So Fallout 4 will introduce a new (vanilla) feature - city building.

And, once more, we slip into the role of a freshly hatched Vault Dweller.

So, what are the chances we get to see a G.E.C.K. (Garden of Eden Creation Kit) in action this time?

Seems like an obvious connection when you think about it. Newly opened Vault, single survivor - there's a good chance a G.E.C.K. was waiting in one of the storage rooms.

And for reference: http://fallout.wikia.com/wiki/Garden_of_Eden_Creation_Kit

The G.E.C.K. used in the east coast vaults doesn't work, as stated by Bruan in Fallout 3.

Highly doubt it... Plus, in all previous fallouts, a GECK was just some seeds and a few other things I believe. The fallout 3 GECK made no sense at all... And didn't work

http://fallout.wikia.com/wiki/Fallout_Bible_6#Life_with_the_GECK

"I'll start by saying that the GECK is a plot device. A McGuffin. It had the ability to save Arroyo when in the hands of the Chosen One or a learned member of the wastes.

As a crude plot device, it may also be used as seen fit to create plots and plant new and exciting adventure seeds as needed. As a result, all of the material in this section is subject to change based on the whims of whoever wants to play with the GECK. If you want it to be a magic box of 1950s science, that's cool - we might do it, too. However, my current take on it is, it's not some miracle device, it's a little more down to earth - more like a deconstruction kit, if you will."

What about Vault city?

As above stated, I guess the GECK isn't really a specific thing or does it do anything specific, rather is something that the developers can use as a plot device. In advertisemants in fallout it states that it contains seeds and fertilizer, while it is also stated to terraphorm things.