Homework for Math 435: Topology (Fall 2006)

Group Assignments

κ   Kristin McNamara, Duane Mueller, and Catie Schwartz
λ   Derek Cole, Matt Lowman, and Sara Toosarvandani
δ   Matthew DiGiosaffatte, Jessica Rice, and Josh Stowers
ρ   Jorge Bruno, Nhat Nguyen, and Eddie Rhodes
φ   Laurel Booth, Shannon Mercadante, and Brandon Strawn

Homework Policy

I've written short verbal descriptions of each assigned problem below, with the hope of illuminating the "big picture purpose" of each problem. By the end of the semester this should be a nice list representing what you have learned, and hopefully it will make it easier for you to look up problems when studying for exams.

Homework List

Chapter 1: Deformations

* read the section and work through all examples
2ab. every equivalence relation defines a partition
5. relations that are not equivalence relations
-- TYPO: change "|a - b| < 1" to "|a - b| ≤ 1"
6. more non-equivalence relations
-- find one example of each type listed in parts (a), (b), and (c) of number 5
-- BONUS: let the universe set be X = P(S), the set of subsets of some set S
7. is relexivity guaranteed by symmetry and transitivity?
9. an equivalence relation on polynomials
11. sets of measure zero and almost-everywhere-equal functions

Presenting Thursday, 8/31:
Σ: 6   ε: 2ab, 11   μ: 5   ρ: 7   φ: 9 (note Section 1.2 is also due on 8/31; see assignment below)

* read the section and work through all examples
-- TYPO: In Definition 1.15, the identity function should map from X to X, not from X to x as is written.
-- General note for all problems in this section: Don't worry about finding equations; just give geometric/picture descriptions.
2. a bijection from (0,1) to [0,1]
5. a bijection from a cylinder to an annulus
7. a bijection from a punctured sphere to the plane
10. a bijection from lines in 3-space to a subset of the sphere
16. identifying a square region to make a cylinder
17. identifying a square region to make a torus

Presenting Thursday, 8/31:
Σ: 10, 16   ε: 2   μ: 5, 17   ρ: 7, 10   φ: 7, 17

* read the section and work through all examples
1. constant functions are continuous
2. compositions of continuous functions are continuous
5. finding a stretching factor bound
6. using a stretching factor bound to prove continuity
9. shrinking functions are continuous
11. continuity under the discrete metric


Presenting Tuesday, 9/5:
Σ: 2, 6   ε: 5, 11   μ: 6, 9   ρ: 2, 11   φ: 1, 11

* read the section and work through all examples
1. homeomorphism is an equivalence relation
2. [0,1) is homeomorphic to [0,∞)
7. constructing a piecewise-linear homeomorphism
-- note you will need Theorem 1.27
10. an open disk is homeomorphic to the plane
11. a punctured disk is homeomorphic to a disk with an open hole of radius 1
13. topological definitions of circle, torus, and theta curve
15. [0,1) x [0,1) is homeomorphic to [0,1] x [0,1)


Presenting Thursday, 9/7:
Σ: 2, 10, 13   ε: 1, 7, 15   μ: 7, 11, 13   ρ: 1, 7, 11   φ: 2, 10, 15

* read the section and work through all examples
-- Notice that the first two homework problems below ask you to present examples from the reading. Work out these examples so that you can explain them to the class in your own words, and with more details/pictures than the book's writeup.
A. Example 1.47: the path components of R² - S¹
B. Example 1.49: (0,1) and [0,1) are not homeomorphic
1. path connectedness defines an equivalence relation on a space
8. removing a point to show that R and R² are not homeomorphic
10. homeomorphism classes of the uppercase alphabet
12. [0,1) and [0,1] x [0,1] are not homeomorphic


Presenting Tuesday, 9/12:
Σ: B, 1, 12   ε: B, 1, 10   μ: A, 8, 12   ρ: B, 8, 10   φ: A, 1, 12

* read the section and work through all examples
1. ambient isotopy is an equivalence relation
3. in the plane, any quadrilateral is ambient isotopic to any triangle
4. which of these spaces are ambient isotopic in the plane? in R³?
7. unknotting an arc between two spheres by ambient isotopy
8. pictures of an ambient isotopy unknotting a dogbone
9. pictures of an ambient isotopy reversing chirality of a figure-8 knot

Presenting Thursday, 9/14:
Σ: 4, 7, 8   ε: 3, 4, 8   μ: 1, 7, 9   ρ: 3, 4, 9   φ: 1, 8, 9

Chatper 2: Knots and Links

* read the section and work through all examples
-- don't stress overly much about the piecewise-linear bits
1. knot equivalence by triangular detours and shortcuts is an equivalence relation
9ab. Borromean rings with three and four components
10. interchanging components of the Whitehead link by ambient isotopy


