isms ▁in ▁- co efficient ▁hom ology ▁ ▁are ▁both ▁zero ▁( e . g . ▁if ▁ ▁is ▁closed ). ▁The ▁intersection ▁form ▁▁ ▁is ▁non - sing ular . ▁( Top olog ists ▁usually ▁write ▁F 2 ▁as ▁. ) ▁A ▁quadratic ▁ref in ement ▁for ▁ ▁is ▁a ▁function ▁ ▁which ▁satisfies ▁ ▁Let ▁ ▁be ▁any ▁ 2 - dimensional ▁sub space ▁of ▁, ▁such ▁that ▁. ▁Then ▁there ▁are ▁two ▁possibilities . ▁Either ▁all ▁of ▁ ▁are ▁ 1 , ▁or ▁else ▁just ▁one ▁of ▁them ▁is ▁ 1 , ▁and ▁the ▁other ▁two ▁are ▁ 0 . ▁Call ▁the ▁first ▁case ▁, ▁and ▁the ▁second ▁case ▁. ▁Since ▁every ▁form ▁is ▁equivalent ▁to ▁a ▁sym ple ctic ▁form , ▁we ▁can ▁always ▁find ▁sub spaces ▁ ▁with ▁x ▁and ▁y ▁being ▁- du al . ▁We ▁can ▁therefore ▁split ▁ ▁into ▁a ▁direct ▁sum ▁of ▁sub spaces ▁is omorphic ▁to ▁either ▁ ▁or ▁. ▁Furthermore , ▁by ▁a ▁clever ▁change ▁of ▁basis , ▁ ▁We ▁therefore ▁define ▁the ▁Ar f ▁invariant ▁▁▁ ▁= ▁( number ▁of ▁copies ▁of ▁ ▁in ▁a ▁decomposition ▁Mod ▁ 2 ) ▁. ▁ ▁Ex amples ▁▁ ▁Let ▁ ▁be ▁a ▁compact , ▁connected , ▁orient ed ▁ 2 - dimensional ▁manifold , ▁i . e . ▁a ▁surface , ▁of ▁genus ▁ ▁such ▁that ▁the ▁boundary ▁ ▁is ▁either ▁empty ▁or ▁is ▁connected . ▁Em bed ▁ ▁in ▁, ▁where ▁. ▁Cho ose ▁a ▁fr aming ▁of ▁M , ▁that ▁is ▁a