Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Types of Induced Maps.

  1. May 25, 2014 #1

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    Hi again, I am trying to get a better grasp of induced maps, and trying to see the
    results that are used/assumed in defining these maps.

    Until recently, I only knew of one type of induced map, described like this:

    We have groups G, G' , with respective subgroups N,N' with ##N \triangleleft G , N'\triangleleft G'##and a homomorphism h:G→ G' . Then this somehow induces a map ##h_*##


    ##h_* G/N → G'/N' : h_*([a]_N ):=([h(a)]_N' ) ## , i.e., the coset class of a ( we can show the map is well-defined, i.e., it is independent of the choice of representative ) is sent to the coset class of the image ##h(a)## .

    This is how induced maps in , e.g., homology, homotopy are defined, or where these maps come from.

    But now I have run into some other induced maps that don't seem to have the same "source".

    These are the maps:

    1) We're given Abelian groups A,B , G, and a homomorphism ##f: A→ B##

    Then this somehow induces a homomorphism ## f_*## with:

    ##f_*: = f\otimes 1 : A \times G → B \otimes G ## defined by:

    ## f_*(a,g):= f(a)\otimes g ##


    2)Same setup, we have Abelian groups A,B,G, a homomorphism ##k: A→B## , then

    we get the induced map ##k^*:=Hom(f,1): Hom(B,G)→ Hom(A,G) ## , defined by:

    ##k^*(\Phi)(a):= \Phi(k(a))## , for ##\Phi## in Hom(B,G), a in A.


    I suspect the map 2 is just an extension to Abelian groups of the induced map on the duals of vector spaces:

    Given a linear map ##L: V→ W ##, where V,W are finite-dimensional vector spaces over the
    same field, we get the induced map ## L*: W*→ V* : L*(w*(w)):=w*(L(v) ##

    Is that it? Where does the induced map 1) come from? I know both maps are functorial (1 is covariant and 2 is contravariant): are all induced maps functorial?

    Thanks.
     
    Last edited: May 25, 2014
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Types of Induced Maps.
  1. Mapping Question (Replies: 1)

  2. Multilinear Maps (Replies: 3)

  3. Injective Mapping (Replies: 3)

  4. Conformal map (Replies: 1)

Loading...