Presenting Tuesday, 9/19:
Σ: 10   ε: 1, 9ab   μ: 9ab   ρ: 9ab   φ: 1, 10     (note Section 2.2 is also due on 9/19; see assignment below)

* read the section and work through all examples
-- same comment as in Section 2.1
5. there are no nontrival knots with just one or two crossings
6. how many knots (up to equivalence) have three crossings?
-- and of course, why?
12. unknotting knots in R⁴


Presenting Tuesday, 9/19:
Σ: 5, 6   ε: 6   μ: 5, 12   ρ: 6, 12   φ: 5

* read the section and work through all examples
A. Example 2.29: linking number is an invariant of oriented links
3. computing linking numbers
4. computing more linking numbers
8. there exists a link with linking number n for all integers n
11. reducing crossings by Reidemeister moves
-- feel free to photocopy your answer to hand out, so you don't have to try to recreate this on the board!

Presenting Thursday, 9/21:
Σ: A, 3   ε: A, 4   μ: 3, 8   ρ: 4, 11   φ: A, 3     (note Section 2.4 is also due on 9/21; see assignment below)

* read the section and work through all examples
A. proof of Theorem 2.36: colorability is preserved by Reidemeister moves
-- Draw pictures! Type I moves are "obvious"; see Exercises 3 and 4 for the idea of how to handle Types II and III.
2. determining tricolorability in three examples
8. determining 5-colorability in three examples
9. p-colorability is a knot invariant
-- Follow the same type of argument as in the proof of Theorem 2.36 and Exercises 3 and 4.

Presenting Thursday, 9/21:
Σ: A, 8   ε: 2, 9   μ: A, 8   ρ: A, 2   φ: 8, 9

Chapter 3: Surfaces

* read the section and work through all examples
-- Be sure to read this section especially carefully.
2. the Klein bottle can be cut into two Möbius bands
-- Use the web as a reference if you like, but make sure you can explain the answer well with a series of pictures.
3. the projective plane can be cut into a Möbius band and a disk
-- same comment as #2
6. visualizing the projective plane using lines in R²
7. (ADDED LATE - KNOW FOR EXAMS) the projective plane as the set of lines through the origin in 3-space


Presenting Tuesday, 9/26:
Σ: 3   ε: 3, 6   μ: 2 ,6   ρ: 2   φ: 3     (note Section 2.4 is also due on 9/26; see assignment below)

* read the section and work through all examples
-- Be sure to read this section especially carefully.
1. gluing a polygonal disk together along pairs of edges results in a surface
2. S² acts as the identity for connected sums
4. gluing an octagonal disk to get the 2-holed torus
5. gluing polygonal disks to get connected sums of tori and connected sums of projective planes
7. P² # P² = K²
-- as part of your solution to 7b, consider example 3.15

Presenting Tuesday, 9/26:
Σ: 1, 7   ε: 4   μ: 2, 4   ρ: 1, 5   φ: 5, 7

1. representing various surfaces with polygonal disks with glued edges
2. what kinds of surfaces can be obtained by connected sums of S², T², K², and P²?


Presenting Thursday, 9/28:
Σ: 1, 2   ε: 1, 2   μ: 1, 2   ρ: 1, 2   φ: 1, 2

* read the section and work through all examples
A. Example 3.21: χ(S²), independent of triangulation.
-- include an explanation of why the Euler characteristic of a connected polygonal region of the plane with no holes is 1 (in other words, explain the "magic trick")
B. Example 3.23: χ(T²) and χ(K²), independent of triangulation
-- this includes explaining any previous example or argument that is needed to complete the argument in this example
3. χ(M²) and χ(P²)
-- do this two ways: first with an n-gon example and a particular triangulation, and then as a more general argument (as those given in Examples 3.21 and 3.23).
4. χ(S # T) = χ(S) + χ(T) - 2
-- don't use Theorem 3.25, of course; that is what you are proving here!
5. the Euler characteristic after gluing two surfaces along their boundaries
6. orientability of S², P², and connected sums
8. a surface is nonorientable if and only if it contains a Möbius band
-- notice that "if and only if"; there are two directions to prove here

Presenting Tuesday, 10/2:
Σ: A, 4, 6, 8   ε: B, 4, 5, 6   μ: A, 3, 5, 8   ρ: B, 3, 5, 6   φ: A, 4, 5, 8

* read the section
-- Pay particular attention to the proof of the "Classification Theorem for Closed Surfaces" (Theorem 3.33); that will be the focus of the first day we will spend on Section 3.4. In fact, all your homework for today is based on the steps of the proof as outlined below.
I. proof of Lemma 3.32: every closed path-connected surface can be obtained by gluing pairs of edges of a polyhedral disk

