Contentshossein/memoire... · 2.2. Exterior di erentiation of the Weil algebra5 2.3. Characteristic...

MAYER-VIETORIS FOR SIMONS-SULLIVANDIFFERENTIAL K-THEORY

ALI ABBAS

Contents

1. Introduction 12. Chern Weil Theory 42.1. Weil algebra 42.2. Exterior differentiation of the Weil algebra 52.3. Characteristic classes 73. Simons-Sullivan Differential K-Theory 103.1. Construction of the Chern-Simons transgression form. 103.2. Structured Bundles 123.3. The Stably Trivial Case 193.4. K(X) 23References 29

1. Introduction

According to the axioms of Eilenberg and Steenrod an OrdinaryCohomology theory H is a collection H i

i∈Z of contravariant functors fromthe category of pairs of spaces to abelian groups that satisfies the followingconditions :

• (Homotopy Invariance) If f is homotopic to g through maps of pairs,then Hn(f) = Hn(g).

• (Preserves Products) Hn(∐Xα) =

∏Hn(Xα) for all n.

• For each pair (X,A), there exists a long exact sequence

· · · // H i(X,A) // H i(X) // H i(A) // H i+1(X,A)δ // · · ·

Such that the boundary map δ is natural.

• (Excision) If Z ⊂ A ⊂ X and Z ⊂ Int(A) then the induced map

H i(X,A) −→ H i(X − Z,A− Z)

Date: July 11st 2013.1

2 ALI ABBAS

is an isomorphism for each i.

• (Dimension) H i(∗) = 0 for i 6= 0.

A Generalized Cohomology theory satisfies all of the aboveexcept the dimension axiom.

A smooth extension of a Cohomology Theory H is a functor X −→ H∗(X)from the category of smooth manifolds to Z-graded rings , that comes withnatural transformations :1. R : H∗(X) −→ ∧∗cl(X) (Curvature).

2. I : H∗(X) −→ H∗(X) (forget smooth data).

3. a : ∧∗−1(X)/im(d) −→ H∗(X) (action of forms).

Here ∧∗(X) and ∧∗cl(X) denote the space of smooth differential forms andits subspace of closed forms. The map a is of degree 1. Furthermore, onehas the following properties :

1. The following diagram commutes

H∗(X)I //

R

H∗(X)

ch

∧∗closed(X)deR // H∗(X,R)

where deR is the de Rham homomorphism, and ch is a natural transfor-mation that depends on the Cohomology theory.2. R a = d.3. R and I are ring homomorphisms.4. The sequence

H∗−1(X)ch // ∧∗−1(X)/im(d)

a // H∗(X)I // H∗(X) // 0

is exact.An example of ordinary cohomology theory is the cohomology with inte-

ger coefficients, i.e H∗(X) = H∗(X,Z). According to Cheeger and Simons,differential characters is a geometric model for the Smooth Cohomology ex-tension of ordinary cohomology theory with integer coefficients, in this casech is induced by the inclusion of coefficients Z −→ R. Below, we give a quickreview of differential characters.

Differential CharactersLet X be a C∞ manifold with Bk ⊂ Zk ⊂ Ck denote the groups of bound-aries, Cycles and Singular k-chains, and let ∂ : Ck −→ Ck−1 be the boundaryoperator.Let Ck = Hom(Ck, R) be the singular cochain, and δ : Ck −→ Ck+1 be the

MAYER-VIETORIS FOR SIMONS-SULLIVAN DIFFERENTIAL K-THEORY 3

coboundary operator defined by :

δ(φ)(σ) = φ(∂(σ)),∀φ ∈ Ck,∀σ ∈ Ck+1.

If ω ∈ ∧k(X) then we can regard it as a real k- cochain by taking

ω(σ) :=

∫∆k

σ∗ω and denoted by

∫σ

ω, where σ ∈ Ck.

After reducing the values of ω mod Z we get ω ∈ Ck(X,R/Z),ω = 0 means that all the values of ω lies only in Z, which is true only if ω = 0,Then the map ∧k(X) −→ Ck(X,R/Z) that associates for each ω −→ ω isan injection, and we may regard ∧k(X) ⊆ Ck(X,R/Z).

Definition: A Differential character is a pair (φ, ω), a homomorphismφ : Zk(X) −→ R/Z and a closed differential form ω ∈ ∧k+1(X), such thatfor any boundary b = ∂c, c ∈ Ck+1(X), we have

φ(b) = [

∫c

ω]R/Z.

Let Hk(X,R/Z) denote the set of such elements, and

H∗(X,R/Z) = ⊕Hk(X,R/Z)

is the graded Z−module of differential characters.

Note that if φ ∈ Hom(Zk,R/Z), there is a real cochain T such that its

reduction T mod Z satisfies T |Zk = φ, Then φ is a differential character if

its coboundary δT = δT = φ ∂ is the mod Z reduction of some differentialform ω ∈ ∧k+1(X), i.e δT = ω + α, for some α ∈ Ck+1(X,Z), it follows thatω is closed and uniquely determined by φ, So we get the couple (φ, ω).

The K-Theory denoted by K, is a well known generalized cohomologythoery, and a differential K-theory K is a Smooth extension of K.

In this project we are going to introduce a Differential K-Theory whichis Simons-Sullivans differential K-theory, and to prove the Mayer-Vietorisproperty in this case.

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2. Chern Weil Theory

2.1. Weil algebra.let π : P −→M be a principal G-bundle, with connection form ω ∈ ∧1(P, g),where g is the Lie algebra of the Lie group G.

Corollary 2.1.1. Let B1, ...Bn be a basis of g, then ω can be written as

ω =∑i

ωiBi where ωi are real 1-forms on P .

The bracket [ω, ω] is defined as a g valued 2-form on P by :

[ω, ω] =∑i,j

ωi ∧ ωj[Bi, Bj] =∑i,j,k

cki,j ωi ∧ ωjBk.

(cki,j are called the stucture constants).

Definition 2.1.2. The curvature Ω ∈ ∧2(P, g) of ω is defined by the struc-ture equation :

(2.1.3) Ω = dω +1

2[ω, ω].

The curvature Ω satisfies the Bianchi’s identity dΩ = [Ω, ω], and Ω can bewritten as Ω =

∑ΩiBi for some Ωi ∈ ∧2(P ).

Let ω be the map g∗ −→ ∧1(P ) induced by the connection form, which isdefined by ω(α) = αω, for every α ∈ g∗, and Let θ1, ..., θn be the dual basisof B1, ..., Bn for g∗,(g∗ ' set of left invariant differential forms on G), itis clear that ω(θi) = ωi, so we extend ω to the exterior algebra of g∗ as amorphism of algebras to get a linear map, ω : Λ∗g∗ −→ ∧∗(P ), explicitly wehave :

ω(θi1 ∧ ... ∧ θik) = ωi1 ∧ ... ∧ ωik .Similarly the curvature form Ω defines a linear map : g∗ −→ ∧2P , with

Ω(α) = α Ω.

Proposition 2.1.4. For any α ∈ g∗ we have Ω(α) = (dω − ωd)(α).

Proof. Since dθi = −12

∑j,k

cij,kθj ∧ θk and using the structure equation we get

Ω(θi) = Ωi = dωi − ω(dθi) = (d ω − ω d)(θi)

and since ω and d are linear and any element α ∈ g∗ is a linear combinationof the θi then we get the result.

