EEQUIVARIANT DIFFERENTIAL COHOMOLOGY
ANDREAS KÜBEL AND ANDREAS THOM
Abstract.
The construction of characteristic classes via the curvature form ofa connection is one motivation for the refinement of integral cohomology by deRham cocycles – known as differential cohomology. We will discuss the analogin the case of a group action on the manifold: The definition of equivariantcharacteristic forms in the Cartan model due to Nicole Berline and MichèleVergne motivates a refinement of equivariant integral cohomology by all Cartancocycles. In view of this, we will also review previous definitions critically, inparticular the one given in work of Kiyonori Gomi.
Contents
1. Introduction 21.1. Equivariant cohomology and simplicial manifolds 31.2. Equivariant characteristic classes and forms 41.3. Equivariant differential cohomology 42. Models for equivariant cohomology 52.1. Simplicial manifolds and differential forms 52.2. The Cartan model 102.3. Getzlers resolution 112.4. A quasi-isomorphism 143. The definition of equivariant differential cohomology 153.1. The case of finite groups 163.2. A version for Lie groups by Gomi 203.3. The new version for Lie groups 244. Equivariant differential characteristic classes 294.1. Definitions 294.2. Multiplicative structures 355. Examples for equivariant differential cohomology 365.1. Free actions 365.2. Conjugation action on S G -Representations 445.5. Towards obstruction to immersions? 45Acknowledgements 45References 45 Date : November 10, 2015. a r X i v : . [ m a t h . DG ] N ov ANDREAS KÜBEL AND ANDREAS THOM Introduction
The interplay between geometry and topology is a widely occurring theme inmodern mathematics, whose most elementary appearance is the formula of
Hopf’sUmlaufsatz : Let c : [0 , a ] → R be a closed smooth curve in the plane. Then thewinding number of the curve is given by the integral over the curvature: n c = 12 π Z a κ ( t ) k c ( t ) k dt. This result is surprising: The quantity on the left-hand side is an integer and purelytopological – vividly speaking this means: it does not depend on small alterationsof the curve. Whereas, on the right-hand side, one integrates a real-valued function,which does depend on the geometry – how long the curve is and how strongly it iscurved.A first generalization of the Umlaufsatz is known as the Gauss-Bonnet theorem,which states that for any compact surface M of genus g in R :2( g −
1) = 12 π Z M κ, where now κ denotes the Gaussian curvature of the surface.The generalizations of these statements by characteristic classes are based on de Rham cohomology : The differential forms on a smooth manifold form a chaincomplex, which depends on the geometry of the space, but the cohomology ofthis chain complex is isomorphic to any real cohomology theory, e.g., to singularcohomology with real coefficients. This means that any real cohomology class – atopological object – can be represented by a closed differential form, a geometricobject.In these terms, the left-hand side of the equations above will be generalized bythe image of an integral cohomology class in real cohomology; the curvature onthe right-hand side will be replaced by a closed differential form (depending on thecurvature) and the integral will be expressed by taking the cohomology class of thisform.In general, characteristic classes associate cohomology classes to (isomorphismclasses of) vector bundles. For smooth bundles, there are two well-known proceduresto construct them, one which applies the geometric structure and one which usestopology only:The Chern-Weil-Homomorphism starts with a connection on the bundle andevaluates an invariant symmetric polynomial on the associated curvature form,which leads to a closed differential form, the characteristic form. As the difference ofthe characteristic forms of two connections is an exact form – the exterior derivativeof the transgression form – one gets a class in de Rham cohomology which isindependent of the chosen connection and called the characteristic class of thebundle.On the other hand one may also obtain these classes by pulling back universalcharacteristic classes via the classifying map of the bundle.Both construction have their own strengths: The characteristic form containsgeometric data, while the class is purely topological. The class itself actually is notan element in real, but in integral cohomology, where algebraic torsion may deliverfiner information, which cannot be reflected by the characteristic form, as there isno algebraic torsion over the field of real or complex numbers.To use both, the geometric information of the characteristic form and its transgres-sion and the algebraic torsion information from integral cohomology in one object,one defines differential cohomology and differentially refined characteristic classes.This was done first by Jeff Cheeger and James Simons in [11]. The differential QUIVARIANT DIFFERENTIAL COHOMOLOGY 3 cohomology theory extends integral cohomology by closed differential forms. Anotable result is that – while the classical first Chern class classifies complex linebundles up to isomorphism – the first differential Chern class classifies complex linebundles with connection up to isomorphism.From this starting point there are various ideas of differential refinements ofcohomology theories: Besides the differential characters of Cheeger and Simons, thereis an isomorphic model by smooth Deligne cohomology (see [5, 6]). On the otherhand there are various models for differential K-theory (see [8] for a survey, whichincludes a discussion of the literature). A general framework for these differentialrefinements is given in [9] and [7].We want to go back to the starting point and generalize the idea of the differentialrefinement to an equivariant setting, i.e., we have a Lie group G acting on a smoothmanifold M and ask for a theory which enables differential refinements of equivariantcharacteristic classes of G -equivariant vector bundles over M .To do so, we need a differential form model for equivariant cohomology, whichis capable to receive a homomorphism from integral cohomology. Moreover, thereshould be two constructions of real/complex equivariant characteristic classes, onevia equivariant characteristic forms and one via integral equivariant characteristicclasses, which should coincide under the homomorphism between the cohomologytheories.The construction of the differential refinement, which we will give, is an equivariantversion of smooth Deligne cohomology, but to stress that it fits into the picture ofdifferential refinements, we will use the term equivariant differential cohomology,even if we will not discuss equivariant differential refinements in general.1.1. Equivariant cohomology and simplicial manifolds.
