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