Editing Chern class (section)

== Chern polynomial ==
A Chern polynomial is a convenient way to handle Chern classes and related notions systematically. By definition, for a complex vector bundle ''E'', the '''Chern polynomial''' ''c''<sub>''t''</sub> of ''E'' is given by:
<math display="block">c_t(E) =1 + c_1(E) t + \cdots + c_n(E) t^n.</math>

This is not a new invariant: the formal variable ''t'' simply keeps track of the degree of ''c''<sub>''k''</sub>(''E'').<ref>In a ring-theoretic term, there is an isomorphism of graded rings:
<math display="block">H^{2*}(M, \Z) \to \oplus_k^\infty \eta(H^{2*}(M, \Z)) [t], x \mapsto x t^{|x|/2}</math>

where the left is the cohomology ring of even terms, η is a ring homomorphism that disregards grading and ''x'' is homogeneous and has degree |''x''|.</ref> In particular, <math>c_t(E)</math> is completely determined by the '''total Chern class''' of ''E'': <math>c(E) =1 + c_1(E) + \cdots + c_n(E)</math> and conversely.

The Whitney sum formula, one of the axioms of Chern classes (see below), says that ''c''<sub>''t''</sub> is additive in the sense:
<math display="block">c_t(E \oplus E') = c_t(E) c_t(E').</math>
Now, if <math>E = L_1 \oplus \cdots \oplus L_n</math> is a direct sum of (complex) line bundles, then it follows from the sum formula that:
<math display="block">c_t(E) = (1+a_1(E) t) \cdots (1+a_n(E) t)</math>
where <math>a_i(E) = c_1(L_i)</math> are the first Chern classes. The roots <math>a_i(E)</math>, called the '''Chern roots''' of ''E'', determine the coefficients of the polynomial: i.e.,
<math display="block">c_k(E) = \sigma_k(a_1(E), \ldots, a_n(E))</math>
where σ<sub>''k''</sub> are [[elementary symmetric polynomials]]. In other words, thinking of ''a''<sub>''i''</sub> as formal variables, ''c''<sub>''k''</sub> "are" σ<sub>''k''</sub>. A basic fact on [[symmetric polynomial]]s is that any symmetric polynomial in, say, ''t''<sub>''i''</sub>'s is a polynomial in elementary symmetric polynomials in ''t''<sub>''i''</sub>'s. Either by [[splitting principle]] or by ring theory, any Chern polynomial <math>c_t(E)</math> factorizes into linear factors after enlarging the cohomology ring; ''E'' need not be a direct sum of line bundles in the preceding discussion. The conclusion is
{{block indent | em = 1.5 | text = "One can evaluate any symmetric polynomial ''f'' at a complex vector bundle ''E'' by writing ''f'' as a polynomial in σ<sub>''k''</sub> and then replacing σ<sub>''k''</sub> by ''c''<sub>''k''</sub>(''E'')."}}

'''Example''': We have polynomials ''s''<sub>''k''</sub>
<math display="block">t_1^k + \cdots + t_n^k = s_k(\sigma_1(t_1, \ldots, t_n), \ldots, \sigma_k(t_1, \ldots, t_n))</math>
with <math>s_1 = \sigma_1, s_2 = \sigma_1^2 - 2 \sigma_2</math> and so on (cf. [[Newton's identities#Expressing power sums in terms of elementary symmetric polynomials|Newton's identities]]). The sum
<math display="block">\operatorname{ch}(E) = e^{a_1(E)} + \cdots + e^{a_n(E)} = \sum s_k(c_1(E), \ldots, c_n(E)) / k!</math>
is called the Chern character of ''E'', whose first few terms are: (we drop ''E'' from writing.)
<math display="block">\operatorname{ch}(E) = \operatorname{rk} + c_1 + \frac{1}{2}(c_1^2 - 2c_2) + \frac{1}{6} (c_1^3 - 3c_1c_2 + 3c_3) + \cdots.</math>

'''Example''': The [[Todd class]] of ''E'' is given by:
<math display="block">\operatorname{td}(E) = \prod_1^n {a_i \over 1 - e^{-a_i}} = 1 + {1 \over 2} c_1 + {1 \over 12} (c_1^2 + c_2) + \cdots.</math>

'''Remark''': The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let ''G''<sub>''n''</sub> be the [[infinite Grassmannian]] of ''n''-dimensional complex vector spaces.  This space is equipped with a tautologous vector bundle of rank <math>n</math>, say <math>E_n \to G_n</math>.  <math>G_n</math> is called the [[classifying space]] for rank-<math>n</math> vector bundles because given any complex vector bundle ''E'' of rank ''n'' over ''X'', there is a continuous map 
<math display="block">f_E: X \to G_n</math>
such that the pullback of <math>E_n</math> to <math>X</math> along <math>f_E</math> is isomorphic to <math>E</math>, and this map <math>f_E</math> is  unique up to homotopy. [[Borel's theorem]] says the cohomology ring of ''G''<sub>''n''</sub> is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials σ<sub>''k''</sub>; so, the pullback of ''f''<sub>''E''</sub> reads:
<math display="block">f_E^*: \Z [\sigma_1, \ldots, \sigma_n] \to H^*(X, \Z ).</math>
One then puts:
<math display="block">c_k(E) = f_E^*(\sigma_k).</math>

'''Remark''': Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let <math>\operatorname{Vect}_n^{\Complex}</math> be the contravariant functor that, to a CW complex ''X'', assigns the set of isomorphism classes of complex vector bundles of rank ''n'' over ''X'' and, to a map, its pullback. By definition, a [[characteristic class]] is a natural transformation from <math>\operatorname{Vect}_n^{\Complex } = [-, G_n]</math> to the cohomology functor <math>H^*(-, \Z ).</math> Characteristic classes form a ring because of the ring structure of cohomology ring. [[Yoneda's lemma]] says this ring of characteristic classes is exactly the cohomology ring of ''G''<sub>''n''</sub>:
<math display="block">\operatorname{Nat}([-, G_n], H^*(-, \Z )) = H^*(G_n, \Z ) = \Z [\sigma_1, \ldots, \sigma_n].</math>