Defining equivariantcohomology H ∗ G ( M ) is a simple business using two expected properties of this functor:homotopy invariance and that, for free actions, the equivariant cohomology shouldcoincide with the cohomology of the quotient. Namely, let EG be a contractiblespace with a free G -action, then the diagonal action of G on EG × M is free andthe map EG × M → {∗} × M is a homotopy equivalence. Hence, we have describedthe well-known Borel construction, which is in formulas H ∗ G ( M ) = H ∗ G ( {∗} × M ) = H ∗ G ( EG × M ) = H ∗ ( EG × G M ) , for any cohomology theory and any coefficient group, e.g. singular cohomology withvalues in Z , R or C . Here EG × G M is the quotient of the diagonal G -action on EG × M .As short and easy this construction is, it creates a task for us: EG is even in simplecases not a finite-dimensional manifold, hence we have no de Rham cohomology. But EG is something similar to a manifold: Namely there is a simplicial manifold ([12,14]), i.e., a simplicial set such that the set of p -simplices forms a smooth manifold foreach p and all face and degeneracy maps are smooth, and the geometric realizationof this simplicial manifold is EG × G M . This will be introduced in Section 2.1and we will explain how one defines (simplicial) differential forms on a simplicialmanifold. They lead to a complex, which is bi-graded: by the form degree and thesimplicial degree. The cohomology of this double complex calculates equivariantcomplex cohomology. In fact, simplicial differential forms also form a (graded)simplicial sheaf Ω • , ∗ C .Using the language of simplicial sheaf cohomology, the de Rham homomorphismis induced by the inclusion of the locally constant simplicial sheaf Z → Ω • , ∗ C , aslocally constant functions.In Section 2.2, we will introduce the reader to a more famous model of equivariantcohomology using differential forms, known as the Cartan model. This is given ANDREAS KÜBEL AND ANDREAS THOM by the so-called equivariant differential forms, i.e., equivariant polynomial maps g → Ω ∗ ( M ), where the differential d C on ( C [ g ] ⊗ Ω ∗ ( M )) G is given by( d C ω )( X ) = d ( ω ( X )) + ι X ] ( ω ( X )) , i.e. the sum of the exterior differential and the contraction with the fundamentalvector field of X , and hence increases the grading given throughtwice the polynomial degree + the differential form degreeby one.The Cartan model has the advantage that its cochain complex is substantiallyslimmer than the double complex Ω • , ∗ defined above, but it is not directly capableto receive a homomorphism from integral cohomology. Therefore we apply ideas of[16] to compare the different models of equivariant cohomology. This comparisonwill enable our construction of a differential refinement of equivariant integralcohomology.1.2. Equivariant characteristic classes and forms.
Let G act on the vectorbundle E → M , i.e., we have an action on the total space and the base space,such that the projection is equivariant. Via the Borel construction, one can defineequivariant characteristic classes easily: Take the usual characteristic classes of EG × G E → EG × G M !There is also a characteristic form construction (see [1]) which does not onlydepend on the curvature, but also uses the moment map µ ∇ of the connection ∇ .This is a map from the Lie algebra of the acting group to the endomorphisms of thevector bundle (see Definition 4.1 for details). In this way, one obtains an equivariantcharacteristic form, which is a closed equivariant differential form, i.e., an elementin the Cartan complex.Both paths lead to tho same class in equivariant complex Borel cohomology. Wediscuss this in [21], since for this compatibility, although generally assumed to hold,there exist only a proof for special cases (compare [2]) in the literature.1.3. Equivariant differential cohomology.
