MAST90029 Di?erential Topology and Geometry 2015

Assignment 3

Due: 5pm Wednesday May 27

Please aim for clear, complete but concise explanations in all questions.

Some marks will be allocated for overall presentation and clarity of solutions.

1. Let M be a compact smooth n-manifold. Recall that two maps f0, f1 : M ! S1 are

smoothly homotopic if there exists a smooth map F : [0, 1] ? M ! S1 such that

F(0, x) = f0(x) and F(1, x) = f1(x) for all x 2 M. Write f0 ? f1.

(a) Prove that

f0 ? f1 ) f0 + g ? f1 + g

for any g : M ! S1. (Use the group structure on S1 induced from S1 ?= R/Z.)

(b) Hence prove that the set of homotopy classes of maps

[M,S1

] := {f | f : M ! S1

}/ ?

is a group, i.e. a Z-module.

(c) Define H1(M; Z) to consist of all ? 2 H1(M) satisfying R

N ? 2 Z for any

smooth compact 1-dimensional submanifold N ? M. (So N is di?eomorphic

to a disjoint union of circles. ) Prove that H1(M, Z) is a group, i.e. a Z-module.

(d) Prove that the map f 7! df defines an isomorphism of groups

[M,S1

] ?= H1

(M, Z)

where df uses the (local) lift of f to R ! R/Z = S1.

2. Consider the upper half-plane U = {(x1, x2) : x2 0} with the Poincar´e metric:

g = 1

x2

2 (dx1

2 + dx2

2

).

(This is one of the standard models for 2-dimensional hyperbolic geometry.)

(a) Show that each of the following maps is an isometry from U onto U.

(i) f(x) = ax + b where a 0, b = (b1, 0) 2 R ? {0}.

(ii) f(x) = x/||x||2 where ||x||2 = x2

1 + x2

2.

(iii) f(x1, x2)=(#x1, x2).

Conclude that there is an isometry from U to U taking any point to any other

point.

(b) Find the length L(c) in the Poincar´e metric of the curve c : [a, b] ! U defined

by c(t) = (0, t), where 0 a b.

(c) Show that the length L(¯c) % L(c) where ¯c is any other curve from (0, a) to

(0, b). Hence find the distance from (0, a) to (0, b) in the Poincar´e metric.

Assignment 3

Due: 5pm Wednesday May 27

Please aim for clear, complete but concise explanations in all questions.

Some marks will be allocated for overall presentation and clarity of solutions.

1. Let M be a compact smooth n-manifold. Recall that two maps f0, f1 : M ! S1 are

smoothly homotopic if there exists a smooth map F : [0, 1] ? M ! S1 such that

F(0, x) = f0(x) and F(1, x) = f1(x) for all x 2 M. Write f0 ? f1.

(a) Prove that

f0 ? f1 ) f0 + g ? f1 + g

for any g : M ! S1. (Use the group structure on S1 induced from S1 ?= R/Z.)

(b) Hence prove that the set of homotopy classes of maps

[M,S1

] := {f | f : M ! S1

}/ ?

is a group, i.e. a Z-module.

(c) Define H1(M; Z) to consist of all ? 2 H1(M) satisfying R

N ? 2 Z for any

smooth compact 1-dimensional submanifold N ? M. (So N is di?eomorphic

to a disjoint union of circles. ) Prove that H1(M, Z) is a group, i.e. a Z-module.

(d) Prove that the map f 7! df defines an isomorphism of groups

[M,S1

] ?= H1

(M, Z)

where df uses the (local) lift of f to R ! R/Z = S1.

2. Consider the upper half-plane U = {(x1, x2) : x2 0} with the Poincar´e metric:

g = 1

x2

2 (dx1

2 + dx2

2

).

(This is one of the standard models for 2-dimensional hyperbolic geometry.)

(a) Show that each of the following maps is an isometry from U onto U.

(i) f(x) = ax + b where a 0, b = (b1, 0) 2 R ? {0}.

(ii) f(x) = x/||x||2 where ||x||2 = x2

1 + x2

2.

(iii) f(x1, x2)=(#x1, x2).

Conclude that there is an isometry from U to U taking any point to any other

point.

(b) Find the length L(c) in the Poincar´e metric of the curve c : [a, b] ! U defined

by c(t) = (0, t), where 0 a b.

(c) Show that the length L(¯c) % L(c) where ¯c is any other curve from (0, a) to

(0, b). Hence find the distance from (0, a) to (0, b) in the Poincar´e metric.

Assignment Cover SheetQualification Module Number and TitleHigher National Diploma in Computing & Software Engineering COM5221 - Business AnalyticsStudent Name & No. Assessor Prepared by Mr.Induranga...Technologies for Web Applications 300582 Web Application Assignment Due: Wednesday 11:59pm 5th of February 2020 – Assessment Weight: 30% Note: · Include comments for your student ID, Name, and Practical...my part is competitive advantage and company strategycompany generic strategiesyou have to do just no 3 and 43. Company’s competitive advantage over its rivals.4. Is this company following any Generic...PPMP20009 Leading Lean Projects Term 3, 2019PPMP20009 Leading Lean ProjectsAssessment 1 SpecificationIndividual Assessment (40%)Due Date: 23:55 AEST Week 9, Friday (24 January, 2020) Length: 2500 – 3000...Name: ____Janit Anand___________________________ Assessment criteria for BSBHRM506Manage Recruitment Selection and Induction ProcessesIn this document, you will find the foundation of what is required...CIS5200ASSIGNMENT 2OVERALL ASSESSMENT WEIGHTING: 20%The overall objective, of assignments 1 and 2 combined, is that the students should acquire an overview of principles, methods and techniques of systems...ITNE3007Advanced RoutingAssignmentITNE3007 AssignmentAssignment Type Group Assignment (3 students in each group)Week Issued 6Total Marks 20Submission Deadline Week 11 (19-January-2019) via MoodleSubmission...**Show All Questions**