Exercise Set III MATHS 441
Due: September 18, 2009 Dr. Fischer
1. Consider the three spaces described in Exercise 2.2 on page 30 of the text.
For each of these three spaces Xanswer the following questions:
(i) Is Xa manifold? If yes, of what dimension?
(ii) Is Xconnected?
(iii) Is Xcompact?
2. Do Exercise 2.3 on page 34 of the text.
3. Consider the four plane models of Figure 2.11 on page 40 of the text.
For each of these plane models answer the following questions about the space X,
which they represent:
(i) The vertices of the plane model end up being identified to how many
distinct points of X?
(ii) Is Xa manifold? If yes, is it orientable or non-orientable?
4. The following collection of 10 triangles (each denoted by their three vertices)
represents a triangulation of a compact surface X:
T1={v1, v2, v4}, T2={v1, v3, v5}, T3={v2, v3, v5},
T4={v2, v4, v5}, T5={v3, v4, v6}, T6={v1, v2, v6},
T7={v4, v5, v6}, T8={v1, v5, v6}, T9={v1, v3, v4},
T10 ={v2, v3, v6}.
Assemble these triangles into a plane model for X.
Can you tell what surface it is?
5. Suppose that Yis a compact surface with the property that X#Yis topologi-
cally equivalent to Xfor every compact surface X.
Show that Ymust be topologically equivalent to a 2–dimensional sphere.
6. Prove directly (without using Example 3.8) that the connected sum of two
projective planes is topologically equivalent to a Klein bottle: P2#P2≈K2.
[See the reverse side for hints!]