After we have achieved this under-standing of equivariant characteristic forms, we can review previous definitionscritically to obtain a more satisfactory one.There is a definition of equivariant smooth Deligne cohomology ˆ H ∗ G ( M, Z ) in [18],and Kiyonori Gomi shows there that ˆ H G ( M, Z ) classifies G -equivariant line bundleswith connection. We will show that his definition fits, for actions of compact groups,into a differential cohomology hexagon (Theorem 3.14) and thus can be interpretedas a model for equivariant differential cohomology. But this definition neglects thesecondary information of the moment map and is, thus, only satisfactory in the caseof finite groups, where there is no moment and in low degrees, where the momentmap does not play a role. There are also other, less elaborated, definitions (seeRemark 3.29), which are all unsatisfactory from our insight to characteristic forms.Therefore, in Section 3.3, we define (full) equivariant differential cohomology b H ∗ G ( M, Z ) (using a mapping cone construction similar to the non-equivariant casein [6]) and show (see Theorem 3.22) that for any compact Lie group G , one has thecommutative diagram Ω n − G ( M ) (cid:30) ( d + ι )Ω n − G ( M ) Ω nG ( M ) cl H n − G ( M, C ) b H nG ( M, Z ) H nG ( M, C ) H n − G ( M, C / Z ) H nG ( M, Z ) → a → d + ι → → → → R (cid:16) I , → → → − β → QUIVARIANT DIFFERENTIAL COHOMOLOGY 5 where the line along the top, the one along the bottom and the diagonals are exact.In the case of the trivial group one obtains the classical differential cohomology.In degree up to two, our definition coincides with the one of Gomi. In higherdegrees one has additional geometric data, e.g., in the case of the conjugationaction of S = SU (2) on itself, as discussed in Section 5.2, one has ˆ H S ( S , Z ) = H S ( S , C / Z ) ⊕ H S ( S , Z ) = C / Z ⊕ Z , while we have a short exact sequence0 → Ω ( S ) S (cid:30) dC ∞ ( S ) S → b H S ( S , Z ) → ˆ H S ( S , Z ) → , hence we have additional transgression data.From the hexagon, one concludes that equivariant differential cohomology is theright group to define equivariant differential characteristic classes in, since theycan refine both, the equivariant integral characteristic class and the equivariantcharacteristic form. The details of this constructions are worked out in Section 4.2. Models for equivariant cohomology
Let M be a smooth manifold acted on from the left by a Lie group G . Todefine equivariant cohomology one uses two properties which one expects from sucha theory: it should be homotopy invariant and for free actions, the equivariantcohomology should be the cohomology of the quotient. Recall that the total spaceof the classifying bundle EG is a contractible topological space with free G -action.Hence EG × M has the homotopy type of M and the diagonal action is free. Henceone defines H ∗ G ( M ) := H ∗ ( EG × G M ) , where EG × G M is the quotient of EG × M by the diagonal action. We are interestedin differential form models for equivariant cohomology, but in general EG is not afinite-dimensional manifold, hence we cannot use the usual de Rham cohomology.But there is a model for EG , which consist of finite dimensional manifold:2.1. Simplicial manifolds and differential forms.
The model of EG × G M weare going to use is a given by a simplicial manifold. Definition 2.1 (see, e.g., [14, p.89]) A simplicial manifold is contra-variant functorfrom the simplex category ∆ to the category of smooth manifolds.Explicitly this is an N -indexed family of manifolds with smooth face and degen-eracy maps satisfying the simplicial relations, i.e. ∂ i ◦ ∂ j = ∂ j − ◦ ∂ i , if i < jσ i ◦ σ j = σ j +1 ◦ σ i , if i ≤ j∂ i ◦ σ j = σ j − ◦ ∂ i , if i < j id , if i = j, j + 1 σ j ◦ ∂ i − , if i > j + 1 Example 2.2
