MATH3061 Geometry and Topology
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CONFIDENTIAL EXAM PAPER
This paper is not to be removed from the exam venue
Mathematics and Statistics
EXAMINATION
Semester 2 - Main, 2019
MATH3061 Geometry and Topology
2 hours
10 minutes
EXAM CONDITIONS:
This is a CLOSED book examination - no material permitted
During reading time - writing is not permitted at all
MATERIALS PERMITTED IN THE EXAM VENUE:
(No electronic aids are permitted e.g. laptops, phones)
Calculator - non-programmable
MATERIALS TO BE SUPPLIED TO STUDENTS:
2 x 12-page answer book
INSTRUCTIONS TO STUDENTS:
Please answer the questions for the Geometry and Topology components of the course in separate booklets.
Please tick the box to confirm that your examination paper is complete. 
I. GEOMETRY — please answer in a separate booklet
1. a) Let a be an odd isometry, and l、m are distinct nonparallel lines in e. Show that 9（l → m) = z - 9（a（l) → a（m)). (You may use the fact from class that the derivative of an odd transformation maps positive bases to negative bases).
b) Show that the result of conjugating a translation r" by an isometry a is a translation.
2. a) Let o = (（x : y : z) | x2 - 2xy + y2 = 0﹔ .
i) Classify o as one of the 5 types of conics we studied.
ii) Find a symmetric matrix A such that o = oA .
iii) Find the tangent oP for all points P ∈ o.
b) Let o = (（x : y : z) | x2 -2xz+y2 = 0﹔ , o0 = (（x : y : z) | xy = z2 ﹔ . Let wM be a collineation
with wM（1 : 0 : 0) =（0 : 0 : 1)、wM（0 : 1 : 0) =（1 : 1 : 1)、wM（1 : 1 : 1) =（2 : 0 : 1).
i) Find a symmetric matrix A such that o = oA .
ii) Find a matrix M such that wM（o0 ) = o.
iii) Find o ∩ l& and classify o according to its intersection with ez (i.e. is it an ellipse or parabola or hyperbola).
3. a) Let a = rⅤ aP、M , where Ⅴ = ┌ -22、, P =（2、1), M = 
、. (Recall, aP、M is the unique affine trasnformation fixing P with derivative M.)
i) Find the coordinate expression for a.
ii) Is a an isometry?
iii) Find the coordinate expression for a2 = a Oa .
b) Let o1 and o2 be two conics that are parabolas in ez . Is there always an affine transformation of
ez mapping ez ∩ o1 to ez ∩ o2 ? Explain. (Hint: l& is tangent to both.)
II. TOPOLOGY — please answer in a separate booklet
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 at most five vertices is planar.
b) Every connected graph G has a subgraph T that is a tree and such that T and G have the same vertex set.
c) Every graph has Euler characteristic at most 1.
d) Every map on D2 can be coloured using 3 colours.
e) If a surface is connected and closed, then it embeds in R3 .
f) There is no regular polygonal decomposition of the torus T by pentagons (5-sided polygons).
g) Every surface with no boundary circles and Euler characteristic zero is homeomorphic to the torus.
5. a) Consider the Klein bottle K.
i) Draw a polygonal decomposition of K.
ii) Use surgary to write K as a standard surface. (To receive marks for this part you must use surgery.)
b) Let Z be the surface given by the word a b c d e b d f g f .
i) Is Z orientable? Justify your answer.
ii) Draw a polygonal decomposition of Z.
iii) How many boundary circles does Z have?
iv) Compute the Euler characteristic of Z.
v) Describe Z as a standard surface.
6. a) Define what it means for a knot to be n-colourable.
b) Compute the knot determinant of the following knot:
c) Determine whether or not the above knot is 5-colourable.
2021-11-28