Dror Bar-Natan: Classes: 2000-01: Fundamental Concepts in Differential Geometry:

# The Final Exam

There will be a 3-hours long final exam. There will be some choice; you'll have to solve 4 out of 5 questions, or 5 out of 6, or something like that. The questions will be multi-part, with the all but one part of each questions routine and simple (though comprehensive), and the remaining part requiring some creativity. The material taught in this class was important and significant, and thus the main purpose of the exam, as I see it, is to encourage you to review that material and to verify that you've understood the main points. Ergo the emphasis on the comprehensive though routine and simple; the bit about creativity will have relatively small overall weight and will be there mostly to make happy those of you who really do need a challenge.

The exam will take place on Thursday February 22, 2001, at 10AM at Mathematics 110. I hope there will be no need for Moed Beth. I remind you that the final grade will be a weighted average of the final exam grade f and the homework grade h, with weights 0.85f+0.15h if h>f and 0.93f+0.07h if h<f (i.e., the carrot has been bigger than the stick).

In principle, all that was discussed in class or in the problem session or in the homework sheets is fair game. Topics that we discussed with precision, I expect you to know with precision. Topics on which I gave no details (and didn't indicate the details as "easy homework exercises"), you are only supposed to know to the same level. For your convenience, here is a list of the topics covered in the course, drawn on my own notes and on the homework sheets:

 The differential for f:V-->W. The differential matrix. The chain rule. The inverse function theorem, the "jelly cube" principle. The implicit function theorem. Manifolds through atlases. Uniqueness of the extension to a maximal atlas. Functional structures and morphisms between them. Manifolds through functional structures. The equivalence of the two definitions of manifolds. Smooth maps between manifolds. Simple examples: Tn, Sn, RPn, submanifolds, product manifolds. The classification of 1-dimensional manifolds (formulation only). Tangent vectors as equivalence classes of paths. Tangent vectors as derivations. Hadamard's lemma and the equivalence of the two definitions of tangent vectors. The tangent space and its dimension. Pushing forward tangent vectors and the differential. Definitions and examples of immersions, submanifolds, embeddings, submersions. The local structure of immersions. The functional structure induced on an embedded submanifold. The local structure of submersions. Regular values and Sard's theorem. The inverse image of a regular value is an embedded submanifold. The orthogonal group O(m,n) as a manifold. Transversal submanifolds and the local structure of transverse intersections; the "lens spaces" example. The "partition of unity" theorem (formulation only). The "smooth Tietze theorem". The "tangent bundle" TM is a manifold. Existence of proper functions. Closedness of proper functions. Whitney's theorem for compact manifolds into RN. Whitney's theorem for compact manifolds into R2n+1. Whitney's theorem for arbitrary manifolds into RN. Whitney's theorem for arbitrary manifolds into R2n+1. The proof of Sard's theorem. Manifolds with boundary. The classification of 1-manifolds with boundary. The boundary of g-1(y) when the domain manifold has boundary. Sn is not a smooth retract of Dn+1. The smooth approximation theorem where the target is RN. The normal bundle to an embedding in RN. The tubular neighborhood theorem. The smooth approximation theorem where the target is a compact manifold. Sn is not a continuous retract of Dn+1. The Brower fixed point theorem (and simple applications). Smooth maps of a low dimensional manifold into Sn are homotopic to constant maps. Sn is not contractible. Sn is not homemomorhic to Sm for n

Good Luck!!!

The exam itself: Moed A.