Notes on the Chern-Character Maakestad H* Department of Mathematics, NTNU, Trondheim, Norway

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t Journal of Generalized Lie r DOI: 10.4172/1736-4337.1000253

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s e G ISSN: 1736-4337 Theory and Applications Research Article Open Access Notes on the Chern-Character Maakestad H* Department of Mathematics, NTNU, Trondheim, Norway

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 Chern-character from complex K-theory to any cohomology Lie theory with values in graded Q-algebras equipped with a theory of characteristic classes is given. It respects the Adams and Steenrod operations.

Keywords: Chern-character; Chern-classes; Euler classes; Singular H* : Top→−Q algebras cohomology; De Rham-cohomology; Complex K-theory; Adams operations; Steenrod operations from the category of topological spaces to the category of graded commutative Q-algebras with respect to continuous maps of topological Introduction spaces. We say the theory satisfy the projective bundle property if the The aim of this note is to give an axiomatic and elementary following axioms are satisfied: For any rankn complex continuous treatment of Chern-characters of vectorbundles with values in a vectorbundle E over a compact space B There is an Euler class. class of cohomology-theories arising in topology and algebra. Given ∈ 2 uE H (P(E)) (1) a theory of Chern-classes for complex vectorbundles with values in → singular cohomology one gets in a natural way a Chern-character from Where π:P(E) B is the projective bundle associated to E. This complex K-theory to singular cohomology using the projective bundle assignment satisfy the following properties: The Euler class is natural, ′→ theorem and the Newton polynomials. The Chern-classes of a complex i.e for any map of topological spaces f:B B it follows: vectorbundle may be defined using the notion of an Euler class [1] fu* = u E fE* (2) and one may prove that a theory of Chern-classes with values in ⊕n singular cohomology is unique. In this note it is shown one may relax For = i 1= LE i where Li are linebundles there is an equation: the conditions on the theory for Chern-classes and still get a Chern- n (uu− π *2 ) = 0in HEn (P ( )) (3) character. Hence the Chern-character depends on some choices. ∏ ELi i=1 Many cohomology theories which associate to a space a graded The map π∗ induce an injection π∗:H∗(B)→ H∗(P(E)) and there is commutative Q-algebra H∗ satisfy the projective bundle property for an equality, complex vectorbundles. This is true for De Rham-cohomology of a * * 21n− real compact manifold, singular cohomology of a compact topological H(P ( E )) = HB ( ){1, uuEE , ,.., u E }. space and complex K-theory. The main aim of this note is to give a self Assume H∗ satisfy the projective bundle property. There is by contained and elementary proof of the fact that any such cohomology definition an equation, theory will recieve a Chern-character from complex K-theory nn−1 n respecting the Adams and Steenrod operations. uEE− c1() Eu + +− (1)()=0c n E Complex K-theory for a topological space B is considered, and in H∗(P(E)). characteristic classes in K-theory and operations on K-theory such as Definition 2.1: The classc (E)∈H2i(B) is the i’th characteristic class the Adams operations are constructed explicitly, following [2]. i of E. The main result of the note is the following (Theorem 4.9): Example 2.2: If P(E)→B is the projective bundle of a complex ∗ Theorem 1.1: Let H be any rational cohomology theory satisfying ∈ 2 vector bundle and uE=e(λ(E)) H (P(E),Z) is the Euler classe of the the projective bundle property. There is for all k≥1 a commutative tautological linebundle (E) on P(E) in singular cohomology as defined diagram. in Section 14 [1], one verifies the properties above are satisfied [4]. One k ψ k ∈ 2i **Chψ even H Ch even gets the Chern-classes ci(E) H (B,Z) in singular cohomology. KBCC() H () B KB () H () B Where Ch is the Chern-character for H∗, ψk is the Adams operation ψ k and H is the Steenrod operation. *Corresponding author: Maakestad H, Dept. of Mathematics, NTNU, Trondheim, Norway, Tel: +47 73 59 35 20; E-mail: [email protected] The proof of the result is analogous to the proof of existence of the Chern-character for singular cohomology. Received December 05, 2016; Accepted January 16, 2017; Published February 06, 2017

Euler Classes and Characteristic Classes Citation: Maakestad H (2017) Notes on the Chern-Character. J Generalized Lie Theory Appl 11: 253. doi:10.4172/1736-4337.1000253 In this section we consider axioms ensuring that any cohomology theory H∗ satisfying these axioms, recieve a Chern-character for Copyright: © 2017 Maakestad H. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted complex vectorbundles [3]. By a cohomology theory we mean a use, distribution, and reproduction in any medium, provided the original author and contravariant functor. source are credited.

