HW #12:
Problem 1: Let $\alpha \in \Pi(X, x_0)$. We proved $c_{x_0} \cdot \alpha ~ \alpha$ in class. For homework, prove $\alpha \cdot c_{x_0} ~ \alpha$.
Problem 2: Let $\alpha, \beta, \gamma \in \Pi(X, x_0)$. Prove $(\alpha \cdot \beta)\cdot \gamma ~ \alpha \cdot (\beta \cdot \gamma)$.
(You may check your answer with the homotopy given in the book, but make sure you explain where the limits of the piecewise function come from and where the arguments of $\alpha, \beta, \gamma$ come from in the homotopy.)
In Section 6.2, problem 6
In Section 6.3, problems 1, 2, 6efgh, 7.
Due: Tuesday, Nov 30
Week 1: Chapter 1, Sections 1.1, 1.2, 1.3 (Equivalence, Bijections, Continuous Functions)
Week 2: Chapter 1, Sections 1.3, 1.4 (Continuous Functions, Topological Equivalence)
Week 3: Chapter 1, Section 1.5, 1.6 (Topological Invariants, Ambient Isotopy), Chapter 2, Section 2.1 (Knots, Links and Equivalence)
Unlinking video for Dogbone Toy
Week 4: Chapter 2, Sections 2.2, 2.3, 2.4 (Knot Diagrams, Reidemeister Moves)
Week 5: Chapter 2, Sections 2.4, 2.5 (Colorings, Alexander Polynomial)
Week 6: Chapter 2, Sections 2.6, 2.7 (Skein Relations, Jones Polynomial)
Week 7: Chapter 2, Section 2.7 (Jones Polynomial), Review, EXAM 1
Week 8: Chapter 3, Sections 3.1, 3.2, 3.3 (Surfaces, Cut and Paste, Euler Characteristic), Midsemester Day, Friday Oct 22
Week 9: Chapter 3, Sections 3.4 (Classification of Surfaces)
Week 10: Chapter 4, Sections 4.1, 4.2, 4.3 (3D Manifolds, Shape of Space, Euler Characteristic, Glueing Polyhedra Solids)
Week 11: Chapter 6, Sections 6.1, 6.2, 6.3 (Deformations with Singularities, Invariance of Fundamental Group)
Week 12: Chapter 6, Sections 6.4, 6.5 (The Sphere and the Circle), Chapter 7, Section 7.2 (Topological Spaces), Review
Week 13: EXAM 2, Thanksgiving Break, Nov 25-26
Week 14: Chapter 7, Sections 7.3, 7.4 (Connectedness, Compactness)
Week 15: Chapter 7, Section 7.5 (Quotient Spaces), Review
FINAL EXAM: Tuesday, Dec. 14, 2010, 8-11 am
HW #11:
In Section 4.3, do problem 1, 4, 5, 11b
In Section 6.1, do problems 4, 7
In Section 6.2, do problems TBA.
Due: Tuesday, Nov 16
HW #10:
In Section 4.1, do problem 2, 8bc
In Section 4.2, do problems 8
Due: Tuesday, Nov 9
HW #9:
In Section 3.3, do problems 7, 8, 6(this problem is easier after doing 7 & 8), 9, 11(tee-hee-hee)
In Section 3.4, do problems 1, 2adf, 4ef, 6, 7, 9
(problem 11 in 3.4 is not officially assigned, but it is so fun I hope that some of you try it!! want some extra credit?! )
Also part of this week's homework:
Go to Jeff Week's site Topology Software. Download and play the torus games. Write a short paragraph about your game play. Find a winning strategy and win a prize!
HW #8:
In Section 3.1, do problem 2
In Section 3.2, do problems 7, 8
HW #7:
In Section 2.6, do problems 7, 9, 10
In Section 2.7, do problems 2, 4, 5, 7
HW #6:
In Section 2.5, do problems 10, 13a, 14a.
Prove that if the index of the region between the two arcs emanating from a crossing X is a (Figure 1), then the index of the other three regions are a+1, a and a-1 as encountered counter-clockwise around X (Figure 2).
Complete and turn in Worksheet 1. (handed out in class on Thursday, Oct 7)
Due: Tuesday, Oct 12.
HW #5:
In Section 2.4, do problems 2, 6, 8, 9
In Section 2.5, do problems 4, 6, 7
Due: Tuesday, Oct 5.
HW #4:
In Section 2.1, do problems 9ab, 10(use colored pencils)
In Section 2.2, do problems 5, 6, 11, 14, 15
In Section 2.3, do problems 3, 4, 8, 9, 11
Due: Tuesday, Sept 28
HW #3:
In Section 1.5, do problems 3, 7, 8, 10
In Section 1.6, do problems 1, 4, 7, 8, 9
(Note: Problem 1.6.11, although not formally assigned, is quite interesting and leads to a possible final project topic.)
Due: Tuesday, Sept 21
HW #2:
In Section 1.3, do problems 1, 2, 6, 7, 10, and
Prove the following statements using the epslion-delta definition of continuity:
Problem I: Suppose f and g are functions from the real numbers to the real numbers. If f and g are continuous, then f+g is continuous.
Problem II: Suppose f is a function from R^2 to R^2 such that f(x,y) = ( f_1(x,y), f_2(x,y) ). If f_1 and f_2 are continuous, then f is continuous.
In Section 1.4, do problems 1, 2, 5, 6, 8, 11
Due: Tuesday, Sept 14.
HW #1:
In Section 1.1, do problems 1abcde, 2, 5, 7, 9
In Section 1.2, do problems 2, 4, 5, 16, 18, 23
Use the epsilon-delta definition of continuity to prove that the linear function f(x)= mx+b is continuous at the real number x=a.
Due: Tuesday, Sept 7.
