Math 8062 Homework 1
Due Thursday, 1/27/11
1. Consider a cell complex X, with one 0cell, one 1cell, and one triangular 2cell. The three
sides of the triangle are identied to the same edge, with the following orientations:
Prove that X is contractible.
2.
Math 8062 Homework 4
Due Thursday, 2/24/11
1. Let X = S 2 A, where A is an axis connecting the north and south poles of S 2 . Describe
the universal cover X of X and the action of 1 (X) on the universal cover.
2. A covering map p : Y X is called regular i
Math 8062 Homework 3
Due Thursday, 2/17/11
1. There is a standard way to glue together two (connected) manifolds M and N of the same
dimension. Remove an open ball B n from each of M and N, and glue M B n to N B n along
the two (n 1) dimensional boundary
Math 8062 Homework 9
Due Thursday, 4/21/11
1. Do problem 29 on page 158 of Hatcher.
2. Do problem 42 on page 159 of Hatcher.
3. Let M and N be closed, connected, oriented manifolds of dimension n. Let M#N be their
connected sum. Prove that
Hk (M#N) Hk (M)
Math 8062 Homework 8
Due Thursday, 4/14/11
1. Do problem 2 on page 155 of Hatcher.
2. Do problem 3 on page 155 of Hatcher.
3. Sort out the diagramchasing proof of problem 38 on page 159 of Hatcher, and be ready to
explain it orally.
4. Let M n and N n be
Math 8062 Homework 5
Due Thursday, 3/3/11
1. Which of the following covering spaces of S 1 S 1 are regular, i.e. correspond to normal
subgroups?
2. Let X be the cell complex obtained by identifying opposite faces of a cube with a 90 twist
(as in problem 1
Math 8062 Homework 10
Due Thursday, 5/5/11
1. Do problem 145(a) on page 383 of Lee.
2. Do problem 153 page 407 of Lee.
3. Do problem 156 page 408 of Lee.
4. Do problem 1511 page 408 of Lee.
5. Do problem 161 page 431 of Lee.
6. Let M be a compact, oriente