@LSpice: I was the one who was insulted, but somehow I'm out of line? And now a moderator has deleted my second comment? I suppose this is all consistent with this website's reputation, but I am still disappointed.

Oh, that's really helpful, thanks! So you really can define it just like the usual $d$ operator. I'll check out Besse's book. Strangely enough, I have it on my shelf but have never opened it -- I inherited it from a retiring colleague several years ago.

@AlexM.: You're right, I fixed the notation. As far as the level of this question, it arose while reading an advanced monograph (in fact, one I am teaching a grad topics course from right now), and I am a tenured full professor working in geometry/topology but do not know the answer. So it seems a little rude to point me instead to a website devoted to helping undergrads cheat.