J Generalized Lie Theory Appl, an open access journal Volume 11 • Issue 1 • 1000253 ISSN: 1736-4337 Citation: Maakestad H (2017) Notes on the Chern-Character. J Generalized Lie Theory Appl 11: 253. doi:10.4172/1736-4337.1000253

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Definition 2.3: A theory of characteristic classes with values in a rank with generator u - the euler class of the tautological line-bundle. cohomology theory H∗ is an assignment. The elements {1,u,u2,..,un−1} is a free basis. → ∈ 2i E ci(E) H (B) Proof: See Theorem IV.2.16 in [2]. for every complex finite rank vectorbundle E on B satisfying the As in the case of singular cohomology, we may define characteristic following axioms: classes for complex bundles with values in complex K-theory using the projective bundle theorem: The element un satisfies an equation, f ∗ci(E)=ci(f∗E) (4) nn−−−12 + n + +− n − 1+− n ≅ u c12() Eu c () Eu  (1)cnn−1 () Eu (1)()=0 c E If E F it follows ci(E)=ci(F) (5) in K∗(P(E)). One verifies the axioms defined above are satisfied, hence c( E⊕ F ) = c ( Ec ) ( F ). * k ∑ ij (6) one gets characteristic classes cEi ()∈ KC () B for all i=0,…,n. i+ jk= Theorem 2.7: The characteristic classes c (E) satisfy the following ∗→ ∗ i Note: if φ: H H is a functorial endomorphism of H which is a properties: ring-homomorphism and c is a theory of characteristic classes, it follows ∗ ∗ the assignment E→ cEi ()=(())ϕ cEi is a theory of characteristic f ci(E)=ci(f E) (7) classes. ck( E⊕ F )= cij ( Ec ) ( F ) k ∑ (8) Example 2.4: Let k∈Z and let ψ H be the ring-endomorphism of i+ jk= Heven defined by ψ kr(x )= kx where x∈H2r(B). Given a theory c (E) H i c1(L)=1−Lc (L)=0,i>1 (9) k i satisfying Definition 2.3 it follows cEi ()=ψ Hi (()) cE is a theory satisfying Definition 2.3. where E is any vectorbundle, and L is a line bundle [6].

1 Proof: See Theorem IV.2.17 in [2]. Note furthermore: Assume γ1 is the tautological linebundle on P . Since we do not assume c (γ )=Z where Z is the canonical generator 1 1 Adams Operations and Newton Polynomials of H2(P1,Z) it does not follow that an assignment E→ci(E) is uniquely determined by the axioms 4-46. We shall see later that the axioms 4-46 We introduce some cohomology operations in complex K-theory is enough to define a Chern-character [5]. and Newton-polynomials and prove elementary properties following the book [2]. Theorem 2.5:Assume the theory H∗ satisfy the projective bundle ∗ Φ ∑ property. It follows H has a theory of characteristic classes. Let (B) be the abelian monoid of elements of the type ni[Ei] with n ≥0. Consider the bundle λi(E)∧iE and the association. Proof: We verify the axioms for a theory of characteristic classes. i ii Axiom 4: Assume we have a map of rank n bundles f:F→E over a map λλt ()=E∑ () Et of topological spaces g:B′→B. We pull back the equation, i≥0 nn−1 n giving a map. uEE− c1() Eu + +− (1)()=0c n E λ =Φ (X ) →+ 1 tK* ( B )[[ t ]] in H2n(P(E)) to get an equation, t C nn*1− n * One checks, uFF− fcEu1() + +− (1)fc n ()=0 E ⊕ ∗ λt(E F)=λt(E)λt(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 5 is ok. Axiom 6: Assume ELii=1 hence the map λt is a map of abelian monoids, hence gives rise to ()uu− a map, is a decomposition into linebundles. There is an equation ∏ ELi hence we get a polynomial relation. λ :KB** ( )→+ 1 tKBt ( )[[ ]] nn−1 n t CC uELE− su1() u + +− (1)()=0snL u * ii from the additive abelian group KBC() to the set of powerseries 2n c (L ) = −u in H (P(E)). Since 1 i Li it follows, with constant term equal to one [7]. Explicitly the map is as follows: ∏ ∏ n −m (c(Li))= (1+c1(Li))=c(E) λt(n[E]−m[F])= λt(E) λt(F) . and this is ok. When n denotes the trivial bundle of rank n we get the explicit formula. Given a compact topological space B. We may consider KB* () −n the Grothendieck-ring C of complex finite-dimensional λt([E]− n)= λt(E) (1+t) . vectorbundles. It is defined as the free abelian group on isomorphism- Let u=t/1−t. We may define the new powerseries, classes [E] where E is a complex vectorbundle, modulo the subgroup ii generated by elements of the type [E⊕F]−[E]−[F]. It has direct sum as γλtu()=E ()= E∑ λ (). Eu additive operation and tensor product as multiplication. Assume E is a k≥0 complex vectorbundle of rank n and let: It follows. ⊕ ⊕ π:P(E)→B γt(E F)=λu(E F)=λu(E)λu(F)=yt(E)γt(F). be the associated projective bundle. We have a projective bundle We may write formally, theorem for complex K-theory: ii* γγt (E ) = ( Et )∈ KC ( B )[[ t ]] . ∗ ∗ ∑ Theorem 2.6: The group K (P(E)) is a free K (B) module of finite k≥0

