It wasn't his wife, but a lover and a roommate, it's explained on page 9. Right before he leaves for his new job at the airport (starting at page 19) they are having the discussion that they're basically "splitting up", because she's getting to get her life back together and she wants to do it on her own. She also agrees that he has to find his own "happiness" in life, so they both agree to separate and live their own lives from then on. At the very end right before Arthur sends his suitcase back to Earth (page 117) he says "there are actually a few people I"m going to miss", I'm assuming Kate is one of them.
Now, what I'm actually wondering is, are his parents still alive? Because it's explained they moved back to Anubia, yet there's no mention of them after that, even when Arthur gets to Anubia.