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NOTES ON THE CHERN-CHARACTER

arXiv:math/0612060v1 [math.AG] 4 Dec 2006

HELGE MAAKESTAD Abstract. Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the note a proof of the existence of a Cherncharacter from complex K-theory to any cohomology theory with values in graded Q-algebras equipped with a theory of characteristic classes is given. It respects the Adams and Steenrod operations.

Contents 1. Introduction 2. Euler classes and characteristic classes 3. Adams operations and Newton polynomials 4. The Chern-character and cohomology operations References 1 2 4 8 12

1. 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 de?ned 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].

Date: November 2006. Key words and phrases. Chern-character, Chern-classes, Euler classes, singular cohomology, De Rham-cohomology, complex K-theory, Adams operations, Steenrod operations.

1

2

HELGE MAAKESTAD

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

? KC (B) ψk Ch

/ Heven (B)

k ψH

? KC (B)

Ch

/ Heven (B)

k where Ch is the Chern-character for H? , ψ k is the Adams operation and ψH is the Steenrod operation.

The proof of the result is analogous to the proof of existence of the Cherncharacter for singular cohomology. 2. 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 satis?ed: For any rank n complex continuous vectorbundle E over a compact space B There is an Euler class (2.0.1) uE ∈ H2 (P(E))

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 : B ′ → B it follows (2.0.2) f ? uE = uf ? E

For E = ⊕n Li where Li are linebundles there is an equation i=1

n

(2.0.3)

?

(uE ? π ? uLi ) = 0 in H2n (P(E))

i=1

The map π induce an injection π ? : H? (B) → H? (P(E)) and there is an equality

n?1 H? (P(E)) = H? (B){1, uE , u2 , .., uE }. E

Assume H? satisfy the projective bundle property. There is by de?nition an equation n?1 un ? c1 (E)uE + · · · + (?1)n cn (E) = 0 E in H? (P(E)) . De?nition 2.1. The class ci (E) ∈ H2i (B) is the i’th characteristic class of E.

NOTES ON THE CHERN-CHARACTER

3

Example 2.2. If P(E) → B is the projective bundle of a and uE = e(λ(E)) ∈ H2 (P(E), Z) is the Euler classe of the λ(E) on P(E) in singular cohomology as de?ned in Section properties above are satis?ed. One gets the Chern-classes singular cohomology.

complex vector bundle tautological linebundle 14 [6], one veri?es the ci (E) ∈ H2i (B, Z) in

De?nition 2.3. A theory of characteristic classes with values in a cohomology theory H? is an assignment E → ci (E) ∈ H2i (B) for every complex ?nite rank vectorbundle E on B satisfying the following axioms: (2.3.1) (2.3.2) (2.3.3) f ? ci (E) = ci (f ? E) If E ? F it follows ci (E) = ci (F ) = ck (E ⊕ F ) =

i+j=k

ci (E)cj (F ).

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 → ci (E) = φ(ci (E)) is a theory of characteristic classes.

k Example 2.4. Let k ∈ Z and let ψH be the ring-endomorphism of Heven de?ned 2r k r by ψH (x) = k x where x ∈ H (B). Given a theory ci (E) satisfying De?nition 2.3 k it follows ci (E) = ψH (ci (E)) is a theory satisfying De?nition 2.3.

Note furthermore: Assume γ1 is the tautological linebundle on P1 . Since we do not assume c1 (γ1 ) = z where z is the canonical generator of H2 (P1 , Z) it does not follow that an assignment E → ci (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 de?ne a Chern-character. Theorem 2.5. Assume the theory H? satisfy the projective bundle property. It follows H? has a theory of characteristic classes. 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

n?1 un ? c1 (E)uE + · · · + (?1)n cn (E) = 0 E

in H2n (P(E)) to get an equation

n?1 un ? f ? c1 (E)uF + · · · + (?1)n f ? cn (E) = 0 F

and by unicity we get f ? ci (E) = ci (F ). It follows ci (E) = ci (F ) for isomorphic bundles E and F , hence Axiom 2.3.2 is ok. Axiom 2.3.3: Assume E ? ⊕n Li is a = i=1 decomposition into linebundles. There is an equation (uE ? uLi ) hence we get a polynomial relation

n?1 un ? s1 (uLi )uE + · · · + (?1)n sn (uLi ) = 0 E

in H2n (P(E)). Since c1 (Li ) = ?uLi it follows (c(Li )) = and this is ok. (1 + c1 (Li )) = c(E)

