▁bund les ▁In ▁general ▁if ▁V ▁is ▁a ▁vector ▁bundle ▁on ▁a ▁space ▁X , ▁with ▁constant ▁fib re ▁dimension ▁n , ▁the ▁n - th ▁exterior ▁power ▁of ▁V ▁taken ▁fib re - by - f ib re ▁is ▁a ▁line ▁bundle , ▁called ▁the ▁determin ant ▁line ▁bundle . ▁This ▁construction ▁is ▁in ▁particular ▁applied ▁to ▁the ▁cot ang ent ▁bundle ▁of ▁a ▁smooth ▁manifold . ▁The ▁resulting ▁determin ant ▁bundle ▁is ▁responsible ▁for ▁the ▁phenomen on ▁of ▁tensor ▁dens ities , ▁in ▁the ▁sense ▁that ▁for ▁an ▁orient able ▁manifold ▁it ▁has ▁a ▁non van ishing ▁global ▁section , ▁and ▁its ▁tensor ▁powers ▁with ▁any ▁real ▁exponent ▁may ▁be ▁defined ▁and ▁used ▁to ▁' tw ist ' ▁any ▁vector ▁bundle ▁by ▁tensor ▁product . ▁ ▁The ▁same ▁construction ▁( t aking ▁the ▁top ▁exterior ▁power ) ▁applies ▁to ▁a ▁finit ely ▁generated ▁project ive ▁module ▁M ▁over ▁a ▁No ether ian ▁domain ▁and ▁the ▁resulting ▁invert ible ▁module ▁is ▁called ▁the ▁determin ant ▁module ▁of ▁M . ▁ ▁Character istic ▁classes , ▁universal ▁bund les ▁and ▁class ifying ▁spaces ▁The ▁first ▁St ief el – Wh it ney ▁class ▁class ifies ▁smooth ▁real ▁line ▁bund les ; ▁in ▁particular , ▁the ▁collection ▁of ▁( equ ival ence ▁classes ▁of ) ▁real ▁line ▁bund les ▁are ▁in ▁correspond ence ▁with ▁elements ▁of ▁the ▁first ▁coh om ology ▁with ▁Z / 2 Z ▁coefficients ; ▁this ▁correspond ence ▁is ▁in ▁fact ▁an ▁isomorphism ▁of ▁ab elian ▁groups ▁( the ▁ ▁group ▁operations