Math 690 Introduction to Algebraic Topology Spring 2004

EXAM 2

10 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. Provide a diagrammatic proof of the fact MATH

  2. Is the following simplicial complex a surface: ABD, BCD, ACD, ABE, BCE, ACE?

  3. Identify the surface which has planar diagram with outer edges labeled as follows: $aabcb^{-1}c^{-1}.$

  4. Compute the Euler characteristic of the union of two touching spheres.

  5. Compute the homology of the projective plane.

  6. Give the definition of the homology of a simplicial function. State the relevant theorems.

  7. The boundary of the Mobius band $M$ is a circle. Let MATH be the function that wraps the circle onto this boundary. Compute the homology of $f.$

  8. Suppose $f:K\rightarrow L$ is a deformation retraction. Prove that MATH is an isomorphism.

  9. Compute the homology groups of MATH-axis $\cup$ $y$-axis $\cup$ $z$-axis$).$

  10. Prove the Brouwer Fixed Point Theorem.

Bonus Problem (5 points): What is the relation between the Lefschetz number and the Euler characteristic?

This document created by Scientific WorkPlace 4.1.