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 March 2nd, 2011, 02:02 AM #1 Newbie   Joined: Nov 2010 Posts: 28 Thanks: 0 Algebraic Topology Does anyone have any help with these questions? (1) Let S^1 = {z element of Complex no. : |z| = 1} and let f : S1 -> S1 be the map defined by f(z) = z^3. What is the induced homomorphism f* : Pi1(S^1) -> Pi1(S^1)? (2) Let X be a space and let A be a subspace of X. Let i : A -> X be the inclusion map. True or false: The induced homomorphism i* : Pi1(A) -> Pi1(X) must be injective.(Justify your answer by providing either a proof or a counterexample.) (3) Let G be a finitely generated abelian group. Using the previous problem (or otherwise), construct a space X whose fundamental group is isomorphic to G. (4) Construct an example of a space X such that Pi1(X) is a cyclic group of order n, where n is some positive integer.
 March 3rd, 2011, 07:56 PM #2 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Algebraic Topology Hi Turloughmack. These are questions worth writing out good proofs for, but I can maybe give a couple hints (not that I am a topologist, but I have gone over this material before!). For the first problem, you really just need to figure out what the cubing map does to an element of the unit circle. So, for z ? S¹, write $z \=\ e^{i\theta}$ and apply the cubing map. This should be enough to tell you the homomorphism of fundamental groups. For the second problem, you should think of some very simple examples first. I would take $^{X \=\ \mathbb{R}^2}$ and $^{A \=\ X - \{(0,0)\}}$. Now think about the corresponding fundamental groups and whether the assertion holds. The third and fourth problems are very similar in spirit. Note that any cyclic group is of course abelian and finitely generated. Now, recall the Fundamental Theorem of (Finitely Generated) Abelian Groups: Every finitely generated abelian group is a product of cyclic groups. You use this fact, but the actual construction of a space with a given group requires a bit of work, and is not amenable to a very brief review. -Ormkärr-

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