Updates

Week	Section	Practice Exercises	Problems to be submitted	Due date	#
1	2.1	1-11
	2.2	19-23
2	2.3	24-26
	2.4	28
	2.5	30-31
3	2.6	32-35	Ex.2.35, part (1). Hint: see part (3). And you may need a picture.+	9/14
	3.1	2,6
	3.2	9,10,14	Show that the outer shell of a tetrahedron is homeomorphic to a sphere in R^3. Exhibit the homeomorphism.	9/28
4	3.3		Ex. 3.27+	9/28
	3.4	28
5	3.5	31-36
			Exam 1: Chapters 2,3+	10/5,10/12
6	4.1	1-4	Ex. 4.3. Find the general rule and prove it.+	10/19
	4.2	5-7
	4.3	8-12	Ex. 4.9+; 4.12
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.
	4.6		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.
8	5.1	1,2
	5.2	3
	5.3	5
9	5.4	7,8,10-13,16-18	Ex. 5.11
			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)
10	6.3	16-18	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...
	6.4	19-23	Prove that the 0th homology group of a tree is Z (independent of triangulation).
	6.5	25-27	Ex. 22
11
			Exam 2: Chapters 4-6
12	7.1	1,2
	7.2	3
13	7.3	4-7	Ex. 4 or 7
	[7.4]
14	8.1,8.2	6
	9.1	1-13
	[9.2]
15	10.1	1,2
			Final Exam