Consider the polynomial algebra S∗g∗ generated by the elements of g∗

S∗g∗ =∞∑k=1

Skg∗

with S1g∗ = g∗ and define the degree of such an element to be 2. To distin-guish these elements, we write α for the elements in S1g∗ that corresponds


to α ∈ g∗. So θ1, ..., θn form a basis of S1g∗.A map f : g −→ R is called a polynomial function if it can be expressed asa polynomial of θ1, ..., θn.

S∗g∗ = f : polynomialfunction g −→ R = R[θ1, ..., θn]

Definition 2.1.5. The Weil Algebra of g is defined by

W (g) = Λ∗g∗ ⊗ S∗g∗

Let Ω : S∗g∗ −→ ∧∗(P ) be the map defined by Ω(θ1 · · · θk) = Ωi1 ∧ ...∧Ωik .

Definition 2.1.6. The Chern Weil map W is defined by W = ω ⊗ Ω,which is a linear map that preserves degree and product.

W : W (g) −→ ∧∗(P )

2.2. Exterior differentiation of the Weil algebra. .We want to define a linear map δ : W (g) −→ W (g) that raises degree by 1and makes the following diagram commutative

(2.2.1) W (g)W //

δ

∧∗(P )

d

W (g)W // ∧∗(P )

Let Vu = X ∈ TuP ; π∗X = 0 be the vertical subspace of TuP for u ∈ P ,which can be identified with the lie algebra g. Then an arbitrary elementA ∈ g induces a vector field A∗ on P. Such vector fields are called fun-damental vector fields. In other words, they are the vector fields whichcorresponds to one parameter transformation groups RetA .

First we define the following three actions:

(1) The Interior Product : for A ∈ g define iA : W (g) −→ W (g) by :iAα = α(A) for any element α ∈ g∗ ⊂ W (g)andiAα = 0 for an element α ∈ S1g∗ ⊂ W (g)

Extend i to all of W (g) as an anti-derivation of degree −1. So we get thefollowing commutative diagram.

(2.2.2) W (g)W //

iA

∧∗(P )

iA∗

W (g)W // ∧∗(P )

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(2) The adjoint action g∗ of an element g ∈ G

Let Ad(g−1) : g −→ g be the derivative of cg−1 : G −→ G, cg−1(h) = g−1hgat the identity, and let Ad(g−1)∗ be its dual.

Define g∗ : W (g) −→ W (g) by :

g∗(α) = Ad(g−1)∗α,

and

g∗α = Ad(g−1)∗α

We get the following commutative diagram :

(2.2.3) W (g)W //

g∗

∧∗(P )

R∗g

W (g)W // ∧∗(P )

(3) The Lie derivative on the Weil Algebra :Define LA : W (g) −→ W (g) by

For α ∈ g∗ Define LA(α) ∈ g∗: LA(α)(B) = −α([A,B]) ForA,B ∈ g,and

For α ∈ S1g∗ Define LA(α) ∈ S1g∗ : LA(α) = LA(α),

Extending LA to be an anti-derivation of degree 0, We get the followingcommutative diagram :

(2.2.4) W (g)W //

LA

∧∗(P )

LA∗

W (g)W // ∧∗(P )

Where LA∗ is the usual Lie derivative.

Definition 2.2.5. The exterior differentiation δ : W (g) −→ W (g) isdefined by : δα = dα + α and δα = −δ(dα) α ∈ g∗, α ∈ S1g∗.

Proposition 2.2.6. δα =∑i

θi ⊗ LBiαfor every α ∈ S1g∗

Proof:Applying the definition to θi and using :

dθi = −1

2

∑j,k

cij,kθj ∧ θk


we get :

δθi = −∑j,k

cij,kθj θk, But LBjθi = −∑k

cij,kθk Then:

δθi =∑j

θj ⊗ LBjθi

and because of linearity of δ we get the result.

Theorem 2.2.7. The Exterior differentiation δ of the Weil Algebracommutes with the differential d,(i.e diagram(2.2.1) is commutative).

Proof. It is enough to prove that for α ∈ g∗ and α ∈ S1g∗.Using proposition(2.1.4) we can see that :

For any α ∈ g∗ W(α) = Ω(α) = (dω − ωd)(α) = (dW −Wd)(α)⇒ dW(α) =W(dα + α) =W(δα),

Now take θi ∈ S1g∗, we have δθi =∑

j,k cik,jθj θk then

W(δθi) =∑j,k

cik,jωj ∧ Ωk =∑j,k

cik,jΩk ∧ ωj

= [Ω;ω]i = (dΩ)i, (by Bianchi′s identity)

= d(Ωi) = d(W(θi)).

Then δ and d commutes.

Remark 2.2.8. Exterior differentiation δ of the Weil algebra satisfies δδ = 0.

2.3. Characteristic classes.Given a principal G bundle π : P −→ M , with connection ω, we get ahomomorphism from the Weil algebra W (g) of the lie algebra g into the deRham complex ∧∗(P ), which is commutative with exterior differentiation.

W : W (g) −→ ∧∗(P )

Definition 2.3.1. An element y ∈ ∧∗(P ) is said to be basic if it is in theimage of π∗ : ∧∗(M) −→ ∧∗(P ), which is an injective map.

Corollary 2.3.2. An element y ∈ ∧∗(P ) is basic iff the following twoconditions holds :(i) iA∗y = 0,∀A ∈ g .(ii) R∗g(y) = y,∀g ∈ G.

where Rg is the right action of g on the Principal bundle.

Definition 2.3.3. Let G be a Lie group and g its Lie algebra, then anelement x of the Weil algebra is said to be basic if it satisfies the followingtwo conditons:(i) iAx = 0 ∀A ∈ g.(ii) g∗x = x ∀g ∈ G.

8 ALI ABBAS

we denote by I(G) the set of all basic elements in W (g).Condition (i) shows that I(G) does not contain components of Λ∗g∗, there-

fore I(G) is a subalgebra of S∗g∗. and so :

I(G) = f ∈ S∗g∗such thatg∗f = f= f ∈ S∗g∗such thatf(AdgA) = f(A)for allg ∈ G

We call I(G) the algebra of invariant polynomials, and we denote by Ik(G)the set of all homogenous elements of degree 2k.

By the above corollary, and by the diagrams (2.2.2) and (2.2.3) we can seethat W sends basic elements of W (g) to basic elements of ∧∗(P ), and sinceδf = 0 ∀f ∈ I(G) we get d(W(f)) = W(δf) = 0, then W(f) is closed andso we can consider W as a map from I(G) −→ H∗(M,R), which is calledthe Weil Homomorphism.

Theorem 2.3.4. Principal theorem of Chern-Weil theory:Let ξ = (P, π,M,G) be a principal G-bundle and choose a connection on it,then the obtained Weil homomorphism ω : I(G) −→ H∗(M ;R) is indepen-dent of the choice of the connection.

Definition 2.3.5. Let f ∈ Ik(G) and set f(ξ) = ω(f) ∈ H2k(M ;R) then wesay that f(ξ) is the characteristic class of the principal bundle relative to f .

Invariant polynomials for Lie groups.Let ω : I(G) −→ H∗(M,R) be the Weil homomorphism of a principal G-bundle ξ = (P, π,M,G). Given a connection on ξ with ω and Ω the connec-tion and curvature forms respectively,we can write ω = ω1B1 + ...+ωnBnandΩ = Ω1B1 + ...+ ΩnBn in a given basis B1, ..., Bnof g.

Remark 2.3.6. For an element f ∈ Ik(G) we have:

(2.3.7) f =∑I

aI θi1 ...θik , and ω(f) = [∑I

aIΩi1 ∧ ... ∧ Ωik ]

On the right hand side of the last equation, the differential form betweenthe braces is a basic closed 2k form on P . We think of it as a closed form onthe base space M and take its deRham cohomology class.

