Book: The Differential Topology of Loop Spaces
Why: Because they are one of the first examples of spaces that are almost, but not quite, entirely unlike manifolds. They are relatively straightforward spaces which can be fairly conceptually grasped, but still contain enough intricacies to reveal some of the important differences between finite and infinite dimensions (though perhaps I should say between manifolds modelled on Banach spaces and more general manifolds). A book on their differential topology would thus be a gentle introduction to the topic than is (as far as I'm aware) currently available (in particular, although just about everything I'd want to say is covered in Kriegl and Michor's works, it's in such a context and with such generality that "daunting" doesn't quite cut the mustard).
Who For: Me, 10 years ago. That is, I'd try to write the book I wish I'd had when starting out in infinite dimensional differential topology so I wouldn't have made all the mistakes that I made.
Why Me: Because I work in that area and I think I've made just about every wrong assumption about loop spaces possible so I know lots of the traps for unwary differential topologists venturing out into the miasma that is infinite dimensional topology.
Will I Ever Actually Write It: Maybe, maybe not (vote for this answer if you want me to!). I made a start by writing up some seminar notes. I've started transferring them in to the nLab (but in the process I've been generalising them which slightly goes against the purpose of the project as I described it above). I'd certainly like to write it, if only to convince myself that I no longer have all those false assumptions, but whether or not I ever actually do it ... (hey, I've an idea, maybe all the time I put into MO and meta.MO could be reallocated to book-writing. Then it'll be finished next week.).