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Hence it follows that, We have, γk()=E γγ ij () EE (). λ (n)=(1+t)n ∑ t i+ jk= Hence,

We get operations, n n −n γt(n)=λu(n)=(1+u) =(1+t/1−t) =(1−t) . γ i **→ :KBCC () KB () We get:

for all i≥1. We next define Newton polynomials using the −1 −1 γt(L−1)=γt(L)γt(1) =λu(L)(1−t) = elementary symmetric functions. Let u1,u2,u3,.. be independent kk k − −1 − − −1 variables over the integers Z, and let Quukk= 12+ ++ u for k≥1. (1+Lu)(1 t) =(1+L(t/t 1))(1 t) =

It follows Qk is invariant under permutations of the variables ui: for any 1+−tL ( 1) (1−t )=1 +− tL ( 1)=1 − c1 ( Lt ) . σ Sk we have σQk= Qk hence we may express Qk as a polynomial in the 1− t elementary symmetric functions σi: And the proposition follows. ∈ Qk= Qk(σ1,σ2,..,σk). Note: if x=L−1 we get, We define, γγ12 γk cx()=∑ aQkk ( (),(),.., x x ())= x k≥0 Sk(σ)=Qk(σ1,σ2,..,σk) to be the k th Newton polynomial in the variables σ ,σ ,..,σ where aQ(γγ11 ( x ),0,...,0) =a ( x )k = 1 2 k ∑∑kk k σi is the i th elementary symmetric function. One checks the following: kk≥≥11 ′ S (σ )=σ , −−k kk 1 1′ 1 ∑∑akk( L 1) = ( 1)ac1 ( L ) . 2 kk≥≥10 s212(σσ , )= σ 1− 2 σ 2 , 3 and s2(σ 1 , σ 2 , σ 3 )= σ 1−+ 3 σσ 12 3 σ 3 We state a Theorem: and so on. Theorem 3.3: Let E→B be a complex vectorbundle on a compact topological space B. There is a map: B′→B such that π∗E decompose Let n≥1 and consider the polynomial. into linebundles, and the map π∗: H∗(B)→ H∗(B′) is injective [8]. n n−1 p(1)=(1+tu1)(1+tu2)…(1+tun)-t σn+t σn−1+…+tσ1+1 Proof: See [2] Theorem IV.2.15. where, Note: By [2] Proposition II.1.29 there is a split exact sequence. σ =σ (u ,..,u ) *0 i i 1 n 0→→→KBCC′ () KB () HB (,)Z → 0 is the ith elementary symmetric polynomial in the variables u ,u ,..,u . 1 2 n hence the group KBC′ () is generated by elements of the form E−n Lemma 3.1: There is an equality. where E is a rank n complex vectorbundle. Q(σσ ( uu ,.., ), ( uu ,.., ),.., σ ( uu ,.., )) = uukk+ ++ u k. Proposition 3.4: The operationc is additive, i.e for any k11 n 21 nk1 n 1 2  n * xy,∈ KC () B we have, Proof: Trivial. c(x+y)=c(x)+c(y). n Assume we have virtual elements x=E−n= (Li−1) and y=F− p * p= (Rj−1) in complex K-theory KBC(). We seek to define a Proof: The proof follows the proof in [2], Proposition IV.7.11. cohomology-operation c on complex K-theory⊕ using a formal We may by the remark above assume x=E−n and y=F−p where ′ FR= ⊕ p powerseries.⊕ xy,∈ KC () B. We may also from Theorem 3.3 assume j and p 2 3 FR= ⊕ j where LRij, are linebundles. We get the following: f(u)=a1u+a2u +a3u +… Z[[u]]. γt(x+ y )= γγti ( L − 1) t ( R j − 1)= (1 ++tui ) (1 tv j )= We define the element.∈ ∏ ∏ ∏∏ n+p n+p1 1 1 1 2 3 t σn+p(u1,..,un,v1,..,vp)+t σn+p−1(u1,..,un,v1,..,vp)+ c(x)=a1Q1(γ (x))+a2Q2(γ (x),γ2(x))+a3Q3(y (x),γ (x),γ (x))+… − − …+tσ (u ,..,u ,v ,..,v )+1 Proposition 3.2: Let L be a linebundle. Thenγ t(L−1)=1+t(L 1)=1 1 1 n 1 p 1 i c1(L)t. Hence γ (L−1)=L−1 and (L−1)=0 for i>1. Hence,

