Chapters 1-3 and 5 from Spivak. We will not follow either book very
closely, so it is important to attend the lectures or get the notes
from another student.
Other recommended books:
- Michael Spivak, Calculus on Manifolds, Perseus Books, 1965.
- John M. Lee, Introduction to Smooth Manifolds, Springer-Verlag,
2003.
- John W. Milnor, Topology from the Differentiable Viewpoint,
Princeton
University Press, 1997.
- James R. Munkres, Elementary differential topology, Princeton
University Press, 1966.
- Morris W. Hirsch, Differential Topology, Springer-Verlag, 1976.
- Frank W. Warner, Foundations of Differentiable Manifolds and Lie
Groups, Springer-Verlag, 1983.
- Theodor Bröcker, Klaus Jänich, Introduction to differential
topology, Cambridge University Press, 1982.
- Bjorn Ian Dundas, Differential Topology, 2009, available online.
Grading: 50% homework, 50% final.
Homework: Homework will be assigned every other week and
will be due the following Wednesday. The homework assignments will be
handed out in class and will also be posted on the web page. You must hand
in the homework in class each Wednesday. You are encouraged to talk about
the problems with other students, but you should write up the solutions
individually. You should acknowledge the assistance of any book, student
or professor.
Exam: The final exam will be on
Wednesday,
December 9, 8am-11am.