Chern class

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This article defines a characteristic class

Definition

The Chern class is a characteristic class (or collection of characteristic classes in different dimensions) for the topological group GL(n,\mathbb{C}) with coefficients in \mathbb{Z}.

Axiomatically, the Chern class can be defined as associating to every complex vector bundle p:E \to B a class c(E) \in H^*(B;\mathbb{Z}) which lives only in even degrees, such that if c_i(E) denotes the component of c(E) in the (2i)^{th} graded component, the following hold:

  1. c_i(f^*(E)) = f^*(c_i(E)) (this is the condition for being a natural transformation, part of the definition of characteristic class)
  2. c(E_1 \oplus E_2) = c(E_1) \smile c(E_2) where \smile denotes the cap product. This is a Whitney sum formula
  3. c_i(E) = 0 if i is greater than the dimension of E
  4. For the canonical complex line bundle E \to \mathbb{C}P^\infty, c_1(E) is a pre-specified generator of H^2(\mathbb{C}P^\infty; \mathbb{Z})

c is termed the total Chern class and c_i is termed the i^{th} Chern class.

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