▁an ▁exact ▁sequence ▁, ▁K ▁the ▁kernel , ▁which ▁leads ▁to ▁a ▁long ▁exact ▁sequence , ▁dro pping ▁the ▁coefficient ▁ ▁and ▁using ▁: ▁, ▁called ▁the ▁G ys in ▁sequence . ▁ ▁Example ▁ ▁Let ▁ ▁be ▁an ▁obvious ▁projection . ▁First ▁assume ▁ ▁with ▁coordinates ▁ ▁and ▁consider ▁a ▁k - form : ▁ ▁Then , ▁at ▁each ▁point ▁in ▁M , ▁ ▁From ▁this ▁local ▁calculation , ▁the ▁next ▁formula ▁follows ▁easily : ▁if ▁ ▁is ▁any ▁k - form ▁on ▁▁ ▁where ▁ ▁is ▁the ▁restriction ▁of ▁ ▁to ▁. ▁ ▁As ▁an ▁application ▁of ▁this ▁formula , ▁let ▁ ▁be ▁a ▁smooth ▁map ▁( th ought ▁of ▁as ▁a ▁hom ot opy ). ▁Then ▁the ▁composition ▁ ▁is ▁a ▁hom ot opy ▁operator : ▁ ▁which ▁implies ▁ ▁indu ces ▁the ▁same ▁map ▁on ▁coh om ology , ▁the ▁fact ▁known ▁as ▁the ▁hom ot opy ▁in var iance ▁of ▁de ▁R ham ▁coh om ology . ▁As ▁a ▁cor ollary , ▁for ▁example , ▁let ▁U ▁be ▁an ▁open ▁ball ▁in ▁R n ▁with ▁center ▁at ▁the ▁origin ▁and ▁let ▁. ▁Then ▁, ▁the ▁fact ▁known ▁as ▁the ▁Po inc ar é ▁lemma . ▁ ▁Pro jection ▁formula ▁ ▁Given ▁a ▁vector ▁bundle ▁ π ▁: ▁E ▁→ ▁B ▁over ▁a ▁manifold , ▁we ▁say ▁a ▁differential ▁form ▁α ▁on ▁E ▁has ▁vertical - comp act ▁support ▁if ▁the ▁restriction ▁ ▁has ▁compact ▁support ▁for ▁each ▁b ▁in ▁B . ▁We ▁written ▁ ▁for ▁the ▁vector ▁space