Theory an ie d L A p d p Maakestad, J Generalized Lie Theory Appl 2017, 11:1 e l z i i c l a a t Journal of Generalized Lie r DOI: 10.4172/1736-4337.1000253 i o e n n 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 *2n (uu− π ) = 0in HE (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 Page 2 of 6 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].