# Math Help - Induced homomorphism and isomorphism.

1. ## Induced homomorphism and isomorphism.

Does anyone have any help with these questions?

(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.

2. (2) Let $X=S^2$ the standard sphere, $A=S^1$ be the equator. $\Pi_1(X)$ is trivial since X is simply connected. While $\Pi_1(A)=Z$. So the induced homeomorphism is a constant map, not an injective.