Updates

Week	Section	Practice Exercises	Problems to be submitted	Due date (by noon)	#
1	2.1	1-11
	2.2	19-23
2	2.3	24-26	(1) Show that the addition, subtraction, and multiplication operations are continuous functions from RxR to R; and the quotient operation is a continuous function from Rx(R\{0}) to R. Do only subtraction and multiplication.	2/6	1
	2.4	27-28
	2.5	30-31
3	2.6	32-34	(2) Ex.2.35, part (1). Hint: see part (3). And you may need a picture.	2/6	1
	3.1	1-7	(3) Suppose X and Y are metric spaces. Let f_n:X->Y be a sequence of continuous functions uniformly convergent to f. Let x_n be a sequence of points of X converging to x. Show that f_n(x_n) converges to f(x). What happens when the convergence is not uniform? You may need an example.	2/6	1
	3.2	8-17	(1) Show that any closed convex polyhedron is homeomorphic to a closed ball in R^3. Prove this for a tetrahedron. Exhibit the homeomorphism.	2/13	2
4	3.3	22,25-27	(2) Ex. 3.27	2/13	2
	3.4	28
5	3.5	31-36
2/16			Exam 1: Chapters 2,3
6	4.1	1-4	Ex. 4.3. Find the general rule and prove it.	2/20	3
	4.2	5-7
	4.3	8-12	(1) Ex. 4.9; (2) 4.12	2/27	4
7	4.4	13-16
	4.5	17-22	Ex. 4.15 Hint 1: It's not obvious! Hint 2: Find the triangulation of the new space in terms of triangulations of the two old ones.	3/5	5
	4.6		(1) Prove that a torus has two but not three distinct (not disjoint) simple closed curves such that the complement of their union is connected. Generalize to other orientable surfaces.	3/12	6
8	5.1	1,2
	5.2	3
	5.3	5
9	5.4	7,8,10-13,16-18	Ex. 5.11	3/26	7
			Extra credit: Find the homology of a real-life complex (big, higher than one-dimensional).
	6.1	1-6
	6.2	7-9	Ex. 6.18 (8)	4/2	8
10	6.3	16-18	(1) Show that the first Betti number of a surface with boundary equals the first Betti number of the corresponding surface without boundary plus the number of boundary curves minus one (Hint: consider disk&sphere). Prove that by comparing the complexes representing these surfaces, their groups of chains, cycles, boundaries, homology groups...	4/9	9
	6.4	19-23	(2) Prove that the 0th homology group of a tree is Z (independent of triangulation).	4/9	9
	6.5	25-27	Ex. 22	4/16	10
11
4/12			Exam 2: Chapters 4-6
12	7.1	1,2
	7.2	3
13	7.3	4-7	Ex. 4 or 7	4/23	11
	[7.4]
14	8.1,8.2	6
	9.1	1-13
	[9.2]
15	10.1	1,2
			Final Exam: Monday, May 3, 5 - 7 pm.