Math 104: Topology, Spring 2006

Weekly plan ten, for Seminar on Tuesday, March 28

Reading

Read Munkres §80—82 and Hatcher §1.3. At this point some of you may have already read a good deal of the Hatcher section, but I advise all of you to read it from beginning to end. Except for the examples, most of Hatcher is in Munkres, but not in the same order.

Presentations

Jonah

Section 80 is all about "why?" Why is the universal covering space called universal? Why should we care about universal coverings? Why do we have to live with the annoying notion of semi-local simple connectivity? Give us pithy answers to these and any other pertinent questions.

Michael

Section 81 corresponds pretty closely to the two and a half pages of Hatcher starting with the title "Deck transformations and group actions." The language in Hatcher differs in small ways from that in Munkres, but the basic facts presented are the same in both. Make sure we all understand the language, the small differences and the basic facts. Doing all this may leave very little time to discuss the details of proofs. If so, it is fine to omit them. We can discuss any necessary details after your presentation.

Zach. M.

Here is the Main Theorem on covering spaces. In Munkres it is 82.1. In Hatcher it is 1.36. Make sure we understand the theorem and the idea of the proof. Is Hatcher's proof "the same" as Munkres'? If not, what is the fundamental difference? If so, what are the lacunae which allow Hatcher's proof to take less than half a page where Munkres takes four?

Paul

The X_m,n spaces are a running example in Hatcher. They are discussed in the context of torus knots on page 47, as an early example of a space to be covered on pages 65—66, and again on page 76. Make some sense out of some or all this. Just an explanation of one or two bits, such as the assertion on page 76 that "X_m,n has a cell-structure with two vertexes, three edges, and one two cell," could make a great part of a presentation. However if you understand more or less, that is ok, just present something you come to understand.

Common Problems

Higher Priority

Handcrafted problems

1. Let X and Y be surfaces each with one hole. (Cf exercise 78.4.) Let f:S¹ -> X and g:S¹ -> Y be homeomorphisms of the circle onto the boundary components of these surfaces. Let g':S¹ -> Y be the homeomorphism of S¹ onto the boundary component of Y which is obtained by precomposing g with the antipodal map. Define X#Y to be the adjunction space formed by identifying f(t) with g(t) in the disjoint union of X and Y. Define X#'Y to be the adjunction space formed by identifying f(t) with g'(t) in the disjoint union of X and Y. Prove that X#Y is homeomorphic to X#'Y. Can you do this without the classification theorem?
2. Let RPⁿ be the quotient of Sⁿ by the antipodal map. Show that S²×RP³ and S³×RP² both cover a single space, X in such a way that the corresponding subgroups of the fundamental groups of X are isomorphic. (Later we shall prove that S²×RP³ is not homeomorphic to S³×RP².)

Munkres

§80: 1a.
§81: 1, 2, 3, 4 (cf H.1.3.23), 5.
§82: 1, 2.

Hatcher

§1.3: 4, 10, 12, 14, 30, 31.

Medium Priority

Munkres

§80: 1b.
§81: 6.

Hatcher

§1.3: 6.

Food

Toon will bring the food for seminar break.