Math 545/6 Reading Guide

Here are some suggestions for reading Introduction to Smooth Manifolds.

• Preface: at least read the paragraph on notation on p. vii: "I should say ... efficiency later."
• Appendix A: No need to read this, but note that it can serve as a summary of Autumn quarter (at least the parts we need for the rest of the year - for the prelim, more on computing fundamental groups and covering spaces is needed). You may use either this appendix or ITM when you need to cite topological results. There is one result we skipped that we will need to cite at least once, the Baire Category Theorem (A.58). There is also one new topic, Lipschitz continuity (starting after Exer. A.46, through Exercise A.49) which you may consult when and if needed.
• Appendices B & C: Look these over, if any section doesn't seem immediately and completely obvious, read more carefully and do the exercises. Exceptions: You may skip Proposition B.57 and its proof, and may skip the proofs of the Inverse and Implicit Function Theorems. You may postpone the section on multiple integrals until we get to Chapter 16.
• Appendix D: Consult as needed in Chapter 9 and later.
• Chapter 1: There is little or nothing new for us in pp. 3-10, so you may skim this quickly. Starting at the bottom of p. 10, read as usual: Your first read may skip details, but be sure to work through all proofs and exercises sometime soon after that.
• Chapter 6: For many of the results in this Chapter, we will have occasion to cite the results, but the techniques used in the proofs will not be important for us. Here is a guide to what you should take away from this chapter.
  • From the first section "Sets of Measure Zero," you should understand that the concept of a set of measure zero is invariant under diffeomorphism, so makes sense on manifolds.
  • Know the statement of Sard's Theorem, Theorem 6.10. OK to skip proof.
  • Know that the Whitney Embedding Theorem, Theorem 6.15, says every smooth manifold can be embedded in some Euclidean space, and understand the proof for the case of a compact manifold. OK to skip rest of proof and variations of the theorem.
  • Whitney Approximation Theorems: The one for functions, Theorem 6.21, and its proof are so basic that it sometimes appears on the prelim. (The expectation is that students will prove it from scratch, not quote the result.) We will need some of the other approximation theorems in Chapter 21, but you can postpone reading them until we need them.
  • Read the subsection on Tubular Neighborhoods (pp. 137-141) and the section on Transversality (pp. 143-147). These concepts and proof techniques are more central to our course than the rest of the chapter.

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