▁defined ▁for ▁ 4 k - dimensional ▁manif olds ▁( d ou b ly ▁even - dimensional ); ▁this ▁ 4 - fold ▁period icity ▁corresponds ▁to ▁the ▁ 4 - fold ▁period icity ▁of ▁L - theory . ▁The ▁Ar f ▁invariant ▁can ▁also ▁be ▁defined ▁more ▁generally ▁for ▁certain ▁ 2 k - dimensional ▁manif olds . ▁ ▁De fin itions ▁ ▁The ▁Ar f ▁invariant ▁is ▁defined ▁for ▁a ▁quadratic ▁form ▁q ▁over ▁a ▁field ▁K ▁of ▁characteristic ▁ 2 ▁such ▁that ▁q ▁is ▁n ons ing ular , ▁in ▁the ▁sense ▁that ▁the ▁associated ▁bil ine ar ▁form ▁ ▁is ▁non de generate . ▁The ▁form ▁ ▁is ▁altern ating ▁since ▁K ▁has ▁characteristic ▁ 2 ; ▁it ▁follows ▁that ▁a ▁n ons ing ular ▁quadratic ▁form ▁in ▁characteristic ▁ 2 ▁must ▁have ▁even ▁dimension . ▁Any ▁binary ▁( 2 - dimensional ) ▁n ons ing ular ▁quadratic ▁form ▁over ▁K ▁is ▁equivalent ▁to ▁a ▁form ▁ ▁with ▁ ▁in ▁K . ▁The ▁Ar f ▁invariant ▁is ▁defined ▁to ▁be ▁the ▁product ▁. ▁If ▁the ▁form ▁ ▁is ▁equivalent ▁to ▁, ▁then ▁the ▁products ▁ ▁and ▁ ▁differ ▁by ▁an ▁element ▁of ▁the ▁form ▁ ▁with ▁ ▁in ▁K . ▁These ▁elements ▁form ▁an ▁add itive ▁subgroup ▁U ▁of ▁K . ▁Hence ▁the ▁cos et ▁of ▁ ▁mod ulo ▁U ▁is ▁an ▁invariant ▁of ▁, ▁which ▁means ▁that ▁it ▁is ▁not ▁changed ▁when ▁ ▁is ▁replaced ▁by ▁an ▁equivalent ▁form . ▁ ▁Every ▁n ons ing ular ▁quadratic ▁form