课程名称︰拓朴学导论 课程性质︰数学系选修 课程教师︰齐震宇 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰103/5/6 4:00 ~ 103/5/11 3:30 考试时限（分钟）：五天 试题 : 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. --

※ 发信站: 批踢踢实业坊(ptt.cc), 来自: 59.115.109.26 ※ 文章网址: https://webptt.com/cn.aspx?n=bbs/NTU-Exam/M.1438022252.A.EC0.html ※ 编辑: acliv (59.115.109.26), 07/28/2015 02:38:56