課程名稱︰拓樸學導論
課程性質︰數學系選修
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）︰103/5/6 4:00 ~ 103/5/11 3:30
考試時限（分鐘）：五天
試題 :
Topology Midterm Exam 05/06/2015
Chen-Yu Chi
In the following, "* := ..." means that "* is defined to be/as...".
Definition 1. Lex X be a topological space. We say that X is as topological
manifold (with boundary) if for every p ∈ X there exists a homeomorphism
φ
U─→V between an open neighborhood U of p and an open set V of the lower half
space
n
H := {(x ,...,x ) ∈ |R | x ≦ 0}
n 1 n n
(H := {ψ}) for some nonnegative integer n, called the dimension of the chart
0
φ. Such a homeomorphism φ is called a coordinate chart of the manifold X
around p.
Question 1. Let X be a topological manifold and p ∈ X.
(1) Show that any two coordinate charts around p have the same
dimension. We denote this dimension by dim X.
p
(2) When dim X ＞ 0, we call p an interior (resp. a boundary) point of
p
X if there exists a coordinate chart φ around p such that the n-th coordinate
of φ(p) is negative (resp. is zero). Show that a point cannot be both an
interior and a boundary point of X.
Question 2. Consider the following topological spaces. Let X be the image of
1
the map
α 3
[-1,1] × [-1,1] ───────→ |R
　u u u
(u,v) ──────→ ((6 + vcos─)cosu,(6+vcos─)sinu,vsin─)
2 2 2
3
equipped with the subspace topology induced from the euclidean topology of |R .
On the other hand, let X be the quotient of [-1,1] × [-1,1] by the
2
equivalence relation
(s,t) ～ (s',t') iff (they are equal or ({s,s'} = {-1,1} and t = -t'))
equipped with the quotient topology.
(1) Show that X and X are homeomorphic.
1 2 1
(2) Show that X is not homeomorphic to S ×[0,1].
1
Question 3. For positive integers a and b, we let M (|R) be the set of all a
a,b
by b matrices all of whose entries are real numbers. We topologize M (|R) by
ab a,b
indentifying it with R equipped with the standard euclidean topology. All
subspaces 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
positve integer)
GL(n,|R) := {A ∈ M (R ) | detA ≠ 0},
n
B ∈ M (|R)
C B m,n-m
UT(m,n-m,|R) := {( ) | C ∈ GL(m,|R), and }
0 D
D ∈ GL(n-m,|R)
(where m ＜ n is a positive integer),
SL(n,|R) := {A ∈ M (|R) | detA = 1},
n
t
O(n) := {A ∈ M (|R) | AA = I }, and
n n
SO(n) := SL(n,|R) ∩ O(n).
(1) Which of the above are 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).
(4) Equip GL(n,|R)/UT(m,n-m,R) with the quotient topology. Show that
it is compact. (Hint. Consider the natural action of GL(n,|R) on
n
the set of all m-dimensional linear subspaces of |R . What are
the orbits? What is the stabilizer of the space generated by the
standard vectors e ,...,e ? Instead of GL(n,|R), what if we use
1 m
O(n) to act on the same set?
Question4. Let G be a locally compact Hausdorff topological group and μ a left
Haar measure of G. Let μ × μ be the regular Borel product of μ and μ. Show
that if A and B are Borel sets of G with μ(A) ≠ 0 ≠ μ(B), then
(μ × μ)(A × B) = μ(A) ×μ(B).
Question5. Let n be a positive integer and
n n+1 2 2
S := {x = (x ,...,x ) ∈ R | x +...+x = 1}.
1 n+1 1 n+1
n
Consider the following equivalence relation ～ on S :
x ～ y iff x = y or -y.
n n
Denote the quotient map from S to S /～ by p.
n
(1) Show that π (S ,e ) si trivial, where e = (1,0,...,0).
1 1 1
(2) Show that p is a covering map.
n
(3) Compute π (S /～,p(e )).
1 1
φ 2
Question 6. Let [0,∞) ─→ |R be a homeomorphism and denote the image by S.
2 2
Suppose that S is a closed subset of |R . Compute the homology groups of |R \S.