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HELGE MAAKESTAD

Given a compact topological space B. We may consider the Grothendieck-ring ? KC (B) of complex ?nite-dimensional vectorbundles. It is de?ned as the free abelian group on isomorphism-classes [E] where E is a complex vectorbundle, modulo the subgroup generated by elements of the type [E ⊕ F ] ? [E] ? [F ]. 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 ?nite rank with generator u - the euler class of the tautological line-bundle. The elements {1, u, u2, .., un?1 } is a free basis. Proof. See Theorem IV.2.16 in [5]. As in the case of singular cohomology, we may de?ne characteristic classes for complex bundles with values in complex K-theory using the projective bundle theorem: The element un satis?es an equation un ? c1 (E)un?1 + c2 (E)un?2 + · · · + (?1)n?1 cn?1 (E)u + (?1)n cn (E) = 0 in K? (P(E)). One veri?es the axioms de?ned above are satis?ed, hence one gets characteristic classes ci (E) ∈ K? (B) for all i = 0, .., n C Theorem 2.7. The characteristic classes ci (E) satisfy the following properties: (2.7.1) (2.7.2) (2.7.3) f ? ci (E) = ci (f ? E) ck (E ⊕ F ) =

i+j=k

ci (E)cj (F ) ci (L) = 0, i > 1

c1 (L) = 1 ? L

where E is any vectorbundle, and L is a line bundle. Proof. See Theorem IV.2.17 in [5]. 3. 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 ni [Ei ] with ni ≥ 0. Consider the bundle λi (E) = ∧i E and the association λt (E) =

i≥0

λi (E)ti

giving a map λt = Φ(X) → 1 + t K? (B)[[t]] C One checks λt (E ⊕ F ) = λt (E)λt (F ) hence the map λt is a map of abelian monoids, hence gives rise to a map λt : K? (B) → 1 + t K? (B)[[t]] C C

NOTES ON THE CHERN-CHARACTER

5

from the additive abelian group K? (B) to the set of powerseries with constant term C equal to one. Explicitly the map is as follows: λt (n[E] ? m[F ]) = λt (E)n λt (F )?m . When n denotes the trivial bundle of rank n we get the explicit formula λt ([E] ? n) = λt (E)(1 + t)?n . Let u = t/1 ? t. We may de?ne the new powerseries γt (E) = λu (E) =

k≥0

λi (E)ui .

It follows γt (E ⊕ F ) = λu (E ⊕ F ) = λu (E)λu (F ) = γt (E)γt (F ). We may write formally γt (E) =

k≥0 ? γ i (E)ti ∈ KC (B)[[t]]

hence it follows that γ k (E) =

i+j=k

γ i (E)γ j (E).

We get operations γ i : K? (B) → K? (B) C C for all i ≥ 1. We next de?ne Newton polynomials using the elementary symmetric functions. Let u1 , u2 , u3 , .. be independent variables over the integers Z, and let Qk = uk + uk + · · · + uk for k ≥ 1. It follows Qk is invariant under permutations of 1 2 k the variables ui : for any σ ∈ Sk we have σQk = Qk hence we may express Qk as a polynomial in the elementary symmetric functions σi : Qk = Qk (σ1 , σ2 , .., σk ). We de?ne sk (σ) = Qk (σ1 , σ2 , .., σk ) to be the k th Newton polynomial in the variables σ1 , σ2 , .., σk where σi is the i′ th elementary symmetric function. One checks the following:

