MATH 86600
Math 82900 Structures on Manifolds: Fall 2022
I will regularly update the following two items, depending on the material
presented in class.
Course Topics:
- Sept 16th: More on global existence of almost complex structures. First
on surfaces, with discussion of Steifel Whitney class w1 obstructing
orientability. Identification of the twistor bundle of a surfaces as the
orientation double cover. Then an overview of dimensions 4 with
statement of Wu's Theorem regarding obstructions in dimension 4 from signature and Euler characteristic. The universal viewpoint of lifting classifying maps
from BSO(2n) to BU(n). Bockstein homomomorphism and lifting to integral
classes introduced. Homotopy groups of fibration and the examples of the
7-sphere fibiring over CP^3 with fiber the circle. Homotopy groups of CP^3
in low degree.
- Sept 9th: The triality of 2-tensors on smooth manifolds: inner products, non-degenerate
alternating 2-forms, operators of square minus the identity. The question
of existence and uniqueness either pointwise (linear algebra), locally, or
globally. The space of complex structures on a real even dimensional vector
space, orthogonal or not, first in low dimension, then a general viewpoint
as homogeneous spaces fitting in iterated fibrations over spheres.
- Sept 2nd: no class (Labor day)
- Aug 26th: differential forms on manifolds, de Rham complex, metrics,
Poincare duality, Hodge theorem.
References:
- "Complex Geometry" Huybrechts.
- "Cohomological Aspects in Complex Non-Kahler Geometry" Angella.
- "Compact Complex Surfaces" Barth.
- "Hodge Theory and Complex Algebraic Geometry" Voisin.
- "Rational Homotopy Theory" Griffiths and Morgan.
- "Fiber Bundles," Husemoller
- General Background: "Introduction to Smooth Manifolds" Lee.
Course Description:
This course is devoted to the interplay between algebraic topology and geometric structures on differentiable manifolds. By algebraic topology we mean, at least, the cohomology and rational homotopy theory, both of which are accessible via the de Rham complex of differential forms. By geometric structures we mean the triality of 2-tensors given by 1] Riemannian metrics, 2] (almost) complex structures, and 3] (almost) symplectic forms. Finally, by "interplay", we are particularly interested in the topological implications of a manifold possessing such a geometric structure.
The case of metrics has numerous classical results, some of which are discussed in first-year courses, while the latter two cases are more recent fertile grounds and areas of active research. The course will be a self contained treatment of topics in these latter two cases, along with their interactions with some metric discussions. We'll develop tools from algebraic topology as needed, such as characteristic classes, index theory, rational homotopy, Morse theory, the theory of bicomplexes, and various supersymmetry algebras that are represented on the de Rham complex in the presence of such a geometric structure. Prominent examples we explore include Kahler manifolds, complex surfaces and symplectic 4-folds, all of which possess some striking properties. Finally, the course will bring students up to speed on several open problems and directions for future research.
General Information:
- Location: Fridays, 2-4, Room 5382.
- Grading: The final grade will be based on class participation.
Contact Information:
- Instructor: Scott Wilson
- Email: scott dot wilson AT qc.cuny.edu
- Office hours: by appointment.