THE UNIVERSITY OF SYDNEY

MATH3061 Geometry and Topology

Semester 2 Tutorial 10 2010

1. We saw in assignment 2 it is possible to draw K5 without accidental crossings on a torusand on a Moebius band. Is it possible to drawK5 without accidental crossings on a cylinder(annulus)?

2. Determine normal form and the Euler characteristic of an ideal (no thickness) T -shirt.

Determine the normal form and the Euler characteristic of the surface of a padded T-shirt.

3. Solid models of the letters of the alphabet

A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z,

are made from clay, shaped from solid cylindrical pieces.

Classify the surfaces of the resulting solids.

4. Show that the graph K3,3 is not planar.

5. Show that there is no regular polygonal decomposition of the torus by pentagons.

For which n is there a regular polygonal decomposition of the torus into n-gons?

6. A ball is constructed from squares and regular hexagons sewn along edges such that ateach vertex 3 edges meet. Each square is surrounded by hexagons, and each hexagon by 3squares and 3 hexagons.

Determine the number of squares and hexagons in the construction.

7. The Degenerate Regular Decompositions of the Sphere

Show that for each p 2 there is regular decomposition of the sphere into p two sidedpolygons, and dually for each n 2 a regular decomposition of the sphere into 2 polygonswith n sides.

8. Find a formula for the connected sum

A1#A2# #An

of surfaces A1, A2, . . . An, in terms of (A1), (A2), . . ., (An).