Math 545/6 Reading Guide
Here are some comments on last two chapters of
Introduction to Topological Manifolds (ITM)
that I made in class.
In ITM Chapter 12, you should read at least to the middle of p. 316.
The rest of the chapter is optional. If you would like to see some more
examples of covering spaces, I recommend Classification of Torus Coverings
(Prop. 12.20), Lens spaces (Ex. 12.28), and the Universal Covering Spaces
for (Compact) Surfaces (Thm. 12.29 & 12.30).
The latter involves some complex analysis and hyperbolic geometry.
"Proper Actions" are presented again in ISM and we will make serious study
of them there.
ITM Chapter 13 is completely optional. For a very brief view of some
aspects of homology, now or later, read the introduction, pp. 339-340, and the
section on homology of spheres, pp. 364-369 (skipping the proofs).
Glance through the chapter and read anything else that interests you.
Some comments on reading ISM
- Preface of Introduction to Smooth Manifolds (ISM):
At least read the paragraph on notation on p. vii:
"I should say ... efficiency later."
- Appendix A: You may use either this appendix or ITM when
you need to cite topological results.
There is one new topic, Lipschitz continuity (starting after Exer. A.46,
through Exer. 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.
- Whitney Embedding Theorem section: pp. 133-134 are about
reducing the dimension of the ambient space, read only if interested.
Read Theorem 6.15 and the first part of the proof, 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 17,
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.
- Chapter 9: This chapter is long; for the RReport, stop early if you
get saturated. The proofs of Thms. 9.12 and 9.20, each 2 pages, are
technical and probably are better skipped on first read. Instead,
think about what the results mean. The following are optional:
"Flows and Flowouts on Manifolds with Boundary" (mid p. 222 to mid p. 227)
and "Time-Dependent Vector Fields" and "First order PDEs" (p. 236
through end of chapter).
- Chapter 13: We will not discuss parts of Chapter 13 that are
covered with better tools in Math 547: flat metrics and
the Riemannian distance function.
Read at least the following three chunks of the chapter:
p. 327 through the middle of p. 332,
ending with "Riemannian geometry;"
part of "Riemannian submanifolds": mid p. 333 through p. 334;
and the section on "The Tangent-Cotangen Isomorphism"
on pp. 341-343.
- Chapter 14: If reading this all at once is too much,
you could stop this week after the statement of 14.23 or
at the end of p. 368.
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Math 545/6 Assignments.
Most recently updated on April 8, 2018.