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MATH3061

Geometry and topology

Final Examination

Semester 2

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

Fine the image of the line under .

b) Let be a collineation such that

Find the matrix M corresponding to this collineation.

2. a) Show that is a reflection if and only if V is perpendicular to the line .

b) Suppose is the line x = y and . Then  is a glide-reflection . Determine the axis d and vector W.

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 C6. (Please include a clear drawing as part of your example. It does not need to be exact.)

b) A transformation of ε 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 K by heptagons (7-sided polygons).

d) Any two orientable surfaces with the same Euler characteristic are homeomorphic.

5. Let Z be the surface given by the word .

a) Is Z orientable? Justify your answer.

b) Draw a polygonal decomposition of Z.

c) How many boundary circles does Z have?

d) Compute the Euler characteristic of Z.

e) Describe Z as a standard surface.

6. Give a polygonal decomposition for #³P².