Lab 9

Exercise 1

For each of the following sets, decide whether is is a closed surface, a surface with boundary, or not a surface:

• X0 = {(x, y, z) ∈ R^3| x^2 + y^2 + z^2 ≤ 1}
• X1 = {(w, x, y, z) ∈ R^4 | w^2 + x^2 = y^2 + z^2 = 1}
• X2 = {(x, y, z) ∈ R^3 | x, y, z ≥ 0, x + y + z = 1}
• X3 = {(x, y) ∈ R^2 | x^2 + y^2 < 1}
• X4 = {(x, y, z) ∈ R^3 | x^2 + y^2 + z^2 ≤ 1, xyz = 0}
• X5 = {(w, x, y, z) ∈ R^4 | w + z = 0, w^2 + x^2 + y^2 + z^2 = 1}.

Exercise 2

Consider the space
X = {(x, y, z) ∈ R^3| x^2 + y^2 + z^2 = 1, −0.9 ≤ x, y, z ≤ 0.9}.
This is homeomorphic to D#n for some n. What is n?

Exercise 3

Draw several different surfaces that are all homeomorphic to the connected sum of three tori.

Exercise 4

For each of the following words Wk, describe the surface Σ(Wk). In particular, you should state whether Σ(Wk) is orientable, whether it has a boundary, and which of the surfaces in the notes it is homeomorphic to

• W1 = abac
• W2 = abcc^{-1}a^{-1}b^{-1}

Exercise 5

Let V = {1, 2, . . ., n} be a set of n vertices and F ⊆ binom(V)(3) a set of ℓ = card F triangles. Give an algorithm that takes time at most proportional to n + ℓ for the following tasks:

1. decide whether or not every edge is shared by exactly two triangles;
2. decide whether or not every vertex belongs to a set of triangles whose union is a disk