I(G) from another point of view:Let A1, ..., Ak ∈ g and σ any permutation of (1, ..., k).

Definition 2.3.8. Skg∗ := f : g× ...× g︸︷︷︸k

−→ R, symmetric and multilinear

That is : Skg∗ := f, f multilinear and f(Aσ(1) , ..., Aσ(k)) = f(A1, ..., Ak)

Proposition 2.3.9. S∗g∗ ∼= S∗g∗ is an isomorphism between Gradedalgebras. by the correspondance Skg∗ −→ Skg∗ that associates for everyf ∈ Skg∗, f ∈ Skg∗ defined by f(A) = f(A, ..., A), (A ∈ g).


In particular an invariant polynomial f ∈ Ik(G) on G is characterized asa symmetric multilinear map

f : g× ...× g −→ R

that is invariant by Adg : f(AdgA1, ..., AdgAk) = f(A1, ..., Ak) for all g ∈ Gand Ai ∈ g.Now we shall have another look at Weil homomorphism from our new pointof view. For each f ∈ Ik(G), as we know, its image ω(f) ∈ H2k(M,R) isrepresented by the closed form on M between the brackets on the right-handside of (2.3.7)

On the other hand the exterior power of degree k of the curvature formΩ ∈ ∧2(P, g) is written

Ωk = Ω ∧ ... ∧ Ω ∈ ∧2k(P ; g⊗ ...⊗ g︸︷︷︸k

)

then the composite f Ωk, written also f(Ωk), is a 2k form on P. To bespecific we have for X1, ..., X2k ∈ TuP

f(Ωk)(X1, ..., X2k) =1

(2k)!

∑σ∈%2k

sgnσf(Ω(Xσ1 , Xσ2), ...,Ω(Xσ2k−1, Xσ2k))

Lemma 2.3.10. : The two 2k forms on P f(Ωk) and ω(f) coincide,that is :

f(Ωk) = ω(f)

Triviality of the Cohomology of the Weil Algebra.Let g be the Lie algebra of a lie group G and let δ be the exterior differenti-ation on the Weil algebra of G.

Since δ δ = 0 then we obtain a complex and its cohomology group isH∗(W (g); δ).

Theorem 2.3.11. The cohomology group of the Weil algebra W (g) relativeto exterior differentiation δ is trivial.

That is : Hk(W (g); δ) = 0 for k > 0, and H0(W (g); δ) = R

Chern Simons formsGiven an invariant polynomial f ∈ Ik(G) for a lie group G, any principal G-bundle ξ = (P, π,M) gets a characteristic class f(ξ) = ω(f) ∈ H2k(M,R).forthermore, this characteristic class is represented by a basic closed formf(Ωk) induced by the curvature form Ω of an arbitrary connection in P .Since f is basic then δ(f) = 0 and f is closed in the Weil algebra of G butthe cohomology of the Weil algebra is trivial then there exists some elementsay Tf ∈ W (g) such that δTf = f . so we get dω(Tf) = ωδ(Tf) = ω(f).

10 ALI ABBAS

The element Tf described above is called the Chern-Simons form.

Flat bundlesLet ξ = (P, π,M,G) be a principal G bundle over a C∞ manifold M .

Definition 2.3.12. A connection ω on ξ is said to be a flat connection if itscurvature form Ω is identically zero (Ω ≡ 0).

Remark 2.3.13. A principal G-bundle is called a flat G-bundle if it has a flatconnection.

3. Simons-Sullivan Differential K-Theory

3.1. Construction of the Chern-Simons transgression form.

Let V −→ X be a smooth complex vector bundle over a manifold X.

Definition 3.1.1. Define ∧∗(X, V ) to be the space of smooth sections of thetensor product vector bundle Λ∗(T ∗X)⊗ V i.e :∧∗(X, V ) := Γ(Λ∗(T ∗X)⊗ V )

Definition 3.1.2. A connection ∇ on V is a C-linear operator :∇ : Γ(V ) −→ ∧1(X;V ) such that for any f ∈ C∞(X), S ∈ Γ(V ), thefollowing leibnitz rule holds,

∇(fS) = df · S + f∇S

A connection ∇ can be extended canonically to a map, which we stilldenote by ∇,

∇ : ∧∗(X;V ) −→ ∧∗+1(X;V )

such that for any ω ∈ ∧∗(X), S ∈ Γ(V ),

∇(ω · S) = dw · S + (−1)degwω ∧∇S.

Definition 3.1.3. The curvature R of a connection ∇ is defined by :R = ∇ ∇ : Γ(V ) −→ ∧2(X, V ), which is written briefly as R = ∇2.

Proposition 3.1.4. The curvature R is C∞(X)-linear, That is :R(fS) = fR(S),∀f ∈ C∞(X), S ∈ Γ(V ).Then we can conclude that R may be thought as an element of ∧2(X,End(V )).

Similarly we can prove that every compostition

Rk = R ... R︸︷︷︸k

: Γ(V ) −→ ∧2k(X;V )

is a well-defined element in ∧2k(X;End(V )).

Definition 3.1.5. For a smooth section S ∈ Γ(End(V )), define the tracefunction tr(S) to be the fiberwise trace of S.

This induces a map :tr : ∧∗(X,End(V )) −→ ∧∗(X) as follows :tr(ω · S) = tr(S)ω


Definition 3.1.6. Define the bracket of two elements A,B ∈ ∧∗(X;End(V ))by : [A,B] = A ∧B − (−1)(degA)(degB)B ∧ A

Now let A = ω.ψ and B = η.ϕ be two elements of ∧∗(X;End(V )), withω, η ∈ ∧∗(X), and ψ, ϕ ∈ Γ(End(V )), then

tr([A,B]) = ω ∧ η tr(ψ.ϕ)− (−1)deg(ω). deg(η)η ∧ ω tr(ϕ.ψ)

but tr(ψ.ϕ) = tr(ϕ.ψ) and η ∧ ω = (−1)deg(ω). deg(η)ω ∧ η, So tr([A,B]) = 0.

Proposition 3.1.7. If ∇ is a connection on V , then for any A ∈ ∧∗(X,End(V )),one has d tr(A) = tr([∇, A]).

Proof. Let A = ω.ψ, with ω ∈ ∧∗(X), and ψ ∈ Γ(End(V )), First of allA acts on ∧∗(X, V ) as follows, Let α = η.φ ∈ ∧∗(X, V ), with η ∈ ∧∗(X),φ ∈ Γ(V ) then : A ∧ α = (ω.ψ) ∧ (η.φ) = (ω ∧ η).(ψ(φ)).

By definition [∇, A](α) = ∇(A ∧ α) + A ∧ (∇α), Second if ∇ is anotherconnection on V , we know that ∇− ∇ ∈ ∧1(X,End(V )), thentr([∇ − ∇, A]) = 0, and so the right hand side in (3.1.7) is independent ofthe choice of the connection.Now locally the vector bundle is trivial, and by the discussion above we canchoose the trivial connection d, in which the proposition holds automatically.Then the result is true for ∇.

Let f(x) = a0 +a1x+ ...+anxn + ... be a power series in one variable, and

let R be the curvature of a connection ∇ on V .The trace of f(R) = a0 + a1R + ...+ anR

n + ... is an element of ∧∗(X).

