MAT 531: Topology&Geometry, II
Spring 2010
Midterm Solutions
Problem 1 (15pts)
Let f : M N and g : N Z be smooth maps between smooth manifolds. State the chain
rule for the differential of the map g f
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 8
Problem 1 (15pts)
Suppose X is a topological space and P = cfw_SU ; U,V is a presheaf on X. Let
[
SU = (U , f )A : U U open, U =
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 5
Problem 1 (5pts)
Let V be a vector space of dimension n and n V a nonzero element. Show that the homomorphism
V n1 V ,
v iv ,
wher
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 11
Problem 1 (15pts)
Suppose M and N are smooth oriented compact connected n-manifolds. If f : M N is a smooth
map, the degree of f
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 9
Problem 1 (20pts)
We have defined Cech
cohomology for sheaves or presheaves of K-modules. All such objects are
0 and H
1 can be
MAT 531: Topology&Geometry, II
Spring 2006
Final Exam Solutions
Part I (choose 2 problems from 1,2, and 3)
1. Suppose M is a compact manifold and is a nowhere-zero closed one-form. Show that
1
[] 6= 0
MAT 531: Topology&Geometry, II
Spring 2010
Final Exam Solutions
Part I (choose 2 problems from 1,2, and 3)
1. Let M and N be compact oriented connected n-manifolds and f : M N a smooth map. Show
that
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 6
Problem 1 (10pts)
i=n
X
.
xi
i=1
(a) Determine the time t-flow Xt : Rn Rn of X (give a formula).
(b) Use (a) to show directly from
MAT 531: Topology&Geometry, II
Spring 2011
Final Exam Solutions
Part I (choose 2 problems from 1,2, and 3)
1. Let f : RP 3 T 3 (S 1 )3 be a smooth map. Show that f is not an immersion.
Suppose f is an
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 7
Problem 1 (15pts)
Let X be a path-connected topological space and (S (X), ) the singular chain complex of continuous
simplices int
MAT 531: Topology&Geometry, II
Spring 2006
Midterm Solutions
Problem 1 (15 pts)
Suppose M is a smooth manifold and X and Y are smooth vector fields on M . Show directly from
definitions that
[X, Y ] =
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 4
Problem 1: Chapter 1, #13ad (10pts)
(a) Show that [X, Y ] is a smooth vector field on M for any two smooth vector fields X and Y o
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 3
Problem 1: Chapter 1, #5 (10pts)
If (M, F) is a smooth n-manifold, let
G
TM =
Tm M
and
: T M M,
(v) = m if v Tm M.
mM
If (U , ) F
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 1
Problem 1: Chapter 1, #2 (10pts)
Let F be the (standard) differentiable structure on R generated by the one-element collection of
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 2
Problem 1: Chapter 1, #10 (5pts)
Suppose M is a compact nonempty manifold of dimension n and f : M Rn is a smooth map.
Show that f
MAT 531: Topology&Geometry, II
Spring 2011
Midterm Solutions
Problem 1 (15pts)
Suppose M is a smooth manifold, X, Y (M ; T M ) are smooth vector fields on M , and g C (M )
is a smooth function on M .
MAT 531: Topology&Geometry, II
Spring 2011
Solutions to Problem Set 10
Problem 1: Chapter 6, #6 (5pts)
Derive explicit formulas for d, , , and in Euclidean space.
Let x1 , . . . , xn denote the standa
Lecture 1: August 25
Introduction. Topology grew out of certain questions in geometry and analysis
about 100 years ago. As Wikipedia puts it, the motivating insight behind topology
is that some geomet