Historically you would use the umlaut (lit. re-sound in German) to signify that both vowels are pronounced separately and not as a diphthong! I think some publications, like The New Yorker, are pretentious enough to still use it… In this case, cooperate would be spelled coöperate.
Edit: Oops! Meant to reply to Geek_King
Freakin’ Nazca lines over here