Homework Assignment 9
Required reading. Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read Hatcher's pages 97-133 and 149-153.
Solve the following problems. (But submit only the underlined ones). In Hatcher's book, problems 11, 12, 13, 14 on page 132, problem 31 on page 133, problem 28 on page 157 and problems 31, 32, 33 on page 158.
Due date. This assignment is due at the end of our class on Thursday, March 9, 2006.
Just for fun. A maximal tree is chosen within the edges of the integer lattice. Show that there are two neigboring points on the lattice whose distance from each other, measured only along the tree, is at least n.
|Above the line, edit only if you are sure||Feel free to edit below the line|
Problems for submission
Hatcher 2.1, pp. 132-133
- Show that if A is a retract of X then the map induced by the inclusion is injective.
- Determine whether there exists a short exact sequence . More generally, determine which abelian groups A fit into a short exact sequence with p prime. What about the case of short exact sequences ?
- Using the notation of the five-lemma, give an example where the maps α, β, δ and ε are zero but γ is nonzero. This can be done with short exact sequences in which all the groups are either or 0.
Hatcher 2.2, p. 158
- Use the Mayer–Vietoris sequence to show there are isomorphisms if the basepoints of X and Y that are identified in are deformation retracts of neighborhoods and .
- For SX the suspension of X, show by a Mayer–Vietoris sequence that there are isomorphisms for all n.