# Types of Induced Maps.

1. May 25, 2014

### WWGD

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