Introduction to Algebraic andGeometric Topology

Week 3

Domingo Toledo

University of Utah

Fall 2016

Recall Maps f : (X , d) ! (Y , d 0)

I Continuous

I Uniformly continuous.

I Lipschitz

I bi-Lipschitz

I Isometry

Recall equivalences f : (X , d) ! (Y , d 0)

I Homeomorphism

I bi-Lipschitz equivalence

I Isometry

QuestionsI Is there an isometry f : (R2, d(2)) ! (R2, d(1))?

I Is there an isometry f : (R2, d(1)) ! (R2, d(1))?

Tool: Equality set for triangle inequality

I Given x , z 2 (X , d), let

E

d

(x , z) = {y 2 X |d(x , z) = d(x , y) + d(y , z)}

I If f : (X , d) ! (Y , d 0) is an isometry, then f takesequality sets to equality sets.:

E

d

0(f (x), f (z)) = f (Ed

(x , z))

I In particular, equality sets must be homeomorphic.

ExamplesI (R2, d(2)) and (R2, d(1)).

I (R2, d(1)) and (R2, d(1)).

Higher Dimensions

I Are (Rn, d(1)) and (Rn, d(1)) isometric for n � 3?

I Look at unit spheres for n = 3:

In > 3?

I Duality?

I What’s special about n = 2?

Groups of IsometriesI (X , d) metric space. The collection of all isometries is

a group.

I Composition

I Inverse

Isometries of Rn

If : (Rn, d(2)) ! (Rn, d(2)) Euclidean isometry.

I Example: translation f (x) = x + b for fixed b 2 Rn

I Example: rotation f (x) = Ax , A orthogonal matrix,det(A) = 1.

I More general: f (x) = Ax , A orthogonal matrix.

I Composition f (x) = Ax + b, A orthogonal matrix,b 2 Rn,

Orthogonal Matrices

ITheorem: These are all the Euclidean isometries.

IMain Point Euclidean isometries are Affine linear

Back to Euclidean Isometries

I Orthogonal matrices

IA : Rn ! Rn linear.

Ix ! Ax is a Euclidean isometry

() A is orthogonal.

IA : Rn ! Rn linear.

Ib 2 Rn.

I Then x ! Ax + b is called an Affine Linear

transformation.

I Example: if A is orthogonal,

x ! Ax + b

is an isometry of Rn.

ITheorem These are all the isometries of Rn.

I Main point of proof: All isometries are affine-linear.

If Euclidean isometry

I Reduce to f (0) = 0,

I Use the equality sets E(x , z).

I Prove f (rx) = r f (x) for all r 2 R

I Prove f (x + y) = f (x) + f (y) for all x , y 2 Rn.

Structure of the Euclidean group E(n)I Linear Part

I Subgroup of index two:

I Subgroup of translations

I Non-normal subgroups

Fixed Points, Classification

Isometries of the sphere