It seems that you're using an outdated browser. Some things may not work as they should (or don't work at all).
We suggest you upgrade newer and better browser like: Chrome, Firefox, Internet Explorer or Opera

The culture need is fulfilled if a competitor walks along the mansion, regardless of his action, so having a gymnasium send its workers to the stadium alongside the mansion is sufficient. The stadium doesn't have to be located anywhere near the elite housings.
"Just to throw my two cents in... You need to do more than just build the stadium. Gymnasium produce walkers that must visit the house. A bit odd if you think about it, but thems the breaks."

Edit: Corrected. My original point a bit clarified, a designated walker needs to visit the house. Your standard Gym guy will not do it. So if you built a loop where the competitor always takes the north path, the south will never upgrade. This is why in many elite layouts you will see multiple gyms.
or, if youre being lazy / on a low difficulty, just put them in a line. then he has to walk by to get to the stadium.

alternatively, if you want to be super cheesy, you can edit the games files so they dont even need culture / other to evolve all the way. just remember to keep food and some disaster prevention since thats timed to go off no matter what its set to and they would complain about having no food.