▁trivial ization ▁of ▁the ▁normal ▁( m - 2 )- plane ▁vector ▁bundle . ▁( This ▁is ▁possible ▁for ▁, ▁so ▁is ▁certainly ▁possible ▁for ▁). ▁Cho ose ▁a ▁sym ple ctic ▁basis ▁ ▁for ▁. ▁Each ▁basis ▁element ▁is ▁represented ▁by ▁an ▁embedded ▁circle ▁. ▁The ▁normal ▁( m - 1 )- plane ▁vector ▁bundle ▁of ▁ ▁has ▁two ▁trivial izations , ▁one ▁determined ▁by ▁a ▁standard ▁fr aming ▁of ▁a ▁standard ▁embedding ▁ ▁and ▁one ▁determined ▁by ▁the ▁fr aming ▁of ▁M , ▁which ▁differ ▁by ▁a ▁map ▁ ▁i . e . ▁an ▁element ▁of ▁ ▁for ▁. ▁This ▁can ▁also ▁be ▁viewed ▁as ▁the ▁fr amed ▁c ob ord ism ▁class ▁of ▁ ▁with ▁this ▁fr aming ▁in ▁the ▁ 1 - dimensional ▁fr amed ▁c ob ord ism ▁group ▁, ▁which ▁is ▁generated ▁by ▁the ▁circle ▁ ▁with ▁the ▁Lie ▁group ▁fr aming . ▁The ▁isomorphism ▁here ▁is ▁via ▁the ▁Pon tr j agin - Th om ▁construction . ▁Define ▁ ▁to ▁be ▁this ▁element . ▁The ▁Ar f ▁invariant ▁of ▁the ▁fr amed ▁surface ▁is ▁now ▁defined ▁ ▁Note ▁that ▁ ▁so ▁we ▁had ▁to ▁stabil ise , ▁taking ▁ ▁to ▁be ▁at ▁least ▁ 4 , ▁in ▁order ▁to ▁get ▁an ▁element ▁of ▁. ▁The ▁case ▁ ▁is ▁also ▁adm iss ible ▁as ▁long ▁as ▁we ▁take ▁the ▁resid ue ▁mod ulo ▁ 2 ▁of ▁the ▁fr aming . ▁▁ ▁The ▁Ar f ▁invariant ▁ ▁of ▁a ▁fr amed ▁surface ▁detect s ▁whether ▁there ▁is ▁a