Tapping on opponent's turn(context question)

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I know the rules for tap abilities and mana abilities of creatures and such that can be used on your opponent's turn. My question is, say I just attacked and dealt my damage and said "And I'll end my turn", can another player say "At the end of you're turn I'll activate this ability", which would cost 4 mana and put out a token? Or do they have to do it BEFORE I say that my turn is over?

So it would look like this:
Combat phase: I attack. They choose not to block. Take X damage.
I have nothing left to do so I declare my turn over.
THEN they say "In response to the ending of your turn I will use this", tap 4 mana, put a token out, and then their turn starts so they can untap their mana back?

There's been some debate in the house over when to use abilities on opponent's turns so I'm looking to the knowledge of you fine humans.
I am Red/Black
say I just attacked and dealt my damage and said "And I'll end my turn", can another player say "At the end of you're turn I'll activate this ability", which would cost 4 mana and put out a token? Or do they have to do it BEFORE I say that my turn is over?

When you say "i'll end my turn", you're really proposing a shortcut that means "i offer to pass priority at every opportunity until you have priority in my end step". Your opponent can either accept the shortcut and then play something in your end step, accept the shortcut and then do nothing (thus starting his turn), or reject the shortcut and act somewhere in the middle (for example, he could cast something in your 2nd main phase). Either way, the game doesn't progress to a new step, phase, or turn unless you both want it to.
Magic isn't a game of who can speak the fastest.  Your opponent gets a chance to cast instants and activate abililties on your end step.  Saying you're done doesn't change that. 
That's why I was asking. It was a context question. There was debate on whether or not "I'll end my turn" means that they'll end their turn or if it really meant "I'm done unless you have something you want to play right quick so you don't have to tap your mana during your own turn". But thanks for the clarification. Now I can put this to rest amongst the roommates lol
I am Red/Black