NOTES ON THE CHERN-CHARACTER

Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the paper a proof of the existence of a Chern-character from complex K-theory to any cohomology theory with values in graded Q-algebras is given. The Chern-character respects the Adams and Steenrod operations.


Introduction
The aim of this note is to give an axiomatic and elementary treatment of Cherncharacters of vectorbundles with values in a class of cohomology-theories arising in topology and algebra. Given a theory of Chern-classes for complex vectorbundles with values in singular cohomology one gets in a natural way a Chern-character from complex K-theory to singular cohomology using the projective bundle theorem and the Newton polynomials. The Chern-classes of a complex vectorbundle may be defined using the notion of an Euler class (see section 14 in [6]) and one may prove that a theory of Chern-classes with values in singular cohomology is unique. In this note it is shown one may relax the conditions on the theory for Chern-classes and still get a Chern-character. Hence the Chern-character depends on some choices.
Many cohomology theories which associate to a space a graded commutative Qalgebra H * satisfy the projective bundle property for complex vectorbundles. This is true for De Rham-cohomology of a real compact manifold, singular cohomology of a compact topological space and complex K-theory. The main aim of this note is to give a self contained and elementary proof of the fact that any such cohomology theory will recieve a Chern-character from complex K-theory respecting the Adams and Steenrod operations.
Complex K-theory for a topological space B is considered, and characteristic classes in K-theory and operations on K-theory such as the Adams operations are constructed explicitly, following [5].
The main result of the note is the following (Theorem 4.9): Theorem 1.1. Let H * be any rational cohomology theory satisfying the projective bundle property. There is for all k ≥ 1 a commutative diagram where Ch is the Chern-character for H * , ψ k is the Adams operation and ψ k H is the Steenrod operation.
The proof of the result is analogous to the proof of existence of the Cherncharacter for singular cohomology.

Euler classes and characteristic classes
In this section we consider axioms ensuring that any cohomology theory H * satisfying these axioms, recieve a Chern-character for complex vectorbundles. By a cohomology theory we mean a contravariant functor H * : T op → Q − algebras from the category of topological spaces to the category of graded commutative Qalgebras with respect to continuous maps of topological spaces. We say the theory satisfy the projective bundle property if the following axioms are satisfied: For any rank n complex continuous vectorbundle E over a compact space B There is an Euler class where π : P(E) → B is the projective bundle associated to E. This assignment satisfy the following properties: The Euler class is natural, i.e for any map of topological spaces f : The map π * induce an injection π * : H * (B) → H * (P(E)) and there is an equality ., u n−1 E }. Assume H * satisfy the projective bundle property. There is by definition an equation in H * (P(E)) .
for every complex finite rank vectorbundle E on B satisfying the following axioms: Note: if φ : H * → H * is a functorial endomorphism of H * which is a ringhomomorphism and c is a theory of characteristic classes, it follows the assignment E → c i (E) = φ(c i (E)) is a theory of characteristic classes.
) is a theory satisfying Definition 2.3. Note furthermore: Assume γ 1 is the tautological linebundle on P 1 . Since we do not assume c 1 (γ 1 ) = z where z is the canonical generator of H 2 (P 1 , Z) it does not follow that an assignment E → c i (E) is uniquely determined by the axioms 2.3.1 − 2.3.3. We shall see later that the axioms 2.3.1 − 2.3.3 is enough to define a Chern-character. Proof. We verify the axioms for a theory of characteristic classes. Axiom 2.3.1: Assume we have a map of rank n bundles f : F → E over a map of topological spaces g : B ′ → B. We pull back the equation and this is ok.
Given a compact topological space B. We may consider the Grothendieck-ring K * C (B) of complex finite-dimensional vectorbundles. It is defined as the free abelian group on isomorphism-classes [E] where E is a complex vectorbundle, modulo the subgroup generated by elements of the type It has direct sum as additive operation and tensor product as multiplication. Assume E is a complex vectorbundle of rank n and let π : P(E) → B be the associated projective bundle. We have a projective bundle theorem for complex K-theory: Theorem 2.6. The group K * (P(E)) is a free K * (B) module of finite rank with generator u -the euler class of the tautological line-bundle. The elements {1, u, u 2 , .., u n−1 } is a free basis.
As in the case of singular cohomology, we may define characteristic classes for complex bundles with values in complex K-theory using the projective bundle theorem: The element u n satisfies an equation in K * (P(E)). One verifies the axioms defined above are satisfied, hence one gets characteristic classes c i (E) ∈ K * C (B) for all i = 0, .., n Theorem 2.7. The characteristic classes c i (E) satisfy the following properties: where E is any vectorbundle, and L is a line bundle.

