Results 1 to 2 of 2

Math Help - Induced homomorphism and isomorphism.

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    49

    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.
    Last edited by mr fantastic; March 2nd 2011 at 04:39 PM. Reason: Deleted excess questions, re-titled.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23
    (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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Homomorphism & Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 28th 2010, 07:41 AM
  2. Replies: 4
    Last Post: February 14th 2010, 03:05 AM
  3. Isomorphism and homomorphism properties
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: September 30th 2009, 07:56 AM
  4. Ring homomorphism - isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: June 25th 2009, 03:57 AM
  5. Isomorphism of a surjective group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 7th 2008, 08:51 PM

Search Tags


/mathhelpforum @mathhelpforum