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.

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