Proof: We have by definition. i γ (x+y)=σi(u1,..,un,v1,..,vp). γλ λkk λ k− k tu()=()=E E∑∑ () Eu = ()(/1). E t t We get: kk≥≥00 1 k We have that, Qk(γ (x+y),..,γ (x+y))=Qk(σ1(ui,vj),..,σk(ui,vj)) n −m which by Lemma 3.1 equals, γt(nE−mF)=λu(E) λu(F) . k kk k We get, u11+ uvn ++ vQ p= k (σσ 1 ( u i ),.., ki ( u ))+ Q k ( σσ1 ( v j ),.., k ( v j ))= −1 Q (γi(x))+Q (γi(y)). γt(L−1)=λu(L)λu(1) . k k

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We get:++γ i Q (c (E⊕F),..,c (E⊕F))=Q (c (E))+Q (c (F)). cxy( )=∑ aQkk ( ( xy ))= k 1 k K i k i k≥0 Proof: We have, ii aQkk(())γγ x++ aQkk (())=() y cx cy () ⊕ ∑∑ Qk(ci(E F))=Qk(σi(ai,bj))= kk≥≥00 k+ kk ++ k + and the claim follows. a11 an b b p= QcE ki ( ( )) QcF ki ( ( )) We may give an explicit and elementary construction of the and the claim follows. Adams-operations: The Chern-Character and Cohomology Operations ≥ Theorem 3.5: Let k 1. There are functorial operations, We construct a Chern-character with values in singular k ** cohomology, using Newton-polynomials and characteristic classes ψ :KBCC ()→ KB () following [2]. The k′th Newton-classe s (E) of a complex vectorbundle with the properties. k will be defined using characteristic classes of E: c1(E),..,ck(E) and the k k k ′ (x+y)=ψ (x)+ψ (y) (10) k th Newton-polynomial sk(σ1,..,σk). We us this construction to define the Chern-character Ch(E) of the vectorbundle E. ψk(L)=Lk (11) We first define Newton polynomials using the elementary ψk(xy)=ψk(x)ψk(y) (12) symmetric functions. Let u1,u2,u3,.. be independent variables over the k kk+ ++ k ψ (1)=1 (13) integers Z, and let Quukk= 12 u for k≥1. It follows Qk is invariant under permutations of the variables u : for any σ S we have where L is a line bundle. The operations ψk are the only operations that i k σQk=Qk hence we may express Qk as a polynomial in the elementary are ring-homomorphisms - the Adams operations. ∈ symmetric functions σi: Proof: We need: Qk=Qk(σ1,σ2,..,σk). ψk(L−1)= ψk(L)− ψk(1)=Lk−1. We define, We have in K-theory: Sk(σ)=Qk(σ1,σ2,..,σk) k k k k− ii LL−1=( −+ 1 1) − 1= (L− 1) 1 − 1= to be the k′th Newton polynomial in the variables (σ1,σ2,..,σk) where ∑i i≥0 σi is the i′th elementary symmetric function. One checks the following:

