MATH 7520: Algebraic Topology Homework

## MATH 7520   Algebraic Topology

### Homework Problems

1. Show that a Δ-complex is a CW-complex.

2. If X is the torus, the real projective plane, the Klein bottle, your favorite other surface, exhibit a Δ-complex structure on X, and use it to compute the simplicial homology of X.

3. Problem #8 from §2.1 in Hatcher

4. Problem #23 from §2.1 in Hatcher

5. Problems #11 - 13 from §2.1 in Hatcher

6. Problems #16, 17, 20, 22 from §2.1 in Hatcher

7. Compute the local homology groups H*(CT,CT-x), where T is the 2-dimensional torus, CT is the cone on T, and x is the cone point.

8. Look at problems #1 - 8 from §2.2 in Hatcher

9. Compute H*( RPn ; Z/k ).

10. Compute the homology and cohomology of the configuration space of 3 ordered points in the plane.

11. Problems #6, 8, 9 from §3.1 in Hatcher

12. Determine the structure of the cohomology rings H*( RP3 ; Z/2 ) and H*( K ; Z/2 ), where K is the Klein bottle.

13. Problem #1 from §3.2 in Hatcher

Dan Cohen   Fall 2007
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