′

s1 (σ1 ) = σ1 ,

2 s2 (σ1 , σ2 ) = σ1 ? 2σ2 ,

and

3 s2 (σ1 , σ2 , σ3 ) = σ1 ? 3σ1 σ2 + 3σ3

and so on. Let n ≥ 1 and consider the polynomial p(1) = (1 + tu1 )(1 + tu2 ) · · · (1 + tun ) = tn σn + tn?1 σn?1 + · · · + tσ1 + 1 where σi = σi (u1 , .., un ) is the ith elementary symmetric polynomial in the variables u1 , u2 , .., un . Lemma 3.1. There is an equality Qk (σ1 (u1 , .., un ), σ2 (u1 , .., un ), .., σk (u1 , .., un )) = uk + uk + · · · + uk . 1 2 n Proof. Trivial.

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HELGE MAAKESTAD

Assume we have virtual elements x = E ? n = ⊕n (Li ? 1) and y = F ? p = ⊕ (Rj ? 1) in complex K-theory K? (B). We seek to de?ne a cohomology-operation C c on complex K-theory using a formal powerseries

p

f (u) = a1 u + a2 u2 + a3 u3 + · · · ∈ Z[[u]]. We de?ne the element c(x) = a1 Q1 (γ 1 (x)) + a2 Q2 (γ 1 (x), γ 2 (x)) + a3 Q3 (γ 1 (x), γ 2 (x), γ 3 (x)) + · · · Proposition 3.2. Let L be a linebundle. Then γt (L?1) = 1+t(L?1) = 1?c1 (L)t. Hence γ 1 (L ? 1) = L ? 1 and γ i (L ? 1) = 0 for i > 1. Proof. We have by de?nition γt (E) = λu (E) =

k≥0

λk (E)uk =

k≥0

λk (E)(t/1 ? t)k .

We have that γt (nE ? mF ) = λu (E)n λu (F )?m . We get γt (L ? 1) = λu (L)λu (1)?1 . We have λt (n) = (1 + t)n hence γt (n) = λu (n) = (1 + u)n = (1 + t/1 ? t)n = (1 ? t)?n . We get: γt (L ? 1) = γt (L)γt (1)?1 = λu (L)(1 ? t)?1 = (1 + Lu)(1 ? t)?1 = (1 + L(t/t ? 1))(1 ? t)?1 = 1 + t(L ? 1) (1 ? t) = 1 + t(L ? 1) = 1 ? c1 (L)t. 1?t And the proposition follows. Note: if x = L ? 1 we get c(x) =

k≥0

ak Qk (γ 1 (x), γ 2 (x), .., γ k (x)) = ak γ 1 (x)k =

k≥1

ak Qk (γ 1 (x), 0, ..., 0) =

k≥1

ak (L ? 1) =

k≥1 k≥0

k

(?1)k ak c1 (L)k .

We state a Theorem: Theorem 3.3. Let E → B be a complex vectorbundle on a compact topological space B. There is a map π : B ′ → B such that π ? E decompose into linebundles, and the map π ? : H? (B) → H? (B ′ ) is injective. Proof. See [5] Theorem IV.2.15. Note: By [5] Proposition II.1.29 there is a split exact sequence

′ 0 → KC (B) → K? (B) → H0 (B, Z) → 0 C ′ hence the group KC (B) is generated by elements of the form E ? n where E is a rank n complex vectorbundle.

NOTES ON THE CHERN-CHARACTER

7

Proposition 3.4. The operation c is additive, i.e for any x, y ∈ K? (B) we have C c(x + y) = c(x) + c(y). 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 ∈ KC (B). We may also n p from Theorem 3.3 assume E = ⊕ Li and F = ⊕ Rj where Li , Rj are linebundles. We get the following:

n p n p

γt (x + y) =

γt (Li ? 1)

γt (Rj ? 1) =

(1 + tui )