kk  2  kk s1(σ1)=σ1, (LL−+−++− 1)  ( 1)  ( L 1) . 12 k 2    s2 (σ1,σ 2 ) = σ1 − 2σ 2 , We get the series, and, k 3 k k s (σ ,σ ,σ ) = σ − 3σ σ + 3σ c=  uu∈ Z[[ ]]. 2 1 2 3 1 1 2 3 ∑i i=1 and so on. The following operator, Assume we have a cohomology theory H∗ satisfying the projective k bundle property. One gets characteristic classes c (E) for a complex kik 1 i ψ=Qi ( γγ ,..., ) vectorbundle E on B: ∑ i i=1  c (E)∈H2i(B). is an explicit construction of the Adams-operator. One may verify the i ∈ 2k ′ properties in the theorem, and the claim follows. Let the class Sk(E)=sk(c1(E),c2(E),..,ck(E)) H (B) be the k th Newton-class of the bundle E. One gets: Assume E,F are complex vectorbundles on B and consider the k Chern-polynomial. sk (σ1,0,..,0) = σ1 ⊕ ⊕ ⊕ N ≥ ct(E F)=1+c1(E F)t+…+cN(E F)t . for all k 1. Assume E,F linebundles. We see that, ⊕n ⊕ ⊕ 2− ⊕ where N=rk(E)+rk(F). Assume there is a decomposition E= Li S2(E F)=c1(E F) 2c2(E F)= ⊕p and F= Rj into linebundles. We get a decomposition, 2 (c1(E)+c1(F)) -2(c2(E)+c1(E)c1(F)+c2(F))= c (E⊕F)=∏c (L )∏c (R )=(1+a t)(1+a t)…(1+b t)…(1+b t) t t i t j 1 2 1 p 2 2− − − c1(E) +2c1(E)c1(F)+c1(F) 2c2(E) 2c1(E)c1(F) 2c2(F)= where a =c (L ),b =c (R ). We get thus, i 1 i j 1 j 2− 2− c1(E) 2c2(E)+c1(F) 2c2(F)=S2(E)+S2(F). c (E⊕F)=σ (a ,..,a ,b ,..,b ). i i 1 n 1 p This holds in general: Let, Proposition 4.1: For any vectorbundles E,F we have the formula, kk++ σσ Quk= 11 uQ kk= ( ,.., k ) ⊕ Sk(E F)=Sk(E)+Sk(F). where σi is the ith elementary symmetric function in the ui’s. Proof: This follows from 3.6.

* Proposition 3.6: The following holds: Let KBC() be the Grothendieck-group of complex vectorbundles on

