課程名稱︰分析二
課程性質︰數學系選修，可抵必修分析導論二
課程教師︰齊震宇
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/4/13 - 2017/5/4 10:00
考試時限（分鐘）：如上，共22天
試題 :
Analysis II 2017 First Midterm Exam
In the following, "＊:=..." means that "＊ is defined to be/as...". When
talking about a manifold, we always assume it to be path-connected, Hausdorff,
and 2nd countable.
Toolkit
Collected here are some facts which will not be (or have not been) proven in
the lectures which you are allowed to use directly in answering questions
below.
n
I. Invariance of domain. Given subsets A and B of S which are homeomorphic to
n
each other, if A is open in S , so is B.
II. de Rham's theorem. For any smooth oriented manifold M we have a natural
k ～
isomorphism H ( M ; C ) ＝ ( H ( M ) Ⓧ C )* between C-vector space for every
k Z
k, where (‧)* denote the dual space of ‧ (as a C-vector space).
f
III. Lefschetz's fixed point theorem. Let M ———→ M is a smooth map from a
orientable smooth compact manifold M to itself. The Lefschetz number L(f) of f
is defined by
∞ k k f* k
L(f) := Σ (－1) trace( H ( X , C ) ———→ H ( X , C ) ).
k=0
If L(f)≠0, then there exists some p ∈ M such that f(p) = p.
Questions
Question 1. Consider the following topological spaces. Let X be the quotient
of [－1,1]×[－1,1] by the equivalence relation (s,t)～(s',t') ⇔
(s,t) = (s',t') or {(s,t),(s',t')} = {(－1,τ),(1,－τ)} for some τ∈ [－1,1]
equipped with the quotient topology. Show that X is not homeomorphic to
1
S ×[0,1].
n+1
Question 2. Let X:= {(x ,...,x ) ∈ R ∣x …x = 0, (x ,...,x )≠(0,...,0) }.
0 n 0 n 0 n
Compute H (X), k ∈ Z.
k
1 h 3 1
Question 3. Let S ———→ R be a continuous map mapping S homeomorphically
3 1
onto its image, and let Ω:= R \ h(S ). Does the statement
∞ ∞
∀C vector field F on Ω [ curl(F) = 0 ⇔ ∃ C function f on Ω such that
F = grad(f) ]
hold? How about
∞ ∞
∀C vector field G on Ω [ div(G) = 0 ⇔ ∃ C vector field F on Ω such that
G = curl(F) ]?
φ 2
Question 4. Let [0,∞) ———→ R be a homeomorphism onto its image and denote
2
the image by S and assume S to be a closed subset of R . Compute the homology
2
group of R \ S.
4 3
Question 5. (1) Does there exist a (topologial) embedding h：R ———→ S ,
4 4
i.e., a map h which maps R homeomorphically to h(R )? Justify your answer.
3 3
(2) Does there exist an embedding g：S ———→ R ? Justify your answer.
Question 6. For a nonnegative integer n we have the "left" half space of
n
dimension n, H :={(x ,...,x ) ∈ R ∣ x ≦ 0 } ( H := {ψ}).
n 1 n 1 0
Let U (resp. V) be an open set in H (resp. H ). Suppose that φ is a
m n
homeomorphism from U onto V.
(1) Show that m = n.
m－1 m－1
(2) Show that for any p ∈ U we have p ∈ {0}×R ⇔ φ(p) ∈ {0}×R .
Question 7. Let M be a smooth orientable manifold of dimension n.
n
(1) Show that there exists an ω ∈ A (M) which is "nowhere vanishing" in the
φ
sense that for any chart ( U ——→ φ( U ) ) ∈ Φ with coordinates
φ φ M
x ,...,x the local expression ω of ω via φ is of the form
1 n φ
f(x ,...,x ) dx ʌ…ʌdx with f a smooth function on φ( U ) whose value is
1 n 1 n φ
n
never zero. (Hint. Note that when M is an open set in R this clearly holds.
Choose a suitable atlas and use a suitable partition of unity.)
n
(2) (de Rham's theorem is not allowed to be used here.) Show that H (M;C)≠{0}
if M is compact. Assume M has no boundary, a slightly more challenging question
n
is to show that H (M;C) is homomorphic to C.
Question 8. Consider the map F defined by
2 1 x－1 1 x－1 √(3) 2 2 3
(x,y,z) ∈ S —→ (—— e cos(yz),—— e sin(yz),——— cos(y +z ))∈ R
2 2 2
2 2
and the map ρ defined by (x,y,z) ∈ S ——→ (z,x,y) ∈ S .
2
(1) Show that ρ(p)≠－F(p) for all p ∈ S .
2 2
(2) Show that ρ ：H ( S ) ——→ H ( S ) is identity map for every k ∈ Z.
* k k
2
(If you do not like homology, answer the question with all H ( S ) replaced
k
k 2
by the de Rham cohomology H ( S ; C ).)
2 2
(3) Let f：S ——→S be defined by
2 ρ(p) ＋ F(p) 2
f：p ∈ S ——→ f(p) := ───────── ∈ S ,
|ρ(p) ＋ F(p) |
2 2 2
where |(x,y,z)|:=√(x ＋y ＋z ). Does f have any fixed point? Justify your
answer.
Question 9. For positive integers a and b, we let M (R) be the set of all a
a,b
by bmatrices all of whose entires are real numbers. We topologize M (R) by
a,b
ab
identifying it with R equipped with the standard euclidean topology. All
subsets of M (R) will be equipped with the subspace topology. We denote
a,b
M (R) simply by M (R). Consider the following matrix groups: (n being a
a,a a
positive integer)
GL(n,R) := { A ∈ M (R) ∣ det(A)≠0 },
n
╭ C B ╮
UT(m,n－m,R) :={ │ │ ∣ B ∈ M (R), C ∈ GL(m,R), D ∈ GL(n－m,R)}
╰ 0 D ╯ m,n－m
(where m ＜ n is a positive integer),
SL(n,R) := {A ∈ M (R) ∣ det(A) = 1 },
n
t
O(n) := {A ∈ M (R) ∣ AA = I },
n n
SO(n) := SL(n,R)∩O(n).
(1) Which of the above are path-connected? Which are compact? (Write down your
reason.)
(2) Compute the homology groups of SL(2,R).
(3) Show that O(n) is a deformation retract of GL(n,R). (If you did not learn
linear algebra before, do the case n = 2.)