(1 + tvj ) =

tn+p σn+p (u1 , .., un , v1 , .., vp ) + tn+p?1 σn+p?1 (u1 , .., un , v1 , .., vp )+ · · · + tσ1 (u1 , .., un , v1 , .., vp ) + 1 hence γ i (x + y) = σi (u1 , .., un , v1 , .., vp ). We get: Qk (γ 1 (x + y), .., γ k (x + y)) = Qk (σ1 (ui , vj ), .., σk (ui , vj )) which by Lemma 3.1 equals

k k uk + · · · uk + v1 + · · · vp = Qk (σ1 (ui ), .., σk (ui )) + Qk (σ1 (vj ), .., σk (vj )) = n 1

Qk (γ i (x)) + Qk (γ i (y)). We get: c(x + y) =

k≥0

ak Qk (γ i (x + y)) = ak Qk (γ i (y)) = c(x) + c(y)

k≥0

ak Qk (γ i (x)) +

k≥0

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 ψ k : K? (B) → K? (B) C C with the properties (3.5.1) (3.5.2) (3.5.3) (3.5.4) ψ k (x + y) = ψ k (x) + ψ k (y) ψ k (L) = Lk ψ k (xy) = ψ k (x)ψ k (y) ψ k (1) = 1

where L is a line bundle. The operations ψ k are the only operations that are ringhomomorphisms - the Adams operations Proof. We need: ψ k (L ? 1) = ψ k (L) ? ψ k (1) = Lk ? 1. We have in K-theory: Lk ? 1 = (L ? 1 + 1)k ? 1 =

i≥0

k (L ? 1)k?i 1i ? 1 = i

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HELGE MAAKESTAD

k k k (L ? 1) + (L ? 1)2 + · · · + (L ? 1)k . 1 2 k We get the series

k

c=

i=1

k k u ∈ Z[[u]]. i k Qi (γ 1 , ..., γ i ) i

The following operator

k

ψ =

i=1

k

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 ct (E ⊕ F ) = 1 + c1 (E ⊕ F )t + · · · + cN (E ⊕ F )tN where N = rk(E) + rk(F ). Assume there is a decomposition E = ⊕n Li and F = ⊕p Rj into linebundles. We get a decomposition ct (E ⊕ F ) = ct (Li ) ct (Rj ) = (1 + a1 t)(1 + a2 t) · · · (1 + b1 t) · · · (1 + bp t)

where ai = c1 (Li ), bj = c1 (Rj ). We get thus ci (E ⊕ F ) = σi (a1 , .., an , b1 , .., bp ). Let Qk = uk + · · · + uk = Qk (σ1 , .., σk ) 1 k where σi is the ith elementary symmetric function in the ui ’s. Proposition 3.6. The following holds: Qk (c1 (E ⊕ F ), .., ck (E ⊕ F )) = QK (ci (E)) + Qk (ci (F )). Proof. We have Qk (ci (E ⊕ F )) = Qk (σi (ai , bj )) = ak + · · · ak + bk + · · · bk = Qk (ci (E)) + Qk (ci (F )) 1 n 1 p and the claim follows. 4. 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 sk (E) of a complex vectorbundle will be de?ned using characteristic classes of E: c1 (E), .., ck (E) and the k ′ th Newton-polynomial sk (σ1 , .., σk ). We us this construction to de?ne the Chern-character Ch(E) of the vectorbundle E. We ?rst de?ne Newton polynomials using the elementary symmetric functions. Let u1 , u2 , u3 , .. be independent variables over the integers Z, and let Qk = uk + 1 uk +· · ·+uk for k ≥ 1. It follows Qk is invariant under permutations of the variables 2 k ui : for any σ ∈ Sk we have σQk = Qk hence we may express Qk as a polynomial in the elementary symmetric functions σi : Qk = Qk (σ1 , σ2 , .., σk ). We de?ne sk (σ) = Qk (σ1 , σ2 , .., σk )

NOTES ON THE CHERN-CHARACTER

9

to be the k ′ th Newton polynomial in the variables σ1 , σ2 , .., σk where σi is the i′ th elementary symmetric function. One checks the following: s1 (σ1 ) = σ1 ,