Another state of Chern Weil theorem

Theorem 3.1.8. tr(f(R)) is a closed form and if we take another connection∇ on the bundle V with curvature R we will get :

tr(f(R))− tr(f(R)) = dω

for some differential form ω ∈ ∧∗(X)

Let γ(t) = ∇t be a smooth curve of connections joining the two connections∇ and ∇, that is ∇t=0 = ∇ and ∇t=1 = ∇ with curvatures Rt for each∇t, and let At = d∇t

dt∈ ∧1(X,EndV ).

Proposition 3.1.9. ddt

tr(f(Rt)) = d tr(Atf′(Rt))

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Proof.

d

dttr(f(Rt)) = tr(

d

dt(f(Rt)))

= tr(dRt

dtf ′(Rt)) = tr(

d(∇2t )

dtf ′(Rt))

= tr([∇t, At]f′(Rt)) = tr([∇t, Atf

′(Rt)])

= d tr(Atf′(Rt)) by(3.1.7).

Definition 3.1.10. Let ex be the exponential power series, and ∇ be a con-nection on V with curvature R, then the chern character form of ∇ is definedto be :

ch(∇) = tr(e1

2πiR) =

∑j=0

1

j!

( 1

2πi

)jtr(Rj).

Theorem 3.1.11. Let the notation be as above, Then :

(3.1.12) ch(∇1)− ch(∇0) = d

∫ 1

0

∑j=1

1

(j − 1)!

( 1

2πi

)jtr(AtR

j−1t )

Proof.

ch(∇1)− ch(∇0) =

∫ 1

0

d

dtch(∇t)

=

∫ 1

0

d

dttr(e

12πi

Rt)

=

∫ 1

0

d tr(At∑j=1

1

(j − 1)!

1

(2πi)j(Rt)

j−1)(by(3.1.9))

= d

∫ 1

0

∑j=1

1

(j − 1)!

1

(2πi)jtr(At(Rt)

j−1)

Definition 3.1.13. The Chern-Simons transgression form is defined by

cs(γ) =

∫ 1

0

∑j=1

1

(j − 1)!

1

(2πi)jtr(At(Rt)

(j−1)) ∈ ∧odd(X,C)

3.2. Structured Bundles.

Let V be a complex vector bundle with two connections ∇0and ∇1 over asmooth manifold X, and let R ∈ ∧2(X,End(V )) denote its curvature tensor.


We proved that if γ(t) = ∇t is a smooth curve of connections joining ∇0

to ∇1 (which always exists since the set of vector bundles is an affine spaceover ∧1(X,End(V ))) then :

(3.2.1) ch(∇1)− ch(∇0) = dcs(γ)

Proposition 3.2.2. cs(γ) is independent of the choice of the path γ joiningthe two connections ∇0 and ∇1 up to an exact form.

Proof. In order to prove this we will need the following Lemma,

Lemma 3.2.3. Let ψt : X −→ X × [0, 1] be the slice map,Π : X × [0, 1] −→ X be the standard projection, W = Π∗(V ), and ∇ be theconnection obtained by defining ∇s = ∇t

Π∗(s)when s is tangent to the slice

through t, and by making ∇∂/∂t(Π∗(f)) = 0 for f any cross section of V then

cs(γ) =

∫ 1

0

ψ∗t (i∂/∂tch(∇)).

Proof. Let R be the curvature of ∇,It is straightforward that if r, s are tangent to the slice through t thenRr,s = Rt

Π∗(r),Π∗(s).

This gives us : ψ∗t (R) = Rt

Now let ω ∈ W(x,t), and extend it to be of the form Π∗(f), where f is across-section of V . Also extend s to be the lift of a vector field on X. Clearly[s, ∂/∂t] = 0, Thus

R∂/∂t,sω = ∇∂/∂t∇sω − ∇s∇∂/∂tω = ∇∂/∂t∇sω = AtΠ∗(s)ω.

This gives us : ψ∗t (i∂/∂tR) = At

tr(At ∧Rt ∧ ... ∧Rt︸︷︷︸j−1

) = tr(ψ∗t (i∂/∂tR) ∧ ψ∗t R ∧ ... ∧ ψ∗t R︸︷︷︸j−1

)

= tr(ψ∗t ((i∂/∂tR) ∧ R ∧ ... ∧ R︸︷︷︸j−1

))

= ψ∗t (tr((i∂/∂tR) ∧ R ∧ ... ∧ R︸︷︷︸j−1

)

= ψ∗t (tr(1

ji∂/∂t(R ∧ ... ∧ R)))

= ψ∗t (i∂/∂t(1

jtr(R ∧ ... ∧R)))

so we can conclude :

cs(γ) =

∫ 1

0

ψ∗t (i∂/∂tch(∇))

14 ALI ABBAS

Proof of Proposition (3.2.3) Let α and γ be two paths connecting ∇0

to ∇1 then to prove that cs(γ) = cs(α) + exact it is sufficient to prove thatif γ is a closed path of connections, then cs(γ) is exact. It is obvious thatcs(γ) is closed so to show that it is exact we will show that it integrates to0 on every cycle of X.

Let Z be such a cycle then :∫Z

cs(γ) =

∫Z

∫S1

ψ∗t (i∂/∂tch(∇)) =

∫Z×S1

ch(∇) by Fubini

= ch(W )(Z × S1) = Π∗(ch(V ))(Z × S1)

= ch(V )(Π∗(Z × S1)) = 0Thus cs(γ) is exact.

Now we can define

CS(∇0,∇1) = cs(γ) mod exact.

It is easy to see that :

(3.2.4) CS(∇0,∇1) + CS(∇1,∇2) = CS(∇0,∇2).

Definition: Let V be a vector bundle over X and let ∇0 and ∇1 be twoconnections on it, we say that ∇0 is equivalent to ∇1 and we write ∇0 ∼ ∇1,if CS(∇0,∇1) = 0.

Since CS(∇,∇) = 0 and CS(∇, ∇) = −CS(∇,∇) and by equation (3.2.4)we can see that this relation is a well defined equivalence relation.

Definition: A pair V = [V, ∇], where ∇ is an equivalence class ofconnections on V , will be called a Structured bundle.

Now we want to define the notions of the pull back and isomorphism classof a structured bundle,

Proposition 3.2.5. Let ψ : X −→ Y be a C∞ map, and let W be a bundleover Y with connections ∇0 and ∇1, then ψ∗(∇0) and ψ∗(∇1) are connectionson ψ∗(W ), such that :

CS(ψ∗(∇0), ψ∗(∇1)) = ψ∗(CS(∇0,∇1)).

Proof. Let ∇tt∈[0,1] be a curve joining ∇0 to ∇1, and At = d∇tdt

Then ψ∗(∇t)t∈[0,1] is a curve joining ψ∗(∇0) to ψ∗(∇1),

Now let At = ddtψ∗(∇t) = ψ∗( d

dt∇t) = ψ∗(At)


The curvature of ψ∗(∇t) is Rt = ψ∗(Rt) Then :

tr(At ∧ Rt ∧ ... ∧ Rt︸︷︷︸j−1

) = tr(ψ∗(At) ∧ ψ∗(Rt) ∧ ... ∧ ψ∗(Rt)︸︷︷︸j−1

)

= tr(ψ∗(At ∧Rt ∧ ... ∧Rt︸︷︷︸j−1

))

= ψ∗(tr(At ∧Rt ∧ ... ∧Rt︸︷︷︸j−1

))

And so

CS(ψ∗(∇0), ψ∗(∇1)) =

∫ 1

0

∑j=1

1

(j − 1)!

1

(2πi)jtr(At ∧ Rt ∧ ... ∧ Rt︸︷︷︸

j−1

)

=

∫ 1

0

∑j=1

1

(j − 1)!

