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MAST30024: GEOMETRY

ASSIGNMENT 1

DUE: 5PM, FRIDAY AUGUST 26TH 2022, VIA GRADESCOPE

Standard notation: The following notation for subspaces of Rn will be used throughout

this assingment:

Sn?1 := {x ∈ Rn : |x| = 1}

Dn := {x ∈ Rn : |x| ≤ 1}

Some background:

Product spaces (see Tutorial 2 Q18): If X and Y are topological spaces then their

cartesian product X × Y is the topological space with U ? X × Y open if U = ?α Uα × Vα,

where Uα ? X is open, Vα ? Y is open and the union is arbitrary. When X = Y , we will

often write X2 = X ×X.

You may freely use the following facts about product spaces in this assignment.

Fact 0: If (X, dX) and (Y, dY ) are metric spaces defining topological spaces X and Y , then

X × Y is a metric space with

dX×Y

(

(x1, y1), (x2, y2)

)

=

√

dX(x1, x2)2 + dY (y1, y2)2

and the topological space defined by (X × Y, dX×Y ) is X × Y .

Fact 1: If f : W → X and g : Z → Y are continuous maps, then the product map

f × g : W × Z → X × Y, (w, z) 7→ (f(w), g(z))

is continuous. In particular, if f and g are homeomorphisms, then so is f × g.

Connected sums and boundary connected sums: A closed surface S is a compact

surface without boundary. Let S? := S \ D? be the compact surface with boundary, where

D S is a subspace with D ～= D2 and D? ? D in the interior of D. Let ?S? ～= ?D2 = S1

denote the boundary of S?. It can be proven that the homeomorphism type of S? does not

depend on the choice of D ? S.

For closed surfaces S1 and S2, their connected sum is obtained by taking a homeomorphism

f : ?S?1 → ?S?2 and defining the quotient space

S1#S2 = S

1 ∪f S?2 = (S?1 unionsq S?2)/ ～f ,

where S?1 unionsq S?2 is the disjoint union and the equivalence relation ～f is defined by x ～f f(x)

and y ～f f?1(y) for x ∈ ?S?1 and y ∈ ?S?2 (see Definition 1.4.8).

The boundary connected sum of S?1 and S?2 is obtained by restricting f to a closed interval

I S?1 , where I ～= [0, 1]. Then the boundary connected sum of S?1 and S?2 is the quotient

space

S?1\S

2 = S

1 ∪f |I S?2 = (S?1 unionsq S?2)/ ～f |I ,

where f |I is defined similarly to ～f . Similarly to connected sum, one can prove that S?1\S?2 is a

compact surface with boundary and that the homeomorphism type of S?1\S?2 does not depend

1

2 DUE: 5PM, FRIDAY AUGUST 26TH 2022, VIA GRADESCOPE

on the choices made. Similarly to connected sum, (c.f. Fact 1.4.9), the boundary connected

sum operation has the following properties, which you may freely use in this assignment:

(1) S?\D2 ～= S?

(2) S?1\S?2 ～= S?2\S?1

(3) (S?1\S?2)\S?3 ～= S?1\(S?2\S?3)

(4) (S1#S2)

～= S?1\S?2

1. (25 Marks)

Consider the following three subsets of R3 with their subspace topologies:

A = {(x, y, z) ∈ R3 : x2 + y2 + z2 = 1}

B = f([0, 2pi]2), for f : [0, 2pi]2 → R3, (u, v) 7→ ((1 + cos(u))(cos(v), sin(v)), sin(u))

C = {(x, y, z) ∈ R3 : |x|+ |y|+ |z| = 1}

1.1. Draw clear sketches of A, B and C. (These may not be produced by a computer).

Hint 1: To visualise B, you can sketch the sets B ∩ {(x, y, z) ∈ R3 : z = c}, as c ∈ R varies.

Hint 2: To determine the homeomorphism type of B, you can consider the equivalence

relation ～ on [0, 2pi]2 defined by (u, v) ～ (u′, v′)?? f(u, v) = f(u′, v′).

1.2. Determine which (if any) of these spaces are homeomorphic. Justify your answer by

giving explicit homeomorphisms and/or by giving reasons why these spaces are not homeo-

morphic.

2. (15 Marks)

Recall that a space X is called homogeneous if for any pair of points x0, x1 ∈ X there is a

homemorphism f : X → X such that f(x0) = x1. It can be shown that being homogenous,

or not, is a topological property; i.e. for homeomorphic spaces X and Y , X is homogeneous

if and only if Y is homogeneous.

2.1. Prove that S1 is homogeneous.

2.2. Prove that T 2 = S1 × S1 is homogeneous.

Hint 3: You may regard S1 ? C as the space of unit complex numbers. Then complex

multiplication μC : C × C → C is continuous and restricts to define a continuous function

μ : S1 × S1 → S1, (z0, z1) 7→ z0z1.

3. (15 Marks)

Let P = S2/～ be the quotient space of S2 = {x ∈ R3 : |x| = 1} defined by the equivalence

relation

x ～ y ?? x = ±y

and let P ′ = D2/～′ be the quotient space of D2 = {x ∈ R2 : |x| ≤ 1} by

x ～′ y ??

{

x = ±y and |x| = 1

x = y and |x| < 1.

3.1. Prove that P and P ′ are homeomorphic. In your proof, you may assume that P is

metrisable.

MAST30024: GEOMETRY ASSIGNMENT 1 3

4. (20 Marks)

Let S be a closed surface with SS defined above.

4.1. Using the classification of surfaces and Fact 2, or otherwise, prove that S? is homeo-

morphic to a subspace of R3.

Hint 4: Property (4) of the boundary connected sum operation listed above will be helpful.

Total marks available for this assignment: 75

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