Our most important example of a simplicial manifold is the following(compare [18, p.316],[16, section 3.2]): G • × M = { G p × M } p ≥ , where G p stands for the p -fold Cartesian product of G . The face maps G p × M → G p − × M are given as ∂ ( g , . . . , g p , x ) = ( g , . . . , g p , x ) ∂ i ( g , . . . , g p , x ) = ( g , . . . , g i − , g i g i +1 , . . . , g p , x ) for 1 ≤ i ≤ p − ∂ p ( g , . . . , g p , x ) = ( g , . . . , g p − , g p x ) ANDREAS KÜBEL AND ANDREAS THOM and the degeneracy maps for i = 0 , . . . , p by σ i : G p × M → G p +1 × M ( g , . . . , g p , x ) ( g , . . . , g i , e, g i +1 , . . . , g p , x ) . These maps satisfy the simplicial relations. In particular for p = 1 the map ∂ equalsthe group action, while ∂ is the projection onto the second factor, i.e. onto M . Definition 2.3 (see, e.g., [14, p.75]) The (fat) geometric realization of a simplicialmanifold M • , is the topological space k M • k = [ p ∈ N ∆ p × M p / ∼ with the identifications( ∂ i t, x ) ∼ ( t, ∂ i x ) for any x ∈ M p , t ∈ ∆ p − , i = 0 , . . . , n and p = 1 , , . . . . Example 2.4
The geometric realization of the simplicial manifold G • × M is amodel of EG × G M and in particular if M is single point the geometric realizationof G • × pt is a model of the classifying space BG (compare [14, pp.75]).Before giving a differential form model for equivariant cohomology, we will explainsheaves and sheaf cohomology for simplicial manifolds, as this is the technical basisfor all further constructions and definitions. Definition 2.5 (see [12, (5.1.6)]) A simplicial sheaf on the simplicial manifold M • is a collection of sheaves F • = {F p } p ∈ N , where, for each p , F p is a sheaf on M p and there are morphisms ˜ ∂ i : ∂ − i F p → F p +1 and ˜ σ i : σ − i F p +1 → F p satisfyingthe simplicial relations as stated above.The simplicial sheaf cohomology is defined as the right derived functor of theglobal section functor [12, def. 5.2.2.], where global sections of a simplicial sheaf,are the equalizer ker (cid:0) ˜ ∂ − ˜ ∂ : F ( M ) → F ( M ) (cid:1) . This definition opens the question: Are there enough injectives? As Pierre Deligneis quite short on this and there are mistakes in the literature (see Remark 2.8), weshould give an answer.
Lemma 2.6
The category of simplicial sheaves has enough injectives.Proof.
Let F • be a simplicial sheaf. Let P p be the functor from simplicial sheavesto sheaves, which sends a sheaf to its p -th level, i.e., F • is sent to the sheaf F p on M p . Pick for any F p an injective sheaf I p on G p × M , in which F p embeds (forexistence see e.g. [19, section III.2]).Now we construct a right adjoint of P p (analogous to [17, p.409]): Let B be asheaf on G p × M . Define a simplicial sheaf on G • × M as( S p B ) q = Y h ∈ ∆( q,p ) h − B By the adjointness of the functors, injectivity of B implies injectivity of S p B .Moreover the equalityHom F • , Y p S p I p ! = Y p Hom( F • , S p I p ) = Y p Hom( P p F • , I p ) = Y p Hom( F p , I p )shows that the simplicial sheaf F • embeds into Q p S p I p because for each F p thereis an injection into I p . (cid:3) QUIVARIANT DIFFERENTIAL COHOMOLOGY 7
Now let 0 → F • → I • , δ → I • , δ → . . . be an injective resolution. Omitting the first columns and taking global sectionsyields to a double complex I p,q ( M p ) , p X i =0 ( − i ˜ ∂ i + ( − p δ ! , whose cohomology is defined to be the cohomology H ∗ ( M • , F • ) = H ∗ I p,q ( M p ) , p X i =0 ( − i ˜ ∂ i + ( − p δ ! of the simplicial sheaf F • on the simplicial manifold M • .The definition does not depend on the injective resolution chosen. In the non-simplicial case, this is a well-known fact: the identity on the space and the sheaf in-duces a morphism between two chosen injective resolutions, which is an isomorphismin cohomology. In the simplicial case, we need an additional argument: As before weobtain a morphism of the double complexes of global sections from the identity onthe space. When taking cohomology in every horizontal line ( I p, ∗ ( M p ) , ( − p δ ), thismorphism will induce an isomorphism between the bi-graded complexes. Hence wecan apply the following lemma, to see, that we have an isomorphism in cohomology. Lemma 2.7 (see e.g. [14, Lemma 1.19])
