MATH3061
Geometry and topology
Final Examination
Semester 2
Time Allowed: 2 hours + 10 minutes reading time
This examination paper consists of three pages.
INSTRUCTIONS TO CANDIDATES
1. Please answer the questions for the Geometry and Topology components of the
course in separate booklets.
2. This is a closed book exam.
3. The use of electronic devices, including calculators, is not permitted.
4. The paper has six questions, with most questions having multiple parts.
5. This examination paper is worth 66% of your final assessment for MATH3061.
6. Please write carefully and legibly. Your final answers should be written using
ink and not pencil.
Page 1 of 3
Page 2 of 3 MATH3061 EXAM, SEMESTER 2 Semester 2, 2019
GEOMETRY — Upload your solution as an assignment in canvas.
Please write carefully and legibly. Your final answers should be written using ink and
not pencil. In order to get full credit, show all working, and use the notation
introduced in lectures. Present your arguments clearly, using words of
explanation and diagrams where relevant.
1. a) Let be a collineation given by the matrix
=
⎡⎢⎢⎣
1 0 2
0 1 1
1 0 4
⎤⎥⎥⎦
Fine the image of the line ∞ under .b) Let be a collineation such that
(1 ∶ 0 ∶ 0) = (0 ∶ 1 ∶ 1), (0 ∶ 1 ∶ 0) = (1 ∶ 0 ∶ 1),
(0 ∶ 0 ∶ 1) = (0 ∶ 0 ∶ 1), (1 ∶ 1 ∶ 1) = (1 ∶ 1 ∶ 0).
Find the matrix corresponding to this collineation.
2. a) Show that is a reflection if and only if is perpendicular to the line .
b) Suppose is the line = and =
[
1
0
]
. Then is a glide-reflection ,.
Determine the axis and vector .
3. Give your own original examples of each of the following. (Briefly justify your
answers.)
a) A figure whose full symmetry group is of type 6. (Please include a clear drawingas part of your example. It does not need to be exact.)
b) A transformation of  that fixes a line pointwise but is not a reflection or identity.
c) An affine transformation that is not an isometry and does not fix any line.
TOPOLOGY — Upload your solution as an assignment in canvas.
4. Prove or give a counterexample for the following statements. That is, prove the
statement if it is true. If the statement is false, give an example where it fails.
a) Every graph with Euler characteristic -8 is planar.
b) If a surface is connected and closed, then it embeds in ℝ3.
c) There is no regular polygonal decomposition of the Klein bottle by heptagons
(7-sided polygons).
d) Any two orientable surfaces with the same Euler characteristic are homeomorphic.
5. Let be the surface given by the word .
a) Is orientable? Justify your answer.
b) Draw a polygonal decomposition of .
c) How many boundary circles does have?
d) Compute the Euler characteristic of .
e) Describe as a standard surface.
6. Give a polygonal decomposition for #3ℙ2.
This is the last page of the exam paper.