(2) Let the standard sphere, be the equator. is trivial since X is simply connected. While . So the induced homeomorphism is a constant map, not an injective.
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.