▁type ▁of ▁R P ∞ , ▁the ▁real ▁project ive ▁space ▁given ▁by ▁an ▁infinite ▁sequence ▁of ▁hom ogeneous ▁coordinates . ▁It ▁car ries ▁the ▁universal ▁real ▁line ▁bundle ; ▁in ▁terms ▁of ▁hom ot opy ▁theory ▁that ▁means ▁that ▁any ▁real ▁line ▁bundle ▁L ▁on ▁a ▁C W ▁complex ▁X ▁determ ines ▁a ▁class ifying ▁map ▁from ▁X ▁to ▁R P ∞ , ▁making ▁L ▁a ▁bundle ▁is omorphic ▁to ▁the ▁pull back ▁of ▁the ▁universal ▁bundle . ▁This ▁class ifying ▁map ▁can ▁be ▁used ▁to ▁define ▁the ▁St ief el - Wh it ney ▁class ▁of ▁L , ▁in ▁the ▁first ▁coh om ology ▁of ▁X ▁with ▁Z / 2 Z ▁coefficients , ▁from ▁a ▁standard ▁class ▁on ▁R P ∞ . ▁ ▁In ▁an ▁analog ous ▁way , ▁the ▁complex ▁project ive ▁space ▁CP ∞ ▁car ries ▁a ▁universal ▁complex ▁line ▁bundle . ▁In ▁this ▁case ▁class ifying ▁maps ▁give ▁rise ▁to ▁the ▁first ▁Ch ern ▁class ▁of ▁X , ▁in ▁H 2 ( X ) ▁( integr al ▁coh om ology ). ▁ ▁There ▁is ▁a ▁further , ▁analog ous ▁theory ▁with ▁qu atern ionic ▁( real ▁dimension ▁four ) ▁line ▁bund les . ▁This ▁gives ▁rise ▁to ▁one ▁of ▁the ▁Pon try agin ▁classes , ▁in ▁real ▁four - dimensional ▁coh om ology . ▁ ▁In ▁this ▁way ▁found ational ▁cases ▁for ▁the ▁theory ▁of ▁characteristic ▁classes ▁depend ▁only ▁on ▁line ▁bund les . ▁According ▁to ▁a ▁general ▁splitting ▁principle ▁this ▁can ▁determine ▁the ▁rest ▁of ▁the ▁theory