Editing Chern class (section)

== The Chern class of line bundles ==

{{For|a sheaf theoretic description|Exponential sheaf sequence}}

(Let ''X'' be a topological space having the [[Homotopy#Homotopy equivalence|homotopy type]] of a [[CW complex]].)

An important special case occurs when ''V'' is a [[line bundle]]. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of ''X''. As it is the top Chern class, it equals the [[Euler class]] of the bundle.

The first Chern class turns out to be a [[Complete set of invariants|complete invariant]] with which to classify complex line bundles, topologically speaking. That is, there is a [[bijection]] between the isomorphism classes of line bundles over ''X'' and the elements of <math>H^2(X;\Z)</math>, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism):
<math display="block">c_1(L \otimes L') = c_1(L) + c_1(L');</math>
the [[tensor product]] of complex line bundles corresponds to the addition in the second cohomology group.<ref>{{cite book | first1=Raoul | last1=Bott| first2=Loring|last2=Tu |author1-link=Raoul Bott |title=Differential forms in algebraic topology | date=1995|publisher=Springer|location=New York [u.a.]|isbn=3-540-90613-4|page=267ff|edition=Corr. 3. print.}}</ref><ref>{{cite web |last=Hatcher |first=Allen |author-link=Allen Hatcher |title=Vector Bundles and K-theory |url=https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf |at=Proposition 3.10.}}</ref>

In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) [[holomorphic line bundle]]s by [[linear equivalence]] classes of [[Divisor (algebraic geometry)|divisor]]s.

For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.