The proof now proceeds by strong induction on the number k of pairs of edges that must be glued in our disk-representation of the surface.
II. prove the base case (theorem is true when k = 1)

In steps III through VII below, we reduce our k-pair disk to one of the following:
A) a disk with < k pairs
B) (a disk with < k pairs) # (a torus)
C) (a disk with < k pairs) # (a projective plane)
D) (a disk with < k pairs) # (a disk with < k pairs)
In each case below, be sure to state which of (A), (B), (C), or (D) is the result.
III. the case where there exists a pair of adjacent edges with opposing orientations
IV. the case where there exists a pair of adjacent edges with matching orientations
V. the case where there exists a pair of nonadjacent edges with matching orientations
VI. the case where there exists a pair of nonadjacent edges with opposing orientations, type one
VII. the case where there exists a pair of nonadjacent edges with opposing orientations, type two
Now by strong induction, we have written our surface as the connected sum of spheres, tori, and projective planes.
VIII. argue that we can write any surface as described above as either S, or T # T # ... # T, or P # P # ... # P.


Presenting Thursday, 10/4:
Σ: II, III, VI   ε: I, IV, V   μ: IV, VI, VIII   ρ: II, IV, VII   φ: I, III, V

* read the section and work through all examples
2ace. following three surfaces through the classification proof algorithm
2bdf. following three more surfaces through the classification proof algorithm
3. identifying a non-orientable surface from its "word"
4ace. identifying three surfaces via Euler characteristic and orientability
4bdf. identifying three more surfaces via Euler characteristic and orientability
5. identifying a surface built from gluing ten triangles
6. identifying a surface with boundary from its "word"
10. the "Purse of Fortunatus"


Presenting Tuesday, 10/10:
Σ: 2ace, 4bdf, 6   ε: 2bdf, 4ace, 6   μ: 2bdf, 3, 4bdf   ρ: 2ace, 4ace, 10   φ: 2bdf, 4ace, 5

Test I: Thursday, October 12, in class

Chapter 6: The Fundamental Group

* read the section and work through all examples
A. Definition 6.4: homotopy of loops
--- be able to explain the definiton clearly and precisely and explain what it is talking about. pictures might help, but make sure you can explain the maps.
B. Example 6.6: every loop in the disk is homotopic to the constant loop
3. why homotopies of loops are stupid if we are allowed to relax the basepoint condition
4. in a disk, any two lops with a common basepoint are homotopic
7abc. homotopy is an equivalence relation on loops


Presenting Tuesday, 10/17:
Σ: A, B, 7   ε: A, 3, 7   μ: A, 4, 7   ρ: A, B, 7   φ: A, 4, 7

* read the section and work through all examples
A. Explain paragraph above Theorem 6.10 and the meaning of Theorem 6.10: what does it mean to show that an operation on equivalence classes is "well-defined"?
B. Proof of Theorem 6.10: showing that concatenation induces a well-defined operation on homotopy classes of loops
C. Explain Figure 6.13: a guide to defining a homotopy from (α β) γ to α (β γ)
1ab. determine whether certain functions are well-defined on homotopy classes of loops
1cd. determine whether certain other functions are well-defined on homotopy classes of loops
3a. draw a guide for a homotopy from ε &alpha to &alpha
-- make something like what is in Figure 6.13

Presenting Thursday, 10/19:
Σ: B, C, 1ab   ε: A, 1ab, 3a   μ: B, C, 1cd   ρ: A, C, 1cd   φ: B, 1ab, 3a

* read the section and work through all examples
--TYPO: In part (2) of Theorem 6.18, the conclusion should be that (g ο f)_* = g_* ο f_*.
A. Explain what Theorem 6.17 means.
B. Explain what Theorem 6.18 means.
C. Prove Theorem 6.18.
D. Explain what Theorem 6.9 means.
E. Prove Theorem 6.9.
1ace. basic properties of homomorphisms and isomorphisms; isomorphism is an equivalence relation on groups
-- for part (e), can assume all previous parts of the problem
1bde. more basic isomorphism properties; isomorphism is an equivalence relation on groups
-- again, for part (e), can assume all previous parts of the problem
2ab. homomorphisms send identity elements to identity elements, and inverses to inverses
2c. a homomorphism is 1-1 iff the kernel of the homomorphism is trivial
3. < Z, + > is isomorphic to < {integer powers of 2}, mult >
4. < C, + > is isomorphic to < R², + >


Presenting Tuesday, 10/24:
Σ: D, 1ace, 2c, 4   ε: E, 1bde, 2ab, 3   μ: A, 1ace, 2ab, 3   ρ: B, 1bde, 2ab, 4   φ: C, 1bde, 2c, 3

