▁, ▁which ▁depends ▁on ▁the ▁fr aming . ▁For ▁ ▁and ▁ ▁represented ▁by ▁an ▁embedding ▁ ▁the ▁value ▁ ▁is ▁ 0 ▁or ▁ 1 , ▁according ▁as ▁to ▁the ▁normal ▁bundle ▁of ▁ ▁is ▁trivial ▁or ▁not . ▁The ▁Ker va ire ▁invariant ▁of ▁the ▁fr amed ▁- dimensional ▁manifold ▁M ▁is ▁the ▁Ar f ▁invariant ▁of ▁the ▁quadratic ▁ref in ement ▁ ▁on ▁. ▁The ▁Ker va ire ▁invariant ▁is ▁a ▁hom omorphism ▁ ▁on ▁the ▁- dimensional ▁stable ▁hom ot opy ▁group ▁of ▁spher es . ▁The ▁Ker va ire ▁invariant ▁can ▁also ▁be ▁defined ▁for ▁a ▁- dimensional ▁manifold ▁M ▁which ▁is ▁fr amed ▁except ▁at ▁a ▁point . ▁▁ ▁In ▁surg ery ▁theory , ▁for ▁any ▁- dimensional ▁normal ▁map ▁ ▁there ▁is ▁defined ▁a ▁n ons ing ular ▁quadratic ▁form ▁ ▁on ▁the ▁- co efficient ▁hom ology ▁kernel ▁▁▁▁ ▁ref ining ▁the ▁hom ological ▁intersection ▁form ▁. ▁The ▁Ar f ▁invariant ▁of ▁this ▁form ▁is ▁the ▁Ker va ire ▁invariant ▁of ▁( f , b ). ▁In ▁the ▁special ▁case ▁ ▁this ▁is ▁the ▁Ker va ire ▁invariant ▁of ▁M . ▁The ▁Ker va ire ▁invariant ▁features ▁in ▁the ▁classification ▁of ▁ex otic ▁spher es ▁by ▁Michel ▁Ker va ire ▁and ▁John ▁Mil nor , ▁and ▁more ▁generally ▁in ▁the ▁classification ▁of ▁manif olds ▁by ▁surg ery ▁theory . ▁William ▁Brow der ▁defined ▁ ▁using ▁functional ▁Ste en rod ▁squares , ▁and ▁C . ▁T . ▁C . ▁Wall ▁defined ▁ ▁using ▁fr amed ▁imm