2 s2 (σ1 , σ2 ) = σ1 ? 2σ2 ,

and

3 s2 (σ1 , σ2 , σ3 ) = σ1 ? 3σ1 σ2 + 3σ3

and so on. Assume we have a cohomology theory H? satisfying the projective bundle property. One gets characteristic classes ci (E) for a complex vectorbundle E on B: ci (E) ∈ H2i (B). Let the class Sk (E) = sk (c1 (E), c2 (E), .., ck (E)) ∈ H2k (B) be the k ′ th Newton-class of the bundle E. One gets: k sk (σ1 , 0, .., 0) = σ1 for all k ≥ 1. Assume E, F linebundles. We see that S2 (E ⊕ F ) = c1 (E ⊕ F )2 ? 2c2 (E ⊕ F ) = (c1 (E) + c1 (F ))2 ? 2(c2 (E) + c1 (E)c1 (F ) + c2 (F )) = c1 (E)2 + 2c1 (E)c1 (F ) + c1 (F )2 ? 2c2 (E) ? 2c1 (E)c1 (F ) ? 2c2 (F ) = c1 (E)2 ? 2c2 (E) + c1 (F )2 ? 2c2 (F ) = S2 (E) + S2 (F ). This holds in general: Proposition 4.1. For any vectorbundles E, F we have the formula Sk (E ⊕ F ) = Sk (E) + Sk (F ). Proof. This follows from 3.6. Let K? (B) be the Grothendieck-group of complex vectorbundles on B, i.e the C ? free abelian group modulo exact sequences KC (B) = ⊕Z[E]/U where U is the subgroup generated by elements [E ⊕ F ] ? [E] ? [F ]. De?nition 4.2. The class Ch(E) =

k≥0

1 Sk (E) ∈ Heven (B) k!

is the Chern-character of E. Lemma 4.3. The Chern-character de?nes a group-homomorphism Ch : K? (B) → Heven (B) C between the Grothendieck group K? (B) and the even cohomology of B with rational C coe?cients. Proof. By Proposition 4.1 we get the following: For any E, F we have 1 1 Ch(E ⊕ F ) = sk (E ⊕ F ) = (sk (E) + sk (F )) = k! k!

k≥0 k≥0

k≥0

1 sk (E) + k!

k≥0

1 sk (F ) = Ch(E) + Ch(F ). k!

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HELGE MAAKESTAD

We get Ch([E ⊕ F ] ? [E] ? [F ]) = Ch(E ⊕ F ) ? Ch(E) ? Ch(F ) = 0 and the Lemma follows. Example 4.4. Given a real continuous vectorbundle F on B there exist StiefelWhitney classes wi (F ) ∈ Hi (B, Z/2) (see [6]) satisfying the necessary conditions, and we may de?ne a “Chern-character” Ch : K? (B) → H? (B, Z/2) R by Ch(F ) =

k≥0

Qk (w1 (F ), .., wk (F )).

This gives a well-de?ned homomorphism of abelian groups because of the universal properties of the Newton-polynomials and the fact H? (B, Z/2) is commutative. The formal properties of the Stiefel-Whitney classes wi ensures that for real bundles E, F Proposition 3.6 still holds: We have the formula Qk (wi (E ⊕ F )) = Qk (wi (E)) + Qk (wi (F )).

k Since Sk (σ1 , 0, ..., 0) = σ1 we get the following: When E, F are linebundles we have:

