??? general idea (pointed out by fred wilhelm in seminar the other day...) of situation where "mapping cone" of nice interesting geometric map specified up to diffeomorphism (??? ....) is nice geometrically interesting space specified up to diffeomorphism ... ??? .... ???examples given by real and complex projective planes??? .... ???? making me now try to tie this in with semi-recent speculations of mine about ... ???... information about attaching maps involved in schubert stratification encoded in root system ..... ???? .... ????....