* read the section and work through all examples
* all questions we didn't get to last time are still fair game
7. a continuous function between spaces induces a homomorphism between fundamental groups

Presenting Tuesday, 10/30:
Σ: 7   ε: 7   μ: 7   ρ: 7   φ: 7

* read the section and work through all examples
A. Lemma 6.30: give a quick, intuitive, convincing reason why two homotopic loops must have the same degree.
4. finding loops in S¹ of each integer degree
5. what happens to degree when loops are concatenated?
9. extending the idea of homotopy: contractions
10. a contractible space is path-connected
11. a contractible space is simply-connected


Presenting Thursday, 11/1:
Σ: A, 9, 10   ε: A, 4, 11   μ: 5, 9, 10   ρ: 4, 5, 11   φ: A, 9, 10

* read the section and work through all examples
A. Example 6.47: defining a group multiplication on a Cartesian product of groups
6. the fundamental group of a product of spaces is isomorphic to the product of the fundamental groups of those spaces
-- Note that outlines for each part of this problem are given in the proof of Theorem 6.49; your job is to fill in the details and explain exactly what is going on.
7. generating loops for the fundamental group of a torus
10. The Poincaré Conjecture for n = 2


Presenting Tuesday, 11/7:
κ: A, 6, 7, 10   λ: A, 6, 7, 10   δ: A, 6, 7, 10   ρ: A, 6, 7, 10   φ: A, 6, 7, 10

Test II: Tuesday, November 14, in class

Chapter 7: Metric and Topological Spaces

* read the section and work through all examples
A. Proof of Theorem 7.11 (⇐): continuous implies inverse images of open sets are open
-- include an explanation of Figure 7.12
B. Proof of Theorem 7.11 (⇒): inverse images of open sets are open implies continuous
-- include an explanation of Figure 7.13
1abcd. four very basic metrics
2abcd. open balls in the four very basic metrics of problem (1)
5. the image of a function, and what this means for the open ball definition of continuity
-- be sure to relate this to Definition 7.6
8. some basic open (and non-open) sets, and an example where an infinite intersection of open sets is not itself an open set
-- relate this to Theorem 7.8
11. under the discrete metric, every subset of R² is open

Presenting Thursday, 11/14:
κ: A, 1b, 2b, 5   λ: A, 1d, 2d, 8   δ: B, 1c, 2c, 5   ρ: A, 1a, 2a, 11   φ: B, 1b, 2b, 8

* read the section and work through all examples
C. prove Theorem 7.9: in a metric space, an "open ball" is open
7. arbitrary unions and finite intersections of open sets are open
9. the taxicab metric is equivalent to the standard metric
10. the "Roman roads" metric is not equivalent to the standard metric


Presenting Tuesday, 11/28:
κ: C, 7, 9, 10   λ: C, 7, 9, 10   δ: C, 7, 9, 10   ρ: C, 7, 9, 10   φ: C, 7, 9, 10

* read the section and work through all examples
A. Example 7.19: determining continuity with respect to a topology
-- ALSO find non-trivial topologies on X and Y so that f is continuous.
-- ALSO find a map g that is continuous with respect to the original topologies given in the problem.
4. Hausdorff spaces
5. continuity and the discrete and indiscrete topologies
10. closure, interior, and boundary in the standard Euclidean topology
11. closure, interior, and boundary in another topology


Presenting Thursday, 11/30:
κ: A, 4, 5, 10, 11   λ: A, 4, 5, 10, 11   δ: A, 4, 5, 10, 11   ρ: A, 4, 5, 10, 11   φ: A, 4, 5, 10, 11

* read the section and work through all examples
A. proof of Theorem 7.28: the continuous image of a connected space is connected
1. checking if a topological space is connected
2. another connected (or not?) topological space
4. a path-connected space cannot be disconnected, since this would imply a disconnection of [0,1]


Presenting Tuesday, 12/5:
κ: A, 1, 2, 4   λ: A, 1, 2, 4   δ: A, 1, 2, 4   ρ: A, 1, 2, 4   φ: A, 1, 2, 4   (note Section 7.4 is also due on 12/5; see assignment below)

* read the section and work through all examples
B. proof of Theorem 7.35: the continuous image of a compact space is compact
2. checking if a topological space is compact
5. a closed subset of a compact space is compact
7. a continuous bijection from a compact space to a Hausdorff space must be a homeomorphism


Presenting Tuesday, 12/5:
κ: B, 2, 5, 7   λ: B, 2, 5, 7   δ: B, 2, 5, 7   ρ: B, 2, 5, 7   φ: B, 2, 5, 7