Math 437, Spring 2016, Homework 1
Posted Tuesday 2/2. Due Tuesday 2/9.
Problem 1: Define a topology O on R via O cfw_U R : R \ U is finite. Prove or disprove:
(R, O) is connected/compact/Hausdorff. Find the continuous functions on this space.
Problem 2: L

Math 437, Spring 2016, Homework 8
Posted Monday May 2. Due Tuesday May 10.
Problem 1: Set-up as in HW7, #2.
(a) Show that the two charts (S 2 \ cfw_(0, 0, 1), H+1 ) and (S 2 \ cfw_(0, 0, 1), R H1 ) form an
oriented atlas for RS 2 , where R : R2 R2 can be

Problem 3 (Exercise 3.1 in the book): Let V be an n-dimensional vector space. Let be
a nonzero element of Altn V . Show that the map V Altn1 V defined by v 7 v y , where
(vy )(v1 , . . . , vn1 ) = (v, v1 , . . . , vn1 ), is an isomorphism.
Solution: Both

Problems 1 and 3: See class notes/handout, and/or ask me.
1
Problem 2 (Exercise 3.4 in the book): The expression = x2 +y
2 (xdy ydx) defines a 12
2
form on R \ cfw_0. Show that each point p R \ cfw_0 has an open neighborhood U R2 \ cfw_0
such that there e

Problem 1 (Exercise 2.4 in the book): Let f : M N be differentiable, let M0 M be
the preimage of a regular value of f (so that M0 is in particular a submanifold of M ), and let
i : M0 M denote the inclusion map. Show that dip (Tp M0 ) = ker(dfp ) for all

Problem 1: Answer question (10) on p. 21 of the book. Justify your answer.
Solution: #3. Consider the two points p [1] = [i] = cfw_1, i and q [1] = [i] = cfw_1, i
in the quotient space. Since i = ei 2 , every neighborhood of p, q contains a point of the f

Math 437, Spring 2016, Homework 2
Posted Thursday 2/11. Due Thursday 2/18.
Problem 1: Answer question (10) on p. 21 of the book. Justify your answer.
Problem 2: Consider the manifold (S n , ORn+1 |S n , Amax ) from class, where A = cfw_(U, h), (V, k),
U =

Math 437, Spring 2016, Final
Please write your name on this sheet and turn it in. Each problem is worth 25 points.
Problem 1: (a) Define f : R3 \ cfw_z-axis R by f (x, y, z) = 41 (log(x2 + y 2 )2 + z 2 . Show that
1 R is a regular value of f and conclude

Problem 1: Set-up as in HW7, #2.
(a) Show that the two charts (S 2 \ cfw_(0, 0, 1), H+1 ) and (S 2 \ cfw_(0, 0, 1), R H1 ) form an
oriented atlas for RS 2 , where R : R2 R2 can be any reflection, e.g. R(x, y) = (x, y).
(b) Calculate S 2 with respect to th

Math 437, Spring 2016, Homework 6
Posted Tuesday April 5. Due Thursday April 14.
Problem 1: Let V be a vector space, let 1 , 2 , 3 Alt1 V , and let v1 , v2 , v3 V . Using only
the definition of , work out an explicit formula for (1 2 3 )(v1 , v2 , v3 ).
P

Problem 1: Let : U (, ) denote the usual angular polar coordinate function, where
1
U = R2 \ cfw_(x, 0) : x 6 0. Show that d = x2 +y
2 (xdy ydx) on U . Deduce from this that d
2
extends differentiably from U to V = R \ cfw_(0, 0) even though itself does n

Math 437, Spring 2016, Midterm
Each problem is worth 25 points. Justify your answers.
Problem 1: We know that if f : X Y is a continuous bijection of topological spaces with
X compact and Y Hausdorff, then f is a homeomorphism. Show that this may fail if

Math 437, Spring 2016, Homework 7
Posted Tuesday April 19. Due Tuesday April 26.
Problem 1: Let : U (, ) denote the usual angular polar coordinate function, where
1
U = R2 \ cfw_(x, 0) : x 6 0. Show that d = x2 +y
2 (xdy ydx) on U . Deduce from this that

Math 437, Spring 2016, Homework 5
Posted Thursday 3/24. Due Thursday 3/31.
Problem 1: Carefully prove the Cauchy-Binet formula, as follows.
f
g
(a) Show that if U V W are linear maps, then Altk (g f ) = (Altk f ) (Altk g).
(b) If f : Rn Rm is the linear m

Problem 1: Prove that the function f : R R defined by f (0) = 0 and f (x) = exp(1/x2 )
for x 6= 0 is infinitely differentiable everywhere.
Solution: First prove by induction on k that f (k) (x) = pk ( x1 )f (x) for all x 6= 0, where pk (t) is
a polynomial

Math 437, Spring 2016, Homework 4
Posted Thursday 3/3. Due Thursday 3/10 for #1,2, and Tuesday 3/22 for #3,4.
Problem 1 (Exercise 2.4 in the book): Let f : M N be differentiable, let M0 M be
the preimage of a regular value of f (so that M0 is in particula

Math 437, Spring 2016, Homework 1
Problem 1: Define a topology O on R via O cfw_U R : R \ U is finite. Prove or disprove:
(R, O) is connected/compact/Hausdorff. Find the continuous functions on this space.
Correction: O should contain the empty set as wel

Math 437, Spring 2016, Homework 3
Posted Tuesday 2/23. Due Tuesday 3/1.
Problem 1: Prove that the function f : R R defined by f (0) = 0 and f (x) = exp(1/x2 )
for x 6= 0 is infinitely differentiable everywhere.
Problem 2: Define M = cfw_(x, y, ei ) R2 S 1

Problem 1: Let V be a vector space, let 1 , 2 , 3 Alt1 V , and let v1 , v2 , v3 V . Using only
the definition of , work out an explicit formula for (1 2 3 )(v1 , v2 , v3 ).
Solution: Brute force computation. Use
P that for a permutation of cfw_1, 2, 3, si

C = cfw_(x, y, z) R3 : x2 + y 2 = 1
C : C R3 , (x, y, z) = (x, y, 0)
pC (v, w) = (v w) C (p) for all p C, v, w Tp C
K : R2 C, K(t, ) = (cos , sin , t)
C
(K C )p (v, w) = K(p)
(dKp (v), dKp (w) for all p R2 , v, w Tp R2
C
= det(v, w)K(p)
(dKp (e1 ), dKp (e