1

(2πi)jψ∗(tr(At ∧Rt ∧ ... ∧Rt︸︷︷︸

j−1

))

= ψ∗(

∫ 1

0

∑j=1

1

(j − 1)!

1

(2πi)jtr(At ∧Rt ∧ ... ∧Rt︸︷︷︸

j−1

))

= ψ∗(CS(∇0,∇1)).

Thus, if W = [W, ∇] is a structured bundle over Y thenψ∗(W) = [ψ∗(W ), ψ∗(∇)] is well defined as a structured bundle over X.

If F : V −→ W is a bundle isomorphism covering the identity map ofX, then we say that V = [V, ∇V ] and W = [W, ∇W] are isomorphic ifF ∗(∇W) = ∇V .

If ∇V and ∇W are connections on V and W they determine connectionson V ⊕W and on V ⊗W , denoted by ∇V ⊕ ∇W and ∇V ⊗ ∇W . For f, gcross-sections in V and W , and r a tangent vector to X,

(∇V ⊕∇W )r(f, g) = (∇Vr f,∇W

r g)

(∇V ⊗∇W )r(f ⊗ g) = ∇Vr (f)⊗ g + f ⊗∇W

r (g).

Proposition 3.2.6. ch(∇V ⊕∇W ) = ch(∇V ) + ch(∇W ).

Proposition 3.2.7. ch(∇V ⊗∇W ) = ch(∇V ) ∧ ch(∇W ).

Proof. Let R∇V ⊗∇W be the curvature of ∇V ⊗∇W and let f, g cross-sections

in V and W , and r, s tangent vectors to X, Then :

16 ALI ABBAS

R∇V ⊗∇W

r,s (f ⊗ g)

= (∇V ⊗∇W )r(∇V ⊗∇W )s(f ⊗ g)− (∇V ⊗∇W )s(∇V ⊗∇W )r(f ⊗ g)−∇V ⊗∇W[r,s](f ⊗ g)

= (∇Vr ∇V

s f −∇Vs ∇V

r f −∇V[r,s]f)⊗ g + f ⊗ (∇W

r ∇Ws g −∇W

s ∇Wr g −∇W

[r,s]g)

So we getR∇

V ⊗∇W = R∇V ⊗ idW + idV ⊗R∇

W

And so :

ch(∇V ⊗∇W ) = tr(exp(−1

2πiR∇

V ⊗∇W ))

= tr(exp(−1

2πiR∇

V ⊗ idW )) ∧ tr(exp(idV ⊗−1

2πiR∇

W

))

= tr(exp(−1

2πiR∇

V

)) ∧ tr(exp(−1

2πiR∇

W

))

= ch(∇V ) ∧ ch(∇W )

Lemma 3.2.8. Let ∇V ,∇W , ∇V , ∇W be connections on the indicatedbundles then :

a) CS(∇V ⊕∇W , ∇V ⊕ ∇W ) = CS(∇V , ∇V ) + CS(∇W , ∇W )b) CS(∇V ⊗ ∇W , ∇V ⊗ ∇W ) = ch(∇V ) ∧ CS(∇W , ∇W ) + ch(∇W ) ∧

CS(∇V , ∇V )

Proof. Using equation (3.2.4) we get

CS(∇V ⊕∇W , ∇V ⊕ ∇W )

= CS(∇V ⊕∇W ,∇V ⊕ ∇W ) + CS(∇V ⊕ ∇W , ∇V ⊕ ∇W )

= CS(∇V , ∇V ) + CS(∇W , ∇W )andCS(∇V ⊗∇W , ∇V ⊗ ∇W )

= CS(∇V ⊗∇W ,∇V ⊗ ∇W ) + CS(∇V ⊗ ∇W , ∇V ⊗ ∇W )

= ch(∇V ) ∧ CS(∇W , ∇W ) + ch(∇W ) ∧ CS(∇V , ∇V )

Proposition 3.2.9. If V = [V, ∇V ] and W = [W, ∇W] are structured,then by the above lemma the equivalence classes ∇V ⊕∇W and ∇V ⊗∇Ware independent of the choices of ∇V ∈ ∇V and ∇W ∈ ∇W, and so

V ⊕W = [V ⊕W, ∇V ⊕∇W]and

V ⊗W = [V ⊗W, ∇V ⊗∇W]


are well defined structured bundles.

Definition: We define Struct(X) to be the set of isomorphism classesof Structured bundles over X.

Struct(X) is an abelian semi-group under the well defined addition ⊕,with commutative, distributive multiplication ⊗.

Remark: Let V = [V, ∇] be a structured bundle over X, if ∇′ isanother connection in ∇, i.e CS(∇,∇′) = dα for some α ∈ ∧∗(X), thench(∇′) − ch(∇) = dCS(∇,∇′) = 0, and so ch(∇) = ch(∇′), thus we canextend the chern map to Struct(X):

ch : Struct(X) −→ ∧even(X)

V −→ ch(V) = ch(∇).

Thus we can see that :

(3.2.10) ch(V ⊕W) = ch(V) + ch(W)

(3.2.11) ch(V ⊗W) = ch(V) ∧ ch(W)

Let V = [V, ∇V ] be a structured bundle, then we say that V is Flat ifsome ∇ ∈ ∇V is a flat connection, and since any two Flat structuredbundles of the same dimension n are isomorphic to the Product bundlewith its trivial product connection, so we denote their isomorphism classby [n] ∈ Struct(X).

Let ∇ be a connection on V ⊕W with curvature R, and letΠV : V ⊕W −→ V be the projection, then it induces a connection ∇V on Vas follows,Let f be a cross section on V and r some tangent vector to X then define∇Vr (f) = ΠV (∇r(f, 0)), and do the same for W to obtain ∇W .∇V ⊕∇W is a new connection on V ⊕W , This connection is not equal to ∇,but sometimes they are equivalent.

Lemma 3.2.12. Let the notation be as above, and suppose that Rr,s(V ) ⊆ Vand Rr,s(W ) ⊆ W for all tangent vectors r,s at any point of X. Then :

∇V ⊕∇W ∼ ∇

Proof. Let ∇ = ∇V ⊕∇W and let R, d denote its curvature and its exteriordifferentiation operator respectively ,Since the set of connections over V ⊕W is an affine space over∧1(X,End(V ⊕W )) we can write ∇ = ∇+ A for someA ∈ ∧1(X,End(V ⊕W )).

18 ALI ABBAS

For any cross-sections f, g on V,W respectively and r tangent vector wecan see :Arf = ∇rf − ΠV (∇rf) = ΠW (∇rf) ∈ W .andArg = ∇rg − ΠW (∇rg) = ΠV (∇rg) ∈ V ,Then we can say :

(3.2.13) Ar(V ) ⊆ W and Ar(W ) ⊆ V.

Since ∇ preserves V and W and using equation (3.2.13) we see that :

(3.2.14) dAr,s(V ) ⊆ W and dAr,s(W ) ⊆ V.

R = R + A ∧ A+ dA.

Since we have assumed that R preserves V and W. and since R does thissince it is the curvature of a direct sum connection, and so does A ∧A fromequation (3.2.13) then dA = R − R − A ∧ A preserves them as well, butequation (3.2.14) show the opposite, Thus dA = 0 and

(3.2.15) R = R + A ∧ A

Let ∇t = ∇+tA, be a curve joining ∇ to ∇. Letting Rt denote its associatedcurvature, then using equation (3.2.15) we can see that :

(3.2.16) Rt = R + t2A ∧ A

We know that the Chern Simons integrand consists of terms of the form

tr(At ∧Rt ∧ ... ∧Rt︸︷︷︸j−1

) where At = d∇tdt

= A in this case,

So this terms will be tr(A∧Rt ∧ ... ∧Rt︸︷︷︸j−1

) which all vanishes since A reverses

V and W and Rt preserves them.

