Homework Assignment 9

From 0506Topology

A maximal tree in the 9x9 integer lattice.

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 $n\times n$ 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

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Problems for submission

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Hatcher $\S$ 2.1, pp. 132-133

11.: Show that if $A$ is a retract of $X$ then the map $H_n(A)\to H_n(X)$ induced by the inclusion $A \subset X$ is injective.

14.: Determine whether there exists a short exact sequence $0 \to \mathbb{Z}_4 \to \mathbb{Z}_8\times \mathbb{Z}_2 \to \mathbb{Z}_4 \to 0$ . More generally, determine which abelian groups $A$ fit into a short exact sequence $0 \to \mathbb{Z}_{p^m} \to A \to \mathbb{Z}_{p^n} \to 0$ with $p$ prime. What about the case of short exact sequences $0 \to \mathbb{Z} \to A \to \mathbb{Z}_n \to 0$ ?

31.: 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 $\mathbb{Z}$ or $0$ .

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Hatcher $\S$ 2.2, p. 158

31.: Use the Mayer–Vietoris sequence to show there are isomorphisms $\tilde H_n(X\vee Y) \approx \tilde H_n(X)\oplus\tilde H_n(Y)$ if the basepoints of $X$ and $Y$ that are identified in $X\vee Y$ are deformation retracts of neighborhoods $U\subset X$ and $V\subset Y$ .

32.: For $S X$ the suspension of $X$ , show by a Mayer–Vietoris sequence that there are isomorphisms $\tilde H_n(SX) \approx \tilde H_{n-1}(X)$ for all $n$ .