# null-homotopic $\Leftrightarrow$ free-null-homotopic

I have some troubles with a problem, I`ve been dealing for a while, so i hope you can help me. :)

I need to show, that, if $G \subset \mathbb{C}$ is a Domain and $\gamma : [0,1] \rightarrow G$ is a closed path, then $\gamma$ is FEP-null-homotopic if and only if it is free-null-homotopic.

First I will explain the phrases, as we defined it.

Two paths are (FEP-)homotopic, if there exists a continous map $h: [0,1] \times [0,1] \rightarrow G$, such that both of the paths have the same beginning- and the same end-Point. A closed path is called (FEP-)null-homotopic, if it is in the same equivalence-class (formed by the homotopy-relation) as it's start point. This means, that we can shrink the path to it's start-point with this map.

Two closed paths $\gamma, \eta$ are free-homotopic, if the continous map $h: [0,1] \times [0,1] \rightarrow G$ holds $h(0,\tau) = h(1, \tau)$ and $h(.,0)=\gamma$ and $h(.,1)=\eta$

Now after giving all these definitions, I'll explain my thoughts (please correct me, if I'm misstaking):

FEP-null-homotopic means, that the paths can shrink to a point, wheareas the starting (=end)point is fixed. Free-homotopic means, that this point is allowed to be translated in the $\mathbb{C}$-plane.

So I suppose the "$\Rightarrow$" - direction should be finished, due to we have the special-case of a free homotopy without translation.

For the "$\Leftarrow$"-direction I think I need to show, that for each pair of free-homotopic paths there exists a continous map between them, which is homotopic. This should be possible via "moving along" with the corresponding point. The movement/translation of this point is continous, because the free-homotopic map $h$ is continous, so it is also continous for fixed $t$ in the second argument (I think this describes the "movement" of my fixed point.)

So I suppose it should be possible to create such a map, which "deletes" the movement, such that we get the wanted homotopic map.

Well, this has been my idea for a possible solution. My Problem is, that I don't know how to make a feasible proof out of that (providing, that my idea is right) and how to define the claimed homotopy.

I hope some of you can help me! :)

Thanks!