Math 690 Introduction to Algebraic Topology Spring 2004

EXAM 2

7 problems, 10 points each

Instructions:

1. Provide complete proofs (with all the definitions whenever necessary).

2. Write the problems in the given order.

3. Start each problem on a new page.

1. Represent the Mobius band as a simplicial complex, list all the cells, their boundaries, find its Euler characteristic.

2. Provide a diagrammatic proof that

3. Prove that the Euler characteristic of a tree is 1.

4. (a) State the Classification Theorem for Surfaces; classify the following surfaces: (b) ; (c)

5. Compute the homology of the figure eight.

6. Compute the homology of the sphere with two whiskers.

7. Compute the homology of the Klein bottle.