MATH 86600
Math 82900 Structures on Manifolds: Fall 2022
Here is a list of material
presented in the course.
Course Topics:
- Dec: 9th: Several examples of Hodge diamonds for compact complex surfaces;
determination of Hodge numbers from oriented topology, complete zig-zag
structures. Example of compact 4-manifold with almost complex structure but
no complex structure (argument with rational homotopy theory). Some remarks
in dimension 6 on index theory and examples (Calabi-Eckmann manifold). Open
problems. Remarks on linking pairings and secondary-index-theory.
- Dec. 2nd: Index theory, Riemann-Roch, Hirzebruch signature theorem,
application to compact complex surfaces. Determination of zigzags and Hodge
numbers from oriented topology of compact complex surfaces.
- Nov. 18th: (We had some prospective students visiting the program, so the
first part of the class was an elementary discussion of Lie groups and Lie
algebras, and how they fit in the course.) Topological restrictions on
structure of bicomplex of forms of complex manifolds. Dimension, connected
components (max principle), Stokes' theorem and bidegree (n,0), relation to
degeration of Frolicher spectral sequence. Duality of
cohomological functors. Statement of results in 4d to be done next class.
- Nov. 11th: More on bicomplexes: even zig-zags and degeneration, odd
zig-zags and pure hodge structures, notion of E1-isomorphism and relation to
multiplicities of zig-zags. Bott-Chern and Aepppli cohomologies,
applications to complex manifolds: Compactness implies finitely many
zig-zags, symmetry under conjugation, Serre duality and its implications.
- Nov. 4th: Structure of bicomplexes (a la Stelzig) and relation to
ddc-condition. Computation of various cohomologies on zigzags types,
including total cohomology, delbar, del, and first pages of spectral sequence.
- Oct. 28th: The dc-diagram of a complex manifold. ddc-condition implies
formality (revised proof, defn,and examples). Massey products, example of
non-zero massey product on KT. Remarks on other 4-manifolds and Iwasawa
manifold. Minimal models and rational homotopy (statement of some results of Sullivan).
- Oct. 21st: Continued example of Kodair-Thurston (KT) manifold: example of
complex structure, a symplectic structure with compatible J that is not
integrable (explicit calculations). Definition of Kahler (and it
generalizations), examples. Discussion of differential operators in these
contexts. Statement of Kahler identities, derived Laplacian identities, odd Betti numbers are
even (so KT has no Kahler structure). ddc-condition, verification for compact
Kahler manifolds.
- Oct. 14th: Complex manifolds, statement of Newlander-Nirenburg, Nijenhuis
tensor, identification with components of d via dualization (and also N as
measure of failure of lie-subalgebra of vector fields). Introduction of
Kodaira-Thurston nilmanifold, cohomology ring structure, symplectic
structure and compatible J.
- Oct. 7th: Almost complex manifolds, background linear algebra, induced
structure on differential forms (bigraded algebra and components of exterior
d), brief statement of integrable case, Dolbeault cohomology and delbar-Hodge Theorem.
- Sept 30th: Complex projective spaces, cell structure, cohomology ring,
tautological bundle, Chern classes, computation for tangent bundle of CPn,
Pontryagin classes, relations for complexfied bundles (p1=c1^2-2c2,...). Retry
homework problem.
- Sept: 23rd: Cochain complexes, additive decompositions, Kunneth formula,
Euler characteristic, signature of 4k manifolds, connected sums and
cohomology/signature, examples. Stated homework problem with first steps towards:
constructing a closed 4-manifolds which has an almost complex structure but
no complex structure.
- 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 surface 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.
- "Algebraic models in Geometry" Felix, Oprea, Tanre.
- "Cohomological Aspects in Complex Non-Kahler Geometry" Angella.
- "On the structure of
double complexes" Stelzig.
- "Compact Complex Surfaces" Barth.
- "Hodge Theory and Complex Algebraic Geometry" Voisin.
- "Rational Homotopy Theory" Griffiths and Morgan.
- "Fiber Bundles," Husemoller
- "Hodge-de Rham numbers of almost complex 4-manifolds" Cirici, Wilson.
- 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.