Professor Stefan Papadima

MTH G375--Topics in Topology:
Rational Homotopy Theory

Spring 2006

* Course Information

Course:   MTH G375 -- Topics in Topology: Rational Homotopy Theory
Web site:   http://www.math.neu.edu/papadima/top06.html
Instructor:   Stefan Papadima, Stone Visiting Professor
Time and Place:   Mon 4:45pm-6:15pm and Wed 11:45am-1:15pm, in 5 KA
Office Hours:   Mon and Wed 3:00-4:00pm, in 445 LA
Prerequisites:   MTH G121 Topology 1, MTH G221 Topology 2
Textbook:   Rational Homotopy Theory and Differential Forms, by Phillip A. Griffiths and John W. Morgan, Progress in Mathematics, vol 16, Birkhäuser, Boston, MA, 1981
Supplement:   Rational Homotopy Theory, by Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Graduate Texts in Mathematics, vol 205, Springer-Verlag, New York, 2001
Grade:   Based on problem sets and class participation

* Course Description

"The manner in which a closed differential form vanishing in the cohomology of a manifold actually becomes exact contains homotopy information." Informally speaking, Rational Homotopy Theory may be viewed as an elaboration of the above principle. More formally, the theory provides a categorical description of homotopy types of reasonable spaces, modulo torsion, by differential graded algebra models. The course will give an introduction to several basic topics in RHT, such as:

  • De Rham algebras and de Rham theorems for simplicial complexes.
  • Homotopy theory of fibrations, spectral sequences, and Postnikov decompositions of simply-connected spaces.
  • Rationalization of simply-connected spaces.
  • Minimal models of topological spaces and continuous maps; algebraic computation of higher rational homotopy groups (in the simply-connected case).
  • (Pro)-nilpotent groups and Lie algebras; algebraic computation of the rational completion of the fundamental group of a connected space.

Depending on time and the interests of the audience, further applications of Rational Homotopy Theory may also be sketched: existence of isometry-invariant geodesics on simply-connected manifolds and/or properties of fundamental groups of connected, smooth, complex algebraic varieties.