Suppose f : ( C ∗ , ∗ , d + d ) → ( C ∗ , ∗ , d + d ) is homomorphism of double complexes and the induced homomorphism ( H q ( C p, ∗ , d ) , d ) → ( H q ( C p, ∗ , d ) , d ) is an isomorphism, then f induces an isomorphism in the total cohomology of doublecomplexes. Remark 2.8
One could have the idea (e.g. [4, p.3],[18, Section 3.2]) that aninjective resolution on any simplicial level would be sufficient as the maps ˜ ∂ i liftby the injectivity of the sheaf. But as this lift is not unique, it is unclear that thesimplicial relations hold and thus there is no general reason why ∂ = P i ( − i ˜ ∂ i isa boundary operator. In fact one can construct the following counterexample: Takethe trivial group, acting on a point, then all ˜ ∂ i : Z → Z are the identity. A injectiveresolution of the abelian group Z is given by Z → C → C / Z . Beside id : C → C , thecomplex conjugation is also a lift of id Z . Making appropriate choices, for the lifts ˜ ∂ i one finds an example where ∂ ◦ ∂ = 0.In practice, one usually uses acyclic resolutions, instead of injective ones, tocalculate cohomology. This works in the simplicial case, too. Let0 → F • → A • , δ → A • , δ → . . . be an acyclic resolution, i.e., each A • ,k is a simplicial sheaf and all but the zerothcohomology of each sheaf A p,q vanish. Let I • , ∗ be a simplicial injective resolution.The identity map on the simplicial manifold and the sheaf F • induce a homomor-phism of the double complex of global sections (by injectivity of I ), which inducesan isomorphism of the bi-complexes, ( H q ( A p, ∗ , δ ) , ∂ ) → ( H q ( I p, ∗ , δ ) , ∂ ), as acyclicresolutions calculate cohomology. Thus the last lemma implies the isomorphism inthe cohomology of the double complexes.In the examples, which we study later, the simplicial sheaf will actually notjust be a sheaf of abelian groups, but a cochain complexes of simplicial sheaves ofabelian groups. A resolution for a chain complex goes by the name Cartan-Eilenbergresolution and exists for cochain complex in any abelian category with enoughinjectives (compare [32, Section 5.7]). In our context, the resolution of a cochain
ANDREAS KÜBEL AND ANDREAS THOM complex of simplicial sheaves is a triple instead of a double complex. Nevertheless,one can form a total complex of the global sections of the triple complex and thecohomology of the cochain complex of simplicial sheaves is defined as the cohomologyof this total complex.We will now discuss some explicit models for simplicial sheaf cohomology.2.1.1.
Simplicial de Rham cohomology.
This exposition is based on [14, Section 6].Let M • = { M p } be a simplicial manifold. For any p , differential forms on M p forma the cochain complex of sheaves (Ω ∗ M p , d ). The face and degeneracy maps of M • induce, via pullback, face and degeneracy maps between the differential forms on M p and M p ± . Thus, one obtains the simplicial sheaf Ω • , ∗ of differential forms on M • .On the global sections of this sheafΩ p,q ( M ) = Ω q ( M p ) , there is a horizontal differential d : Ω p,q ( M • ) → Ω p,q +1 ( M • ), given by the exteriordifferential and vertical differential ∂ : Ω p,q ( M • ) → Ω p +1 ,q ( M • ) , given by the alternating sum of pullbacks along the face maps(1) ∂ ( ω ) = p +1 X i =0 ( − i ∂ ∗ i ω. Proposition 2.9 (Ω p,q ( M • ) , d + ( − q ∂ ) p,q forms a double complex.Proof. ( d + ( − q ∂ ) = 0, because ∂ = 0 by the simplicial relations, d∂ = ∂d as d is functorial, d = 0 by the well-known property of the exterior derivative. (cid:3) Moreover, since the differential forms form a sheaf of C ∞ -module, they form afine and hence acyclic sheaf.In particular, for the simplicial manifold G • × M , we have the double complexΩ q ( G p × M ), what is a first de Rham type model for equivariant cohomology bythe following Proposition. Proposition 2.10 (Prop. 6.1 of [14])
Let M • be a simplicial manifold. There is anatural isomorphism H ∗ (Ω • , ∗ ( M • ) , d + ( − ∗ ∂ ) ∼ = H ∗ ( k M • k , C ) . Simplicial Čech cohomology.