Theorem 3.2.17. Let V ∈ Struct(X), then there exists W ∈ Struct(X)such that V ⊕W = [n] for some n, W is called the inverse of V.

Proof. The manifold BkGL(n,C) = GL(n+k, C)/GL(n,C)×GL(k, C) comewith natural bundles, V n and V k, of dimension n and k, and connections ∇n

and ∇k induced by the standard flat connection ∇ on V n ⊕ V k. Since ∇is flat then its curvature R∇ = 0, and so it satisfies the condition of lemma(3.2.12), so Vn = [V, ∇n] and Vk = [V, ∇k] are inverses of each other.The theorem of Narasimhan-Ramanan shows that for sufficiently large k, ann-dim V ∈ Struct(X) may be obtained as the pullback of Vn via a C∞ mapfrom X −→ BkGL(n,C). Now let W = ψ∗(Vk), then W is the inverse ofV .


3.3. The Stably Trivial Case.Let GL = lim

nGL(n,C), the stabilized complex general linear group and ℘

its lie algebra. ℘ consists of complex valued matrices, all but a finite numberof whose entries are 0. Let θ ∈ ∧1(GL,℘) denote the canonical left invariant℘-valued form on GL. Set

(3.3.1) Θ =∑j=1

bjtr(θ ∧ ... ∧ θ︸︷︷︸2j−1

) ∈ ∧odd(GL)

wherebj = 1

(j−1)!( 1

2πi)j∫ 1

0(t2 − t)j−1dt

It is well known that Θ is a bi-invariant closed odd form, We define∧GL ⊆ ∧odd by

(3.3.2) ∧GL (X) = g∗(Θ)+ ∧oddexact

Where g : X −→ GL runs through all C∞ maps.

Note that if g, h map X into GL, then g ⊕ h : X −→ GL may be defined,and (g ⊕ h)∗(Θ) = g∗(Θ) + h∗(Θ). Moreover, (g−1)∗(Θ) = −g∗(Θ). Thus∧GL(X) is an abelian group.

Lemma 3.3.3. Let V be a trivial bundle with the two flat connections ∇and ∇. Then

CS(∇, ∇) ∈ ∧GL/∧oddexact

Proof. Every element g ∈ Aut(V ) acts on the space of connections in thefollowing way:g∗(∇) = g−1∇g, Now since ∇ and ∇ are flat connections on a trivial bundlewe can find some element g ∈ Aut(V ) such that ∇ = g∗∇, i.e

∇t(f) = g−1(∇t(g(f))).

Expressing g as a matrix with respect to ∇-parallel framing of V, we see

∇ = ∇+ g−1dg

g ∈ Aut(V ) then for every p ∈ X g(p) ∈ EndVp and so it can be representedby a matrix in GL so we can regard g : X −→ GL, but g−1dg = g∗(θ), so

∇ = ∇+ g∗(θ).

Set ∇t = ∇ + tg∗(θ), a curve joining the two connections ∇ and ∇, we getRt = R+ tdg∗(θ) + t2g∗(θ)∧ g∗(θ), where R is the curvature of ∇, and Rt isthe curvature of ∇t, but R = 0 since ∇ is a flat connection.Now using the Maurer-Cartan equation in the case of GL which is

dθ = −θ ∧ θ

20 ALI ABBAS

we get dg∗(θ) = g∗(dθ) = g∗(−θ ∧ θ) = −g∗(θ) ∧ g∗(θ), then :

Rt = (t2 − t)g∗(θ) ∧ g∗(θ)At = d∇t

dt= g∗(θ) and so

tr(At ∧ Rt ∧ ... ∧ Rt︸︷︷︸j−1

) = tr((t2 − t)j−1 g∗(θ) ∧ ... ∧ g∗(θ)︸︷︷︸2j−1

)

= (t2 − t)j−1g∗(tr(θ ∧ ... ∧ θ︸︷︷︸2j−1

))

and soCS(∇, ∇) = g∗(Θ)

Definition: StructST (X) = [V, ∇] ∈ Struct(X)|V is stably trivial

For V ∈ StructST (X), there exists F and H trivial bundles such thatV ⊕ F = H and let ∇F ,∇H be flat connections on F and H. We define

CS : StructST (X) −→ ∧odd/∧GLby

CS(V) = CS(∇H ,∇⊕∇F ) mod ∧GL /∧oddexact

Proposition 3.3.4. CS is a well defined homomorphism.

Proof. Suppose F , H,∇F ,∇H are another pair of trivial bundles with flatconnections such that V ⊕ F = H.Then we can see that :CS(∇H ,∇⊕∇F ) = CS(∇H ⊕∇F ,∇⊕∇F ⊕∇F )= CS(∇H ⊕∇F ,∇H ⊕∇F ) + CS(∇H ⊕∇F ,∇⊕∇F ⊕∇F )but ∇H ⊕ ∇F and ∇H ⊕ ∇F are two flat bundles then by Lemma (3.3.3)we will have :

CS(∇H ,∇⊕∇F ) = g∗(Θ) + CS(∇H ⊕∇F ,∇⊕∇F ⊕∇F )

Now working mod ∧GL we get :

CS(∇H ,∇⊕∇F ) = CS(∇H ⊕∇F ,∇⊕∇F ⊕∇F ) = CS(∇H ,∇⊕∇F )

Thus CS is well defined. And from the properties of CS we can see that

CS is a homomorphism.

Definition: V ∈ Struct(X) is called stably flat if there exists flatStructured bundles F ,H such that V ⊕F = H. The set of these objects willbe denoted by StructSF (X). Clearly StructSF (X) ⊆ StructST (X) and is asub semi-group.

Proposition 3.3.5. Ker(CS) = StructSF (X).


Proof. Obviously StructSF ⊆ Ker(CS). Now suppose CS(V) = 0. Let

F,H,∇F and ∇H be as in the definition of CS. Now, CS(V) = 0 implies

CS(∇H ,∇⊕∇F ) = g∗(Θ) mod ∧oddexact

for some g : X −→ GL. Again as in the proof of Lemma (3.3.3), Choosinga ∇H-parallel framing of H, We may regard g ∈ Aut(H) and set

∇H = g−1(∇H g)

As in the Lemma we see CS(∇H , ∇H) = g∗(Θ), thus

CS(∇H ,∇H) = −g∗(Θ)

Therefore

CS(∇H ,∇⊕∇F ) = CS(∇H ,∇H) + CS(∇H ,∇⊕∇F )

= −g∗(Θ) + g∗(Θ) = 0 mod exact.

Setting H = [H, ∇H] and F = [F, ∇F] we see V ⊕ F = H and thusV ∈ StructSF (X).

Proposition 3.3.6. Im(CS) = ∧odd(X)/ ∧GL (X)

Proof. If L is a trivialized line bundle over X then any connection on L issimply a complex valued 1-form ω. Since ω∧ω = 0, the associated curvatureRω is dω.

Let Lω = [L, ω]. Using tω as a curve of connections joining ω to the

trivial connection, and noting that Rtω = tRω,, the definition of the CSshows :

(3.3.7) CS(Lω) =∑j=1

1

j!

(1

2πi

)jω ∧ (dω)j−1.