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linebundles and the pull-back map f∗:H∗(B)→H∗(B′) is injective. We B, i.e the free abelian group modulo exact sequences KB* ( )=⊕Z [ EU ]/ C get the following calculation: where U is the subgroup generated by elements [E⊕F]−[E]−[F]. f∗S (E⊗F)=S (f∗E⊗F)=S (⊕L ⊗M ) Definition 4.2: The class, k k k i j hence by Lemma 4.1 we get, 1 even Ch()= E Sk () E∈ H () B ∑ k! SL(⊗⊗ M )= ( SL ( M ))= k≥0 ∑ki j ∑∑ ki j is the Chern-character of E. ij, i j uv+ Lemma 4.3: The Chern-character defines a group-homomorphism, Suiv( LS ) ( M j )= ∑∑ ∑ u * even i juvk+ = Ch: KC () B→ H () B * between the Grothendieck group KBC() and the even cohomology uv+ S( LS ) (⊕ M )= of B with rational coefficients. ∑∑ u uiv j iuvk+ =  Proof: By Proposition 4.1 we get the following: For any E,F we + have, uv Su(⊕⊕ LS iv ) ( M j )= ∑ u 11 uvk+ = ChEF(⊕⊕ )= sE ( F )= ( sE ( ) + sF ( ))= ∑∑kk!!k kk Ch: K* () B→ Heven (). B kk≥≥00 C ∗ 11 and the result follows since f is injective. s() E++ s ()= F Ch () E Ch (). F ∑∑kk!!kk kk≥≥00 Theorem 4.6: The Chern-character defines a ring-homomorphism. * even We get, Ch: KC () B→ H (). B Ch([E⊕F]−[E]−[F])=Ch(E⊕F)−Ch(E)−Ch(F)=0 Proof: From Proposition 4.5 we get: 1 and the Lemma follows. ChEF(⊗⊗ )= SEF ( )= ∑ k! k Example 4.4: Given a real continuous vectorbundle F on B there k≥0 ∈ i exist Stiefel-Whitney classes wi(F) H (B,Z/2) (see [1]) satisfying the 1 ij+ necessary conditions, and we may define a “Chern-character” Sij( ES ) ( F )= ∑∑k! i ** k≥+0= i jk Ch: KR ( B )→ H ( B ,Z / 2) 11 (S ( E ))( S ( F ) = Ch ( E ) Ch ( F ) by ∑∑kk!!kk kk≥≥00 Ch( F ) = Q ( w ( F ),.., w ( F )). ∑ kk1 and the Theorem is proved. k≥0 * This gives a well-defined homomorphism of abelian groups Example 4.7: For complex K-theory KBC() we have for any complex cE()∈ K* () B because of the universal properties of the Newton-polynomials and vectorbundle E characteristic classes i C satisfying the the fact H∗(B,Z/2) is commutative. The formal properties of the Stiefel- neccessary conditions, hence we get a group-homomorphism. Whitney classes w ensures that for real bundles E,F Proposition 3.6 still ** i ChKBZC: ()→ KB C () holds: We have the formula, defined by, ⊕ Qk(wi(E F))= Qk(wi(E))+ Qk(wi(F)). ChZ ( E ) = Qkk ( c1 ( E ),.., c ( E )). Since σσk we get the following: When E,F are ∑ Sk (11 ,0,...,0) = k≥0 linebundles we have: If we tensor with the rationals, we get a ring-homomorphism. ⊗ ⊗ ⊗ k k ** Sk(E F)=Sk(c1(E F),0,..,0)=(c1(E F)) =(c1(E)+c1(F)) = ChKBQC: ()→⊗ KB C () Q ij++ij ij defined by, cEcF11() ()= SESFij() (). ∑∑ii 1 i++ jk==i jk Ch( E ) = Q ( c ( E ),.., c ( E )). ∑ k! kk1 This property holds for general E,F: k≥0 Theorem 4.8: Let B be a compact topological space. The Chern- Proposition 4.5: Let E,F be complex vectorbundles on a compact character, topological space B. Then the following formulas hold: Q * even ij+ Ch:() K B⊗→QQ H(, B ) ⊗ C SEFk( )= Sij() ES () E (14) ∗ ∑ i is an isomorphism. Here H (B,Q) denotes singular cohomology with i+ jk= rational coefficients. 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 Proof: See [2]. of topological spaces such that f ∗E=⊕ L ,f ∗F=⊕ M where L ,M are i i j j i j The Chern-character is related to the Adams-operations in the

Page 6 of 6 following sense: There is a ring-homomorphism. kk ψψHH(exp ( c1 ( L ))) = ( Ch ( L ) k even even ψ H :HBHB ()→ () and the claim follows. defined by, Hence the Chern-character is a morphism of cohomology-theories respecting the additional structure given by the Adams and Steenrod- ψ kr(x )= kx H operations. ∈ 2r when x H (B). The Chern-character respects these cohomology References operations in the following sense: 1. Milnor J (1966) Characteristic Classes. Princeton University Press. ≥ Theorem 4.9: There is for allk 1 a innovative diagram. 2. Karoubi M (1978) K-theory - an introduction. Grundlehren Math Wiss.

k ψ k 3. Dupont J (1978) Curvature and characteristic classes, Lecture Notes in **Chψ even H Ch even KBCC() H () B KB () H () B Mathematics. Springer Verlag V: 640. where ψk is the Adams operation defined in the previous section. 4. Fulton W, Lang S (1985) Riemann-Roch algebra. Grundlehren Math Wiss No: 277. Proof: The proof follows Theorem V.3.27 in [2]: We may assume k k 5. Grothendieck A (1958) Theorie des classes de Chern. Bull Soc Math France L is a linebundle and we get the following calculation: ψ (L)=L and 86: 137-154. c (Lk)=kc (L) hence, 1 1 6. Husemoeller D (1979) Fibre bundles. GTM. 1 Ch(ψ k ())= L exp ( kc ())= L kii c ()= L 7. Steenrod N (1962) Cohomology operations. Princeton University Press. 11∑ i! i≥0 8. End W (1969) Über Adams-operationen. Invent Math 9: 45.

J Generalized Lie Theory Appl, an open access journal Volume 11 • Issue 1 • 1000253 ISSN: 1736-4337