Master Seminar in Algebraic Geometry (Morse Theory)

Seminar on Morse Theory

This is a Bachelor/Masters seminar focusing on aspects of topology of manifolds, with goal the weak Lefschetz theorems for algebraic manifolds over the complex numbers, using Morse theory as the main tool. The seminar will meet once a week and the students will give the lectures. We will have an organisational meeting Tues. April 12, 12 Uhr c.t.,  in WSC-N-U-3.04 to distribute the lectures. Here is a provisional lecture plan.

Course Outline

Section I: A review of some topology. [Hatcher]

  • Lecture 1. Basic notions of topology: topological spaces and continuous maps. Simplicial complexes and CW complexes. The cellular approximation theorem. Homotopy of maps and homotopy equivalence.
  • Lecture 2. Some homological algebra. A review of chain and co-chain complexes, maps of complexes, chain homotopy. Homology and cohomology.
  • Lecture 3. Singular homology and cohomology: the singular chain complex, singular homology and cohomology, cellular (co)homology of CW complexes and the comparison theorem.

Section II: A review of manifolds. (mainly Chapter 1)[Warner] plus the statement of the de Rham theorem (Chapter 5)[Warner].

  • Lecture 4. Basic concepts: This lecture will recall the basic notions involved in the theory of differentiable manifolds: definitions, local charts, differentiable mappings of manifolds. The tangent space, the differential of a mapping, immersions and submersions.
  • Lecture 5. Submanifolds, inverse and implicit function theorems. Vector fields. Differential forms. Vector bundles. The tangent and cotangent bundles.
  • Lecture 6. Cohomology of manifolds and the de Rham theorem.

Section III: Morse theory. Mainly from (Part I)[Milnor].

  • Lecture 7. Structure theorem for real quadratic forms (Sylvesters theorem of inertia). index and nullity.
  • Lecture 8. Introduction to Morse theory, definitions and lemmas (Chapter1, sec. 1 and 2)[Milnor]. Critical points, non-degenerate critical points, Hessian matrix. Morse lemma and corollaries.
  • Lectures 9 and 10: Homotopy type in terms of critical values, and some examples (Chapter 1,sec. 3 and 4)[Milnor]. This is the main construction of Morse theory: how to build up a manifold cell by cell using a function with non-degenerate critical points.
  • Lecture 11: Manifolds in Euclidean space (Chapter1, sec. 6)[Milnor].
  • Lecture 12: The weak Lefschetz theorem (Chapter1, sec. 7)[Milnor].
  • Lecture 13: Extensions of weak Lefschetz: The Lefschetz theorem on the fundamental group.

For our source on topology (Lectures 1-3), we will use [Hatcher], for differentiable manifolds (Lectures 4-6) [Warner] and [Spivak]. The section on Morse theory itself will come mainly out of Milnor's book [Milnor].

Bibliography

[Hatcher] Hatcher, Allen. Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN: 0-521-79160-X; 0-521-79540-0

[Warner] Warner, Frank W. Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983. ix+272 pp. ISBN: 0-387-90894-3

[Spivak] Spivak, Michael. Calculus on manifolds. A modern approach to classical theorems of advanced calculus. W. A. Benjamin, Inc., New York-Amsterdam 1965 xii+144 pp.

[Milnor] Milnor, J. Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963 vi+153 pp.