▁space , ▁the ▁realiz able ▁De hn ▁invari ants ▁do ▁not ▁necessarily ▁form ▁a ▁vector ▁space , ▁because ▁scalar ▁multiplication ▁is ▁no ▁longer ▁possible , ▁but ▁they ▁still ▁form ▁a ▁subgroup . ▁ ▁Dup ont ▁and ▁Sah ▁prove ▁the ▁existence ▁of ▁the ▁exact ▁sequences ▁ ▁and ▁ ▁Here ▁ ▁denotes ▁the ▁special ▁linear ▁group , ▁and ▁ ▁is ▁the ▁group ▁of ▁M ö b ius ▁transformations ; ▁the ▁su pers cript ▁minus - sign ▁" ind ic ates ▁the ▁( − 1 )- e ig ens pace ▁for ▁the ▁in volution ▁induced ▁by ▁complex ▁conjug ation ". ▁ ▁denotes ▁the ▁special ▁unit ary ▁group . ▁The ▁subgroup ▁ ▁in ▁ ▁is ▁the ▁group ▁generated ▁by ▁the ▁whole ▁sphere . ▁Again , ▁the ▁right most ▁non zero ▁group ▁in ▁these ▁sequences ▁is ▁the ▁obst acle ▁to ▁realiz ability ▁of ▁a ▁value ▁in ▁ ▁as ▁a ▁De hn ▁invariant . ▁ ▁This ▁algebraic ▁view ▁of ▁the ▁De hn ▁invariant ▁can ▁be ▁extended ▁to ▁higher ▁dimensions , ▁where ▁it ▁has ▁a ▁motiv ic ▁interpretation ▁involving ▁algebraic ▁K - theory . ▁ ▁Rel ated ▁results ▁An ▁approach ▁very ▁similar ▁to ▁the ▁De hn ▁invariant ▁can ▁be ▁used ▁to ▁determine ▁whether ▁two ▁rect il ine ar ▁pol yg ons ▁can ▁be ▁disse cted ▁into ▁each ▁other ▁only ▁using ▁axis - parallel ▁cut s ▁and ▁transl ations ▁( r ather ▁than ▁cut s ▁at ▁arbitrary ▁angles ▁and ▁rot ations ). ▁An ▁invariant ▁for ▁this ▁kind ▁of ▁dis section ▁uses ▁the ▁tensor ▁product ▁ ▁where ▁the ▁left ▁and ▁right ▁terms