Results 1 to 3 of 3

Math Help - Homotopy Functor

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    3

    Homotopy Functor

    Show that the relative singular homology group is a covariant homotopy functor from the category
    of pairs of topological spaces (X,A) to the category of graded abelian groups.

    Thanks a ton!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by sorrow View Post
    Show that the relative singular homology group is a covariant homotopy(homology ?) functor from the category
    of pairs of topological spaces (X,A) to the category of graded abelian groups.

    Thanks a ton!!
    Let X be a topological space and A be a subspace of X.
    For a pair (X, A), we can consider the following infinite sequence of homology groups and their homomorphisms.

    \cdots H_n(A) \longrightarrow H_n(X) \longrightarrow H_n(X,A)  \longrightarrow H_{n-1}(A) \longrightarrow H_{n-1}(X) \cdots \longrightarrow H_0(X, A) \longrightarrow 0 .

    H_n is a functor from the category of topological spaces to the category of (graded) abelian groups such that H_n:Top \rightarrow Ab given by X \mapsto H_n(X). To check it is a covariant functor, consider the topological inclusion map i:A \rightarrow X. It induces an abelian group homomorphism H_n(i):H_n(A) \rightarrow H_n(X) (If it induces H_n(X) \rightarrow H_n(A) , then H_n is a contravariant functor).

    In a similar vein, consider a topological pair (X, A). H_n assigns (X, A) to H_n(X, A) and X,A) \rightarrow (Y,B)" alt="fX,A) \rightarrow (Y,B)" /> to H_n(f):H_n(X,A) \rightarrow H_n(Y,B).You also need to check H_n satisfies the functoriality (link).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    3
    thank you for your help
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Composition of homotopy equivalences is a homotopy equivalence.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 30th 2011, 08:33 PM
  2. Forgetful Functor
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: September 8th 2011, 04:23 PM
  3. Replies: 2
    Last Post: November 3rd 2010, 05:15 PM
  4. homotopy
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: June 10th 2010, 05:20 AM
  5. Additive functor
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 6th 2009, 10:17 AM

Search Tags


/mathhelpforum @mathhelpforum