Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is 'twisted' — and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is 'twisted' — and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Let G be a topological group, and for a topological space X {displaystyle X} , write b G ( X ) {displaystyle b_{G}(X)} for the set of isomorphism classes of principal G-bundles over X {displaystyle X} . This b G {displaystyle b_{G}} is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map f : X → Y {displaystyle fcolon X o Y} to the pullback operation f ∗ : b G ( Y ) → b G ( X ) {displaystyle f^{*}colon b_{G}(Y) o b_{G}(X)} . A characteristic class c of principal G-bundles is then a natural transformation from b G {displaystyle b_{G}} to a cohomology functor H ∗ {displaystyle H^{*}} , regarded also as a functor to Set. In other words, a characteristic class associates to each principal G-bundle P → X {displaystyle P o X} in b G ( X ) {displaystyle b_{G}(X)} an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(f*P) = f*c(P). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology. Characteristic classes are elements of cohomology groups; one can obtain integers from characteristic classes, called characteristic numbers. Some important examples of characteristic numbers are Stiefel–Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented manifold M of dimension n with fundamental class [ M ] ∈ H n ( M ) {displaystyle in H_{n}(M)} , and a G-bundle with characteristic classes c 1 , … , c k {displaystyle c_{1},dots ,c_{k}} , one can pair a product of characteristic classes of total degree n with the fundamental class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions of n into deg c i {displaystyle {mbox{deg}},c_{i}} . Formally, given i 1 , … , i l {displaystyle i_{1},dots ,i_{l}} such that ∑ deg c i j = n {displaystyle sum {mbox{deg}},c_{i_{j}}=n} , the corresponding characteristic number is: where ⌣ {displaystyle smile } denotes the cup product of cohomology classes.These are notated various as either the product of characteristic classes, such as c 1 2 {displaystyle c_{1}^{2}} or by some alternative notation, such as P 1 , 1 {displaystyle P_{1,1}} for the Pontryagin number corresponding to p 1 2 {displaystyle p_{1}^{2}} , or χ {displaystyle chi } for the Euler characteristic.