2014 Fall Math 5041
Assignment 1.
Due: Sept 26, 2014
Name:
In problems 1 and 2, we will show explicitly that RP 3 is nullcobordant.
1. a) Consider the set of quanternions
H = cfw_x = a + bi + cj + dk  a, b, c, d R, i2 = j 2 = k 2 = 1, ij = k, jk = i, ki
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2014 Fall Math 5041
Assignment 2.
Due: Oct 20, 2014
Name:
1. a) Let R GL(2, R) be a smooth oneparameter family of invertible 2 2 matrices. Show that
dA
d (det A)
= (det A) tr A1
dt
dt
.
b) Show that the special linear group
SL(2, R) = cfw_A Mat22  det A
2014 Fall Math 5041
Assignment 2.
Due: Oct 29, 2014
Name:
1. Show that Brouwer xed point theorem is false if the closed ball Dn = B n is replaced with the open
ball B n . That is, nd a continuous map f : B n B n with no xed point.
2. Let M be the solid to
2014 Fall Math 5041
Assignment 4.
Due: Nov 7, 2014
Name:
1. Let M be a smooth manifold with boundary. Show that there exists a smooth function f : M
[0, ) such that 0 is a regular value of f and M = f 1 (0).
2. a) For the zero vector eld X0 : RP 2 T (RP
2014 Fall Math 5041
Assignment 5.
Due: Nov 24, 3014
Name:
1. Let f : M M be a dieomorphism. For vector elds X and Y on M , show that
f ([X, Y ]) = [f (X), f (Y )]
2. Consider two vector elds X = y and Y = y x z on R3 , where we use the coordinates (x, y,
2014 Fall Math 5041
Assignment 6.
Due: Dec 8, 2014
Name:
1. (a) Let V be a vector space. Let v1 , v2 , . . . , vk V be a nite collection of vectors. Show that
v1 v2 . . . vk = 0
if and only if v1 , v2 , . . . , vk are linearly independent.
(b) For any kf
2014 Fall Math 5041
Test 2
Nov 11, 2014
Name:
1. Let X RN be an embedded submanifold. Show that almost every vector space V of a xed
dimension l in RN intersects X transversely.
2. Let N be a closed embedded submanifold of M . Show that every smooth vecto
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