Sk (E ? F ) = Sk (c1 (E ? F ), 0, .., 0) = (c1 (E ? F ))k = (c1 (E) + c1 (F ))k = i+j c1 (E)i c1 (F )j = i i+j Si (E)Sj (F ). i

i+j=k

i+j=k

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: (4.5.1) Sk (E ? F ) =

i+j=k

i+j Si (E)Sj (E) i

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 Li , f ? F = ⊕j Mj where Li , Mj are linebundles and the pull-back map f ? : H? (B) → H? (B ′ ) is injective. We get the following calculation: f ? Sk (E ? F ) = Sk (f ? E ? F ) = Sk (⊕Li ? Mj ) hence by Lemma 4.1 we get Sk (Li ? Mj ) =

i,j i

(

j

Sk (Li ? Mj )) =

i

j

u+v=k

u+v Su (Li )Sv (Mj ) = u

i

u+v=k

u+v Su (Li )Sv (⊕Mj ) = u

u+v=k

u+v Su (⊕Li )Sv (⊕Mj ) = u

NOTES ON THE CHERN-CHARACTER

11

u+v=k

u+v Su (f ? E)Sv (f ? F ) = f ? u

u+v=k

u+v Su (E)Sv (F ), u

and the result follows since f ? is injective. Theorem 4.6. The Chern-character de?nes a ring-homomorphism

? Ch : KC (B) → Heven (B).

Proof. From Proposition 4.5 we get: Ch(E ? F ) =

k≥0

1 Sk (E ? F ) = k!

k≥0

1 k!

i+j=k

i+j Si (E)Sj (F ) = i 1 Sk (F ) = Ch(E)Ch(F ) k!

(

k≥0

1 Sk (E))( k!

k≥0

and the Theorem is proved. Example 4.7. For complex K-theory K? (B) we have for any complex vectorbundle C E characteristic classes ci (E) ∈ K? (B) satisfying the neccessary conditions, hence C we get a group-homomorphism ChZ : K? (B) → K? (B) C C de?ned by ChZ (E) =

k≥0

Qk (c1 (E), .., ck (E)).

If we tensor with the rationals, we get a ring-homomorphism ChQ : K? (B) → K? (B) ? Q C C de?ned by Ch(E) =

k≥0

1 Qk (c1 (E), .., ck (E)). k!

Theorem 4.8. Let B be a compact topological space. The Chern-character

? ChQ : KC (B) ? Q → Heven (B, Q)

is an isomorphism. Here H? (B, Q) denotes singular cohomology with rational coef?cients. Proof. See [5]. The Chern-character is related to the Adams-operations in the following sense: There is a ring-homomorphism

k ψH : Heven (B) → Heven (B)

de?ned by

k ψH (x) = k r x

when x ∈ H2r (B). The Chern-character respects these cohomology operations in the following sense:

12

HELGE MAAKESTAD

Theorem 4.9. There is for all k ≥ 1 a commutative diagram

? KC (B) ψk Ch

/ Heven (B)

k ψH

? KC (B)

Ch

/ Heven (B)

where ψ k is the Adams operation de?ned in the previous section. Proof. The proof follows Theorem V.3.27 in [5]: We may assume L is a linebundle and we get the following calculation: ψ k (L) = Lk and c1 (Lk ) = kc1 (L) hence 1 i k c1 (L)i = Ch(ψ k (L)) = exp(kc1 (L)) = i!

i≥0 k ψH (exp(c1 (L)))

=

k ψH (Ch(L)

and the claim follows. Hence the Chern-character is a morphism of cohomology-theories respecting the additional structure given by the Adams and Steenrod-operations. References

[1] J. Dupont, Curvature and characteristic classes, Lecture Notes in Mathematics Vol. 640 Springer Verlag (1978) [2] W. Fulton, S. Lang, Riemann-Roch algebra, Grundlehren Math. Wiss. no. 277 (1985) [3] A. Grothendieck, Theorie des classes de Chern, Bull. Soc. Math. France no. 86 (1958) [4] D. Husemoeller, Fibre bundles, GTM (1979) [5] M. Karoubi, K-theory - an introduction, Grundlehren Math. Wiss. (1978) [6] J. Milnor, Characteristic Classes (1966) [7] N. Steenrod, Cohomology operations, Princeton University Press (1962) ¨ [8] W. End, Uber Adams-operationen, Invent. Math no. 9 (1969) NTNU, Trondheim E-mail address: Helge.Maakestad@math.ntnu.no

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