Advanced Geometry 2

Course for UniTS master students and joint SISSA/UniTS diploma 

Aim of the course is to introduce the basic tools of smooth manifolds and differential topology.

Sissa page of the course link.

Main references:

• Introduction to Smooth manifolds, by John M. Lee.

• Lecture notes on basic differential topology, by A. Lerario

Note: the reference by Lee is huge and hyper-detailed. We did not follow it uniformly for what concerns the level of detail (some proofs were left as exercise, some readapted or simplified at the price of lesser generality). A selection has to be made, but you should have at the end of the course a solid working knowledge of smooth manifolds and basic differential topology.

Syllabus (AA 2023/2024):

From the book Introduction to Smooth manifolds, by John M. Lee, FIRST EDITION:

• Chapter 1. Smooth manifolds. Skip connectivity and fundamental groups). Problems 1-1, 1-2, 1-5.

• Chapter 2. Smooth Maps. (we assumed continuity in definition of smooth map to simplify exposition). Only the definition of Lie groups. Skip smooth coverings and  proper maps (but you should know the definitions). Partition of unity: only statement of existence (thm 2.25, no proof), existence of bump functions (prop. 2.26), and extension lemma 2.27.

• Chapter 3. Tangent Vectors. We did not treat in detail the tangent space of manifolds with boundary. We only mentioned with no details the alternative definitions of tangent spaces.

• Chapter 4. Vector Fields. skip vector fields on manifolds with boundary, Lie algebra of a Lie group.

• Chapter 5. Vector Bundles. (skip categories and functors.

• Chapter 6. The Cotangent Bundle. Skip line integrals and conservative vector fields.

• Chapter 7. Submersions, Immersions, and Embeddings. Skip the proof of inverse/rank/implicit function on R^n which was given for granted.

• Chapter 8. Submanifolds. Skip submanifolds of manifolds with boundary, skip Lie subgroups.

From the book Introduction to Smooth manifolds, by John M. Lee, SECOND EDITION:

• Chapter 6. Sard's theorem. Sets of measure zero, Sard Theorem, Whitney embedding theorem, Whitney approximation theorem, Transversality. This chapter contains some of the most importan theorems in the course.

• Chapter 12. Tensors. Skip Lie derivative of tensor fields. Lemma 12.24 is not necessary. Be aware of different notation for pullback (the book uses the notation dF^*_p for the "pointwise pullback" which we never used during the course and may be confusing)

• Chapter 14. Differential forms. Note that we used Einstein convention so, when dealing with k-forms, we sum over non-ordered indices. Keep this in mind when comparing with the notes. Skip sections: Exterior Derivatives and Vector Calculus in R3 and Lie Derivatives of Differential Forms.

• Chapters 15/16. Orientations and Integration on manifolds. In class we actually followed a more succinct presentation as for example in section 4.9-4.10 of A.  Lerario's lecture notes, which is more streamlined than Lee's chapters 15/16 (second edition).

• Chapter 17. De Rham Cohomology. Ex. 17.29 done in class (Compactly Supported Cohomology of R^n). We skipped the proof of Thm. 17.30, 17.31, 17.32, 17.34 on top cohomology in orientable and non-orientable case. Skip degree theory.

• Miscellanea: Poincarè duality and Kunneth theorem(see e.g sections 4.11 and 4.12 of Lerario's lecture notes)

Exercises:

List of exercises solved during the exercise session (taken from Lerario's lecture notes). 86, 87, 88, 89, 29, 45, 47, 78, 53, 54, 57, 58, 59, the absolute value is not an immersed manifold, embedding of product of spheres, 162, 167, 171, 172, 173.

You should be able, at least in principle, to do all exercises in the book by J. Lee corresponding to the chapters/sections listed above. This is not required for passing the exam, but it is an exhaustive collection of problems on the material we have done in class.

Exam:

Written exam (around 3/4 hours) with a selection of exercises/problems and possibly a few quick questions from the theory.

Oral exam (optional, the mark can increase or decrease): correction of the written exam + questions from the theory (with proofs of main theorems).