We first suppose X = Rn. If ω = fdx then ω ∧ dω = 0 and thus

CS(Lfdx) = fdx. Moreover, since CS is a homomorphism

CS

(∑i

⊕Lfixi

)=∑i

fixi

Thus ∧1(Rn)/ ∧GL (Rn) ⊆ Im(CS).

Proceeding by induction on k, suppose

(∑k

j=1 ∧2j−1(Rn))/ ∧GL (Rn) ⊆ Im(CS).Let ω = x1dx2 + x3dx4 + ...+ x2k−1dx2k + fdx2k+1.

22 ALI ABBAS

Claim: ω ∧ (dω)k = (k + 1)!fdx1 ∧ ... ∧ dx2k+1+ exact.

To show this, let γ = dx1 ∧ dx2 + ...+ dx2k−1 ∧ dx2k, and note

dω = γ + df ∧ dx2k+1 ⇒ (dω)k = (γ + df ∧ dx2k+1)k.

Since all powers of df ∧ dx2k+1 vanish,

(dω)k = γk + kγk−1 ∧ df ∧ dx2k+1

= k!dx1 ∧ ... ∧ dx2k+

k![∑k

j=1 dx1∧dx2∧...∧ dx2j−1 ∧ dx2j∧...∧dx2k−1∧dx2k]∧df∧dx2k+1

Thus,ω ∧ (dω)k = k!fdx1 ∧ ... ∧ dx2k+1+

k![∑k

j=1 dx1 ∧ ... ∧ dx2j−2 ∧ x2j−1 ∧ dx2j ∧ ... ∧ dx2k] ∧ df ∧ dx2k+1

= k!fdx1∧...∧dx2k+1−k!∑k

j=1 dx1∧...∧dx2j−2∧x2j−1df∧dx2j∧...∧dx2k+1

= (k + 1)!fdx1 ∧ ... ∧ dx2k+1 + exact.

Thus working mod ∧GL(Rn),

CS(L(2πi)k+1ω) = fdx1 ∧ ... ∧ dx2k+1 + θ,

Where

θ ∈k∑j=1

∧2j−1 (Rn).

By induction, θ = CS(V) for someV ∈ StructST (Rn). Theorem (3.2.17)shows V has an inverse V−1.Clearly V ∈ StructST (Rn) and by proposition CS(V−1) = −θ. Thus

CS(L(2πi)k+1ω ⊕ V−1) = fdx1 ∧ ... ∧ dx2k+1.

The general element of ∧2k+1(Rn) is the sum of such terms, and thus is the

image under CS of the direct sum of the inverse images of each of these terms.

For the general case let ψ : X −→ Rn be an imbedding.ψ∗ : ∧odd(Rn) −→ ∧odd(X) is onto,

Now let g∗(Θ) + dα be an element of ∧GL(Rn) with g : Rn −→ GL, Thenψ∗(g∗(Θ) + dα) = ψ∗(g∗(Θ)) + ψ∗dα = (g ψ)∗(Θ) + d(ψ∗(α)),Where g ψ : X −→ Gl, Then


ψ∗(∧GL(Rn)) ⊆ ∧GL(X) and soψ∗ : ∧odd(Rn)/ ∧GL (Rn) −→ ∧odd(X)/ ∧GL (X) is defined and onto.

Finally : CS ψ∗ = ψ∗ CS since if :V = [V, ∇V ] is in StructST (Rn) with V ⊕ F = H for some F,H two flatbundles with two flat connections ∇F ,∇H respectively, Thenψ∗(V) = [ψ∗(V ), ψ∗(∇V )] is in StructST (X) withψ∗(V )⊕ ψ∗(F ) = ψ∗(H), and so

CS(ψ∗(V)) = CS(ψ∗(∇H), ψ∗(∇V )⊕ ψ∗(∇F ))

= CS(ψ∗(∇H), ψ∗(∇V ⊕∇F ))

= ψ∗CS(∇H ,∇V ⊕∇F )

= ψ∗CS(V)

Let ρ ∈ ∧odd(X)/ ∧GL (X), we can find ρ ∈ ∧odd(Rn)/ ∧GL (Rn) with

ψ∗(ρ) = ρ. and since we proved that CS is surjective in the case of Rn then

ρ = CS(V) for some V ∈ StructST (Rn), and so :

ρ = ψ∗(CS(V)) = CS(ψ∗(V)).

Hence the general case is proved.

3.4. K(X).Let X be a manifold and consider the abelian semi group A = Struct(X)under the addition of structured bundles ⊕.We know that Defining an equivalence relation on A×A as follows leads tothe construction of a group denoted by K(A):

Let(x, y), (x′, y′) ∈ A× A and define:(x, y) is equivalent to (x′, y′) (or simply (x, y) = (x′, y′)) iff there exists someelement a ∈ A such that

x+ y′ + a = x′ + y + a

Now define K(A) to be the set of such equivalence relations with addition(x, y) + (x′, y′) = (x+ x′, y + y′).

K(A) is a group with identity element (a, a) a ∈ A which is independentof the choice of a, since:

(a, a) = (a+ b, a+ b) = (b, b)

The inverse of an element (x, y) is (y, x).

(x, y) + (y, x) = (x+ y, y + x) = (x+ y, x+ y) = 0K(A)

Note: denote the elements of the form (x, 0) just by x, then we get:

(x, y) = (x, 0) + (0, y) = (x, 0)− (y, 0) = x− y

24 ALI ABBAS

so we write sometimes x− y instead of (x, y).

Now let

K(X) = K(Struct(X))

An element of K(X) is of the form V −W where V and W are structuredbundles over X but we know that there exists some structured bundle W∗such that W⊕W∗ = [n] where [n] is the structured bundle consisting of then trivial bundle with its flat connection.Then V −W = (V −W)⊕ (W∗ −W∗) =W ⊕W∗ − [n]

and so : every element of K(X) is of the form V − [n].we already know that ⊗ is a commutative and distributive multiplicationover the addition of structured bundles then define on K(X) a product ⊗ asfollows :(V ,W)⊗ (V ′,W ′) = (V ⊗ V ′ ⊕W ⊗W ′,W ⊗V ′ ⊕ V ⊗W ′)It is a commutative and distributive multiplication over the addition of K(X)

then K(X) is a commutative ring.

It is obvious that V − [n] = 0 in K(X) ⇐⇒ V is stably flat andn = dim(V).Now we define

∧BGL(X) = ch(V) + ∧evenexact

where V ranges over all elements of Struct(X). it is clear that ∧BGL(X) isa commutative ring.clearly, ch extend to K(X), and maps it to ∧BGL(X) :

ch : K(X) −→ ∧BGL(X) such that for every two structured bundles V Wch(V −W) = ch(V)− ch(W).It is well defined sinceV −W = V ′ −W ′⇒ V ⊕W ′ ⊕ [n] =W ⊕V ′ ⊕ [n] for some n ∈ N⇒ ch(V ⊕W ′ ⊕ [n]) = ch(W ⊕V ′ ⊕ [n])⇒ ch(V) + ch(W ′) = ch(W) + ch(V ′)⇒ ch(V)− ch(W) = ch(V ′)− ch(W ′)⇒ ch(V −W) = ch(V ′ −W ′)We also define :

δ : K(X) −→ K(X)

by δ([V, ∇]− [W, ∇]) = V −W .Letting c : K(X) −→ Heven(X,C) be the natural transformation defined bythe chern character, and

deR : ∧BGL(X) −→ Heven(X,C)

be that defined by the de Rham theorem, we see that this diagram is com-mutative.


