Unless I was taught wrong they build using a crane. Then remove the cross section of the crane and use the vertical portion as the elevator shaft. (Skyscrapers) Having watched them build Miller Park (Now American Family Field baseball stadium Milwaukee) I can verify that they have mobile cranes that are actually big/tall enough to build stadiums and arenas. It is VERY dangerous, however, as was proven when wind caused the partial collapse of the roof while they were building Miller Park, killing 3 construction workers.
They also use something called [slip forming.](https://en.wikipedia.org/wiki/Slip_forming) The concrete form for the building is constantly moving upwards. Fresh concrete gets poured up top, hard cured concrete comes the bottom. The forms moves up by jacks secured to the hardened concrete.
[https://www.youtube.com/watch?v=XEAc11FHzpY](https://www.youtube.com/watch?v=XEAc11FHzpY)
[https://www.youtube.com/watch?v=rIMxXsyzzM8](https://www.youtube.com/watch?v=rIMxXsyzzM8)
Cranes can then be attached to the top, like others have mentioned, to haul up all the other stuff like cladding and mechanical equipment.
There is an entire engineering branch that design how to do it. The hardest are bridges. 90% of a bridge design is how to put it there.
Skyscrapers are pretty easy in comparison, you build 10 floors, than put a crane on the 10th floor and make another 10 floors, until you get to the top. Stadiums are even easier, except for the roof, stadium cover is as complex as a bridge to be put in place.
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