▁being ▁tensor ▁product ▁of ▁line ▁bund les ▁and ▁the ▁usual ▁addition ▁on ▁coh om ology ). ▁ ▁Anal og ously , ▁the ▁first ▁Ch ern ▁class ▁class ifies ▁smooth ▁complex ▁line ▁bund les ▁on ▁a ▁space , ▁and ▁the ▁group ▁of ▁line ▁bund les ▁is ▁is omorphic ▁to ▁the ▁second ▁coh om ology ▁class ▁with ▁integer ▁coefficients . ▁ ▁However , ▁bund les ▁can ▁have ▁equivalent ▁smooth ▁structures ▁( and ▁thus ▁the ▁same ▁first ▁Ch ern ▁class ) ▁but ▁different ▁hol omorphic ▁structures . ▁ ▁The ▁Ch ern ▁class ▁statements ▁are ▁easily ▁proven ▁using ▁the ▁exponential ▁sequence ▁of ▁she aves ▁on ▁the ▁manifold . ▁ ▁One ▁can ▁more ▁generally ▁view ▁the ▁classification ▁problem ▁from ▁a ▁hom ot opy - the or etic ▁point ▁of ▁view . ▁ ▁There ▁is ▁a ▁universal ▁bundle ▁for ▁real ▁line ▁bund les , ▁and ▁a ▁universal ▁bundle ▁for ▁complex ▁line ▁bund les . ▁According ▁to ▁general ▁theory ▁about ▁class ifying ▁spaces , ▁the ▁he ur istic ▁is ▁to ▁look ▁for ▁contract ible ▁spaces ▁on ▁which ▁there ▁are ▁group ▁actions ▁of ▁the ▁respective ▁groups ▁C 2 ▁and ▁S 1 , ▁that ▁are ▁free ▁actions . ▁Those ▁spaces ▁can ▁serve ▁as ▁the ▁universal ▁principal ▁bund les , ▁and ▁the ▁quot ients ▁for ▁the ▁actions ▁as ▁the ▁class ifying ▁spaces ▁B G . ▁In ▁these ▁cases ▁we ▁can ▁find ▁those ▁explicitly , ▁in ▁the ▁infinite - dimensional ▁analog ues ▁of ▁real ▁and ▁complex ▁project ive ▁space . ▁ ▁Therefore ▁the ▁class ifying ▁space ▁BC 2 ▁is ▁of ▁the ▁hom ot opy