Definition 2.11 (see [4, 18]) A simplicial cover for the simplicial manifold M • isa family U • = {U ( p ) } of open covers such that(1) U ( p ) = { U ( p ) α | α ∈ A ( p ) } is an open cover of M p , for each p , and(2) the family of index sets forms a simplicial set A • = { A ( p ) } satisfying(3) ∂ i ( U ( p ) α ) ⊂ U ( p − ∂ i α and σ i ( U ( p ) α ) ⊂ U ( p +1) σ i α for every α ∈ A ( p ) . Definition 2.12 (see [4, 18]) Given a simplicial cover U • , one forms the Čech chaingroups ˇ C • , ∗ ( U • , F • ) byˇ C p,q ( U • , F • ) = Y α ( p )0 ,...,α ( p ) q ∈ A ( p ) F p (cid:18) U ( p ) α ( p )0 ∩ · · · ∩ U ( p ) α ( p ) q (cid:19) , with the usual Čech boundary operator δ : ˇ C p,q → ˇ C p,q +1 and the simplicial bound-ary map ∂ : ˇ C p,q → ˇ C p +1 ,q defined as alternating sum as above. QUIVARIANT DIFFERENTIAL COHOMOLOGY 9
Observe, that the third condition of the simplicial cover ensures that ∂ mapsbetween the Čech groups. The simplicial Čech cohomology , denoted byˇ H ∗ ( U • , F • ) , is the cohomology of the double complex ( ˇ C p,q , ∂, ( − p δ ). As in the non-simplicialcase (see [19, section III.4]), any simplicial open cover induces a canonical homo-morphism ˇ H ∗ ( U • , F • ) → H ∗ ( M • , F • ) . Moreover, given a refinement V • of the simplicial open cover U • , then the naturaldiagram ˇ H ∗ ( U • , F • ) H ∗ ( M • , F • )ˇ H ∗ ( V • , F • ) → → → commutes. Thus one can form the limit over all refinements of simplicial open coversand obtains an isomorphismlim U • ˇ H ∗ ( U • , F • ) → H ∗ ( M • , F • ) . For more details see [4, 18].2.1.3.
Simplicial singular cohomology.
Let A be an abelian group. Later, the mostinteresting cases for us will be A ∈ { Z , R , C , C / Z , R / Z } . Then there is the locallyconstant sheaf A δ , consisting of continuous maps to A furnished with the discretetopology, in any simplicial degree. The maps ˜ ∂ i and ˜ σ i are given by pullback along ∂ i respectively σ i . One can calculate H ∗ ( M • , A ) via singular cohomology. Definition 2.13 (see [14, p. 81]) The simplicial singular cochain complex ( C • , • sing ( M • , A ) , ∂, ∂ sing )is the double complex consisting of groups C p,q sing = C q sing ( M p ) = map( C ∞ (∆ q , M p ) , A )of smooth singular cochains on each M p with group structure induced from A , verticalboundary map induced from the simplicial manifold and horizontal boundary mapgiven by the singular boundary operator.To obtain a double complex one has to use the boundary map ∂ + ( − p ∂ sing . Asimplicial map f • : M • → M induces a map of double complexes f ∗• : C • , • sing ( M , A ) → C • , • sing ( M • , A ) . Theorem 2.14 (Theorem 5.15. of [14])
There are functorial isomorphisms H ∗ ( k M k , A ) = H ∗ sing ( M • , A ) := H ∗ (cid:0) C • , • sing ( M • , A ) , ∂ + ( − p ∂ sing (cid:1) To compare singular cohomology with general sheaf cohomology, one can usearguments of [31, pp. 191-200]. Sheafify the singular cochains C q sing ( M p ): Let S q ( M p , A ) be the sheaf associated to the presheaf M ⊂ U map( C ∞ (∆ q , U ) , A ) . Then one has an acyclic resolution0 → A • → S ( M • , A ) → S ( M • , A ) → . . . and hence H ∗ ( M • , A ) = H ∗ ( M • , S ∗ ( M • , A )) . On the other hand, the global sections of S q ( M p , A ) are exactly C q sing ( M p ).Thus we have shown the following theorem. Theorem 2.15 H ∗ ( k M k , A ) = H ∗ sing ( M • , A ) = H ∗ ( M • , S ∗ ) = H ∗ ( M • , A ) . In particular, for M • = G • × M , we obtain: H ∗ G ( M, A ) = H ∗ sing ( G • × M, A ) = H ∗ ( G • × M, A ) . Simplicial cellular cohomology.
The most handy cohomology theory for calcu-lation is cellular cohomology. Recall (compare [29, p. 12]) that a
CW complex is atopological space X with a collection of subspaces, called cellular decomposition, X ⊂ X ⊂ X ⊂ · · · ⊂ X, such that X is discrete, X p is obtained from X p − by attaching p -cells, X = S i X i ,and U ⊂ X is closed, if and only if U ∩ X p is closed in X p for any p ∈ N . A map f : X → Y between cellular complexes is called cellular, if f ( X p ) ⊂ Y p . The cellularchain complex (see [29, pp. 118-122]) is given by C n ( X ) = H n sing ( X n , X n − ; A ) and d n cell is the composition H n ( X n , X n − ) → H n ( X n , ∅ ) → H n +1 ( X n +1 , X n )of the map induced from the inclusion ( X n , ∅ ) ⊂ ( X n , X n − ) and the connectingmorphism of ( X n , ∅ ) ⊂ ( X n +1 , ∅ ) ⊂ ( X n +1 , X n ).By a cellular decomposition of the simplicial manifold G • × M , we understand acollection of topological spaces ( X p,q ) p,q ∈ N , such that X p, ∗ is a cellular decompositionof G p × M and all face and degeneracy maps are cellular. Thus we receive a doublecomplex, the simplicial cellular chain complex ( C q cell ( G p × M ) , d cell + ( − q ∂ ). Wedefine H ∗ cell ( G • × M, A ) to be the cohomology of this double complex.One has the following small proposition, for which I did not find a reference inthe literature.
