Rav Yosef Messas in Mayim Chaim 2:340 writes that chewing gum on a fast is permissible. The basis for his answer comes from the Rama 567:3 who writes that one may chew on cinnamon sticks, and other such spices, except on Yom Kippur. He notes that the MA argues, nevertheless he writes all we have(need) is the Rama.
However, it should be noted that MB learns ...
TL;DR should DEF CYLOR
(having said that, in the interest of learning would like to take a stab at it)
The Shulchan Aruch, Orach Chaim 611:2 establishes the principle that Yom Kippur is like Shabbos:
כל מלאכה שחייבים עליה בשבת חייבים עליה ביום הכפורים וכל שבשבת פטור אבל אסור גם ביום הכפורים כן
Any laborous activity for which one is liable on ...