Adams operations and Newton polynomials
We introduce some cohomology operations in complex K-theory and Newtonpolynomials and prove elementary properties following the book [5].
Let Φ(B) be the abelian monoid of elements of the type hence the map λ t is a map of abelian monoids, hence gives rise to a map from the additive abelian group K * C (B) to the set of powerseries with constant term equal to one. Explicitly the map is as follows: When n denotes the trivial bundle of rank n we get the explicit formula hence it follows that We get operations We next define Newton polynomials using the elementary symmetric functions. Let u 1 , u 2 , u 3 , .. be independent variables over the integers Z, and let Q k = u k 1 + u k 2 + · · · + u k k for k ≥ 1. It follows Q k is invariant under permutations of the variables u i : for any σ ∈ S k we have σQ k = Q k hence we may express Q k as a polynomial in the elementary symmetric functions σ i : We define s k (σ) = Q k (σ 1 , σ 2 , .., σ k ) to be the k ′ th Newton polynomial in the variables σ 1 , σ 2 , .., σ k where σ i is the i ′ th elementary symmetric function. One checks the following: Let n ≥ 1 and consider the polynomial p(1) = (1 + tu 1 )(1 + tu 2 ) · · · (1 + tu n ) = t n σ n + t n−1 σ n−1 + · · · + tσ 1 + 1 where σ i = σ i (u 1 , .., u n ) is the ith elementary symmetric polynomial in the variables u 1 , u 2 , .., u n . Lemma 3.1. There is an equality Q k (σ 1 (u 1 , .., u n ), σ 2 (u 1 , .., u n ), .., σ k (u 1 , .., u n )) = u k 1 + u k 2 + · · · + u k n . Proof. Trivial.
Assume we have virtual elements x = E − n = ⊕ n (L i − 1) and y = F − p = ⊕ p (R j − 1) in complex K-theory K * C (B). We seek to define a cohomology-operation c on complex K-theory using a formal powerseries We define the element Proof. We have by definition And the proposition follows.
We state a Theorem:  Proof. The proof follows the proof in [5], Proposition IV.7.11. We may by the remark above assume x = E − n and y = F − p where x, y ∈ K ′ C (B). We may also from Theorem 3.3 assume E = ⊕ n L i and F = ⊕ p R j where L i , R j are linebundles. We get the following: ., u n , v 1 , .., v p ). We get: and the claim follows.
We may give an explicit and elementary construction of the Adams-operations: Theorem 3.5. Let k ≥ 1. There are functorial operations where L is a line bundle. The operations ψ k are the only operations that are ringhomomorphisms -the Adams operations Proof. We need: We have in K-theory: We get the series The following operator is an explicit construction of the Adams-operator. One may verify the properties in the theorem, and the claim follows.
Assume E, F are complex vectorbundles on B and consider the Chern-polynomial ., σ k ) where σ i is the ith elementary symmetric function in the u i 's.
Proposition 3.6. The following holds: ) and the claim follows.

The Chern-character and cohomology operations
We construct a Chern-character with values in singular cohomology, using Newtonpolynomials and characteristic classes following [5]. The k ′ th Newton-classe s k (E) of a complex vectorbundle will be defined using characteristic classes of E: c 1 (E), .., c k (E) and the k ′ th Newton-polynomial s k (σ 1 , .., σ k ). We us this construction to define the Chern-character Ch(E) of the vectorbundle E.
Assume we have a cohomology theory H * satisfying the projective bundle property. One gets characteristic classes c i (E) for a complex vectorbundle E on B: Let the class S k (E) = s k (c 1 (E), c 2 (E), .., c k (E)) ∈ H 2k (B) be the k ′ th Newton-class of the bundle E. One gets: s k (σ 1 , 0, .., 0) = σ k 1 for all k ≥ 1. Assume E, F linebundles. We see that Proof. This follows from 3.6.
is the Chern-character of E.

Lemma 4.3. The Chern-character defines a group-homomorphism
Ch : K * C (B) → H even (B) between the Grothendieck group K * C (B) and the even cohomology of B with rational coefficients.
Proof. By Proposition 4.1 we get the following: For any E, F we have We get and the Lemma follows.
Example 4.4. Given a real continuous vectorbundle F on B there exist Stiefel-Whitney classes w i (F ) ∈ H i (B, Z/2) (see [6]) satisfying the necessary conditions, and we may define a "Chern-character" Since S k (σ 1 , 0, ..., 0) = σ k 1 we get the following: When E, F are linebundles we have: This property holds for general E, F : Proposition 4.5. Let E, F be complex vectorbundles on a compact topological space B. Then the following formulas hold: Proof. We prove this using the splitting-principle and Proposition 4.1. Assume E, F are complex vectorbundles on B and f : B ′ → B is a map of topological spaces such that f * E = ⊕ i L i , f * F = ⊕ j M j where L i , M j are linebundles and the pull-back map f * : H * (B) → H * (B ′ ) is injective. We get the following calculation: and the result follows since f * is injective.
The Chern-character is related to the Adams-operations in the following sense: There is a ring-homomorphism ψ k H : H even (B) → H even (B) defined by ψ k H (x) = k r x when x ∈ H 2r (B). The Chern-character respects these cohomology operations in the following sense: Theorem 4.9. There is for all k ≥ 1 a commutative diagram where ψ k is the Adams operation defined in the previous section.
Hence the Chern-character is a morphism of cohomology-theories respecting the additional structure given by the Adams and Steenrod-operations.