Proposition 2.16
There is an isomorphism H ∗ cell ( G • × M, A ) = H ∗ sing ( G • × M, A ) . Proof.
Given a map between the singular and cellular chains, Lemma 2.7 wouldimply the result. Hence we are done, if we find such a map for normal, i.e., non-simplicial spaces, in a functorial manner. There is no map between singular andcellular chains in general, but one can construct a complex of so called simplicialsingular chains (see [13, Section V.8]), and functorial quasi-isomorphisms to both,singular and cellular chains. (cid:3)
The Cartan model.
A well-known de Rham-like model for equivariant co-homology goes back to Henri Cartan ([10]). Our Exposition follows [22]. Let G be a compact Lie group acting smoothly on the smooth manifold M and denotethe Lie algebra of G by g = T e G . Let S ∗ ( g ∨ ) be the symmetric tensor algebra ofthe (complex) dual of the Lie algebra g ∨ . The group G acts on this algebra bythe coadjoint action and on Ω ∗ ( M ) by pulling back forms along the map m gm .Hence we have a G -action on S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ). The invariant part of this algebra( S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) G is exactly what one calls the Cartan complex and is denoted byΩ ∗ G ( M ). In other words: The Cartan complex consists of G -equivariant polynomialmaps ω : g → Ω ∗ ( M ). Let ω , ω ∈ Ω ∗ G ( M ), then there is a wedge product( ω ∧ ω )( X ) = ω ( X ) ∧ ω ( X ) . On this algebra one defines a differential as d C ω ( X ) = d ( ω ( X )) + ι ( X ] ) ω ( X ) , QUIVARIANT DIFFERENTIAL COHOMOLOGY 11 for ω ∈ Ω ∗ G ( M )) and X ∈ g , i.e., one takes the differential on the manifold and addsthe contraction with the fundamental vector field. To make this differential raisethe degree by one, the grading on Ω ∗ G ( M ) is given bytwice the polynomial degree + the differential form degree . Lemma 2.17 (Ω ∗ G , d C ) is a cochain complex.Proof. First, observe that d C increases the total degree by one, since d increasesthe differential form degree, and the contraction ι , while decreasing the form degreeby one, increases the polynomial degree by one. Next, one has to check, that thedifferential really maps invariant forms to invariant forms and that it squares tozero.Let ω ∈ Ω ∗ G ( M )) and X ∈ g . d C ω (Ad g X ) = d ( ω (Ad g X )) + ι ((Ad g X ) ] ) ω (Ad g X )= d ( gω ( X )) + ι ( gX ] g − ) g ( ω ( X ))= gd ( ω ( X )) + gι ( X ] ) g − g ( ω ( X ))= gd C ω Thus d C ω is G -equivariant. Moreover, we have d C ω ( X ) = d ω ( X ) + dι ( X ) ω ( X ) + ι ( X ) dω ( X ) + ι ( X ) ω ( X ) = L X ω ( X )and L X ω ( X ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( tX ) ω ( X ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ω (exp( − tX ) X exp( tX )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ω ( X ) = 0 . Thus d C squares to zero, i.e., it is a boundary operator. (cid:3) In the special case of M = pt , i.e., of a single point, the Cartan algebra reducesto the algebra of invariant symmetric polynomials I k ( G ) = (( S ∗ ( g ∨ ) ⊗ Ω ∗ ( pt )) G ) k = ( S k ( g ∨ )) G . Getzlers resolution.