(3.4.1) K(X)

c

''K(X)

δ99

ch

%%

Heven(X,C)

∧BGL(X)

deR77

because if V = [V, ∇] andW = [W, ∇] are two elements of K(X) thenc(δ(V −W)) = c(V −W ) = c(V )−c(W ) where c(V ) and c(W ) are the cherncharacteristic classes of V and W respectively, on the other hand :deR(ch(V−W)) = deR(ch(∇))−deR(ch(∇)) which is equal to c(V )−c(W )by the definition of the characteristic classes.

Proposition 3.4.2. ker(δ) ∼= ∧odd(X)/ ∧GL (X)

Proof. Define Γ : StructST/StructSF −→ K by Γ(V) = V − [dim(V)],which is well defined because if V ⊕H is a left coset in V for someH ∈ StructSF then H − [dim(H)] = 0 since H is stably flat and so

Γ(V ⊕H) = V ⊕H− [dim(V ⊕H)]

= V ⊕H− [dim(V)]⊕ [dim(H)]

= (V − [dim(V)])⊕ (H− [dim(H)])

= V − dim(V)

= Γ(V)

It is clear that Γ is injective since V − [dim(V)] = 0⇐⇒ V is stably flat.then Im(Γ) ∼= StructST/StructSF and it is clear that Im(Γ) = Ker(δ).and previously we have proved that:

CS : StructST/StructSF −→ ∧odd(X)/ ∧GL (X)

is an isomorphism then

i = Γ CS−1

: ∧odd(X)/ ∧GL (X) −→ Ker(δ)

is an isomorphism.

Since δ is clearly onto, we get the following exact sequence

(3.4.3) 0 // ∧odd(X)/ ∧GL (X)i // K(X)

δ // K(X) // 0.

Proposition 3.4.4. ch i = d, and ch is onto.

26 ALI ABBAS

Proof. To show the first, note that from the definition of CS, and given astably trivial V = [V, ∇] of dimension n such that V ⊕ Em = En+m forsome trivial bundles Em and En+m = En ⊕ Em we have :

dCS(V) = dCS(∇n ⊕∇m,∇⊕∇m) = dCS(∇n,∇)

= ch(∇)− ch(∇n) = ch(V)− ch([dim(V)]) = ch(Γ(V))

here we denoted by ∇n and ∇m the canonical connections on En and Em re-spectively, and we always denote by [n]the n flat connection [n] = [En, ∇n].Thus, for θ ∈ ∧odd(X) and θ its equivalence class mod ∧GL(X),

ch(i(θ)) = ch(Γ(CS−1

(θ))) = d CS CS−1

(θ) = d(θ) = d(θ).To show that ch is onto, let µ ∈ ∧BGL(X). By definition, ∃V ∈ Struct(X)and θ ∈ ∧odd so that µ = ch(V) + dθ. By the above, µ = ch(V + i(θ)).

Let deR : Hodd(X,C) −→ ∧odd(X)/∧GL (X) be the map that associates foreach cohomology class [ω], ω ∈ ∧odd(X) the class ω in ∧odd(X)/ ∧GL (X),which is well defined, and since the image of deR consists of closed forms,d deR = 0, which implies ch i deR = 0. Thus,

i deR(Hodd(X,C)) ⊆ ker(ch)

.

Proposition 3.4.5. The following diagram of functors and natural trans-formations is commutative , and its diagonals are exact.

(3.4.6)

0

''

0

ker(ch)δ| //

⊆

$$

K

55

c

((Hodd(C)

ideR77

deR

''

K

δ

88

ch

&&

Heven(C)

∧odd/∧GL

i

::

d // ∧BGL

deR66

((0

77

0

Corollary 3.4.7. The outside sequences :

(3.4.8) Hodd(C)deR// ∧odd/∧GL

d // ∧BGLdeR // Heven(C)


(3.4.9) Hodd(C)ideR// ker(ch)

δ| // Kc // Heven(C)

are exact.

Proof. Exactness of the first follows from the de Rham theorem.For the second one :ker(c) ⊆ im(δ|): let α ∈ K such that c(α) = 0, then there

exists β ∈ K such that α = δ(β)(since δ surjective), and so deR(ch(β)) =c(δ(β)) = 0, hence ch(β) ∈ ker(deR) = im(d), then there exists some γ ∈∧odd/∧GL such that ch(β) = dγ = chi(γ), it follows that β−i(γ) ∈ ker(ch),but δ(β − i(γ)) = δ(β) = α.The other inclusions follows directly.

Theorem 3.4.10. Mayer-Vietoris Let A,B ⊆ X with A ∩ B = D andA ∪ B = X. If µA ∈ K(A) and µB ∈ K(B) with µA|D = µB|D, then there

exists µ ∈ K(X) with µ|A = µA and µ|B = µB

Proof. Following the diagram in proposition (3.4.5), since δ(µA)|D = δ(µB)|D,the Mayer-Vietoris property for K produces V − [n] ∈ K(X) with(V − [n])|A = δ(µA) and (V − [n])|B = δ(µB).

Choose µ ∈ K(X) with δ(µ) = V − [n]. Now, δ(µ)|A = δ(µ)|A = δ(µA), andsimilarly for B. Thus, by the diagramµ|A = µA + i(αA) (∗)µ|B = µB + i(αB)where αA, αB ∈ ∧odd(A),∧odd(B) and αAαB represent their equivalenceclasses mod ∧GL(A),∧GL(B).By the above,

i(αA|D)−i(αB|D) = i(αA)|D−i(αB)|D = (µ|A)|D−(µ|B)|D−µA|D+µB|D.The first pair vanishes since each term is µ|D, and the second pair vanishes

by hypothesis.Since i is an injection,

αA|D = αB|D + ω

Where ω ∈ ∧GL(D).Case I: ω = dρ

Extend ρ to all of A, and set αA = αA − dρ. Thus αA = αA, andα|D = αB|D. The latter equation implies there is a unique α ∈ ∧odd(X)with α|A = αA and α|B = αB. Thus by ∗)

µ|A = µA + i(α)|Aµ|B = µB + i(α)|B

which implies that µ = µ− i(α) satisfies the conditions of the theorem.

Case II: ω = g∗(Θ) + dρ, Where g : D −→ GL, and g∗(Θ) is not exact.


References

[1] James Simons and Dennis Sullivan. Structured Vector Bundles Define DifferentialK-Theory.

[2] Shigeyuki Morita. Translations of Mathematical Monographs. Geometry of Differen-tial Forms

[3] Weiping Zhang. Lectures on chern-Weil Theory And Witten Deformations.[4] Shigeyuki Morita. Geometry of Characteristic Classes[5] Ulrich Bunke. Differential Cohomology[6] Ulrich Bunke. Thomas Schick. Differential K-Theory. A survey.[7] Bunke, U. and Schick, T. “Smooth K-Theory”. arXiv : 0707.0046 (math.KT). 2007.

57 pp.[8] Narasimhan, M.S. and Ramanan, S. “Existence of Universal Connections”. Amer. J.

Math. 83. 1961. pp. 563-572, 85. 1963. pp. 223-231.[9] Cheeger, Jeff and Simons, James. “Differential Characters and Geometric Invariants”.

Notes of Stanford Conference 1973, Lecture Notes in Math. No. 1167. Springer-Verlag,New York. 1985. pp. 50-90.

[10] Atiyah. K-Theory. Advanced Book Classics[11] Ulrich Bunke, Matthias Kreck and Thomas Schick. A geometric description of smooth

cohomology

Ali Abbas, Faculty of sciences, Lebanese University

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