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 : M Z and obtain it directly from the relevant definit
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 =
U ; f SU ;
A
, A, p U U W U U open s.t. p W, W,U f =
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 ,
where iv is the contraction map, is an isomorphism.
Let cfw
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 is the number deg f R such that
Z
Z
E n (N ).
f = (deg
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 defined for sheaves or presheaves of non-abelian groups
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 HdeR
(M ).
We need to show that is not exact, i.e. 6=
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 there exists a unique number (deg f ) R such that
Z
Z
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 the definition of the Lie derivative LX that the homom
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 immersion. Since RP 3 and T 3 have the same dimension,
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 into X with integer coefficients. Denote by H1 (X; Z) the
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 ] = [Y, X].
(You can assume that [X, Y ] is whatever objec
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 on M .
(d) Show that [, ] satisfies the Jacobi identity,
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 and = (x1 , . . . , xn ), define
: 1 (U ) R2n
by
Show
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
charts F0 = cfw_(R, id). Let F be the differentiable st
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 is not an immersion (i.e. df |m is not injective for s
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 . Show directly from the definition that
[gX, Y ] = g[X,
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 standard coordinate functions on Rn . It is sufficient to des
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 geometric problems depend not on the exact shape of the objec