Middle East Technical University Spring 2002 422 Elementary Geometric Topology Instructor: S. Finashin

HOMEWORK IV (DUE TO APRIL 22)

PRESENTATION OF SURFACES BY

GRAPHS. VECTOR FIELDS ON SURFACES.

Please, give your solution for each problem on a separate page. Do not forget to write your name and student’s number on every page that you submit

Problem 1. For the graph shown below

(1) sketch the surface that it represents and determine the number of its boundary components;

(2) argue why this surface, F, is orientable;

(3) choose basic cycles for F and determine their intersection number in F;

(4) determine the Euler characteristic and classify surface F;

(5) determine the Seifert matrix with your choice of basic cycles;

(6) determine the Arf invariant (if it is well-defined) for F.

Problem 2. Let F denote a Klein bottle with a hole.

(1) Choose a suitable gluing scheme for F and sketch a gradient-like vector field outward normal to the boundary of F.

(2) Identify the type of the singularities of this vector field and determine their indices.

(3) Sketch a spine of F determined by your vector field.

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