In order to investigate cohomology of actions of non-compact groups, Ezra Getzler [16, Section 2] defines a bar-type resolution of theCartan complex. We will apply his ideas slightly different: The complex defined byGetzler will allow us to compare equivariant integral cohomology (defined via thesimplicial manifold) with equivariant cohomology defined by the Cartan model.Let, as before, a Lie group G act on a smooth manifold M from the left. Define C -vector spaces C p ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) consisting of smooth maps, from the p -foldCartesian product G p → S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ) , to the space of polynomial maps from g to differential forms on M . We give thesegroups a bigrading: The horizontal grading is the one of S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ) definedabove and the vertical grading is p . The Cartan boundary operator d + ι nowinduces a map ( − p ( d + ι ), which increases the horizontal grading by 1 in any row.As we are not restricted to the G -invariant part of S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ), this map willnot square to zero, but (( − p ( d + ι )) = dι + ιd = L is the Lie derivative (see e.g. [15, Proposition 1.121]). In vertical direction, there isa differential ¯ d : C k ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) → C k +1 ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) defined by( ¯ df )( g , . . . , g k | X ) := f ( g , . . . , g k | X ) + k X i =1 ( − i f ( g , . . . , g i − g i , . . . , g k | X )+ ( − k +1 g k f ( g , . . . , g k − | Ad( g − k ) X )for g , . . . , g k ∈ G and X ∈ g .Note, in particular, that the kernel of¯ d : C ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) → C ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ))is exactly Ω ∗ G ( M ). Moreover, in case of a discrete Group G , g = 0 and thus onechecks, that C p ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) = C p ( G, Ω ∗ ( M )) = Ω p, ∗ ( G • × M )and ¯ d is equal to ∂ .In the case of a compact Lie group, the map ¯ d admits a contraction (compare,e.g., [18, p. 322]): Lemma 2.18
Integration over the group, with respect to a right invariant probabilitymeasure, defines a map Z G : C p ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) → C p − ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ))(2) (cid:18)Z G f (cid:19) ( g , . . . , g p − , m ) = ( − i Z g ∈ G f ( g, g , . . . , g p − , m ) dg such that ¯ d R G f = f if ¯ df = 0 .Proof. This is proven by a direct calculation: (cid:18) ¯ d Z G ω (cid:19) ( g , . . . , g p , m )= (cid:18)Z G f (cid:19) ( g , . . . , g p | X ) + p X i =2 ( − i (cid:18)Z G f (cid:19) ( g , . . . , g i − g i , . . . , g p | X )+ ( − p +1 g p (cid:18)Z G f (cid:19) ( g , . . . , g p − | Ad( g − p ) X )= Z G f ( g, g , . . . , g p | X ) dg + p X i =2 ( − i Z G f ( g, g , . . . , g i − g i , . . . , g p | X ) dg + Z G g p f ( g, g , . . . , g p − | Ad( g − p ) X ) dg = Z G f ( g, g , . . . , g p | X ) + p X i =2 ( − i f ( g, g , . . . , g i − g i , . . . , g p | X )+ ( − p +1 g p f ( g, g , . . . , g p − | Ad( g − p ) X ) ! dg Now we apply ¯ df ( g, g , . . . , g p | X ) = 0= Z G ( f ( g , . . . , g p | X ) − f ( gg , . . . , g p | X ) + f ( g, g , . . . , g p | X )) dg = f ( g , . . . , g p | X ) − Z G f ( gg , g , . . . , g p | X ) dg + Z G f ( g, g , . . . , g p | X ) dg = f ( g , . . . , g p | X ) QUIVARIANT DIFFERENTIAL COHOMOLOGY 13 (cid:3)
Thus, for compact groups, the vertical cohomology of this bi-graded collection ofgroups is the Cartan complex.One can turn the bi-graded collection C p ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) of groups into adouble complex. Therefore Getzler defines another map,¯ ι : C p ( G, S l ( g ∨ ) ⊗ Ω m ( M )) → C p − ( G, S l +1 ( g ∨ ) ⊗ Ω m ( M )) , given by the formula(¯ ιf )( g , . . . , g p − | X ) := p − X i =0 ( − i ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( g , . . . , g i , exp( tX i ) , g i +1 , . . . , g p − | X ) , where X i = Ad( g i +1 . . . g p − ) X . Lemma 2.19 (Lemma 2.1.1. of [16])
The map ¯ ι has the following properties: ¯ ι = 0 and ¯ d ¯ ι + ¯ ι ¯ d = − L. Proof.
This is shown in [16] by recollection of the sums in the definition of ¯ ι and¯ d . (cid:3) Moreover one obtains:
Lemma 2.20 (Corollary 2.1.2. of [16]) d G = ¯ d + ¯ ι + ( − p ( d + ι ) is a boundaryoperator on the total complex L p +2 q + r = n C p ( G, S q ( g ∨ ) ⊗ Ω r ( M )) .Proof. d G increases the total index by one, as ¯ d increases the first index, d increasesthe third index, ι decreases the third, while it is increasing the second index and ¯ ι decreases the first index, while it is increasing the second one.As d and ι are equivariant under the G -action, they commute with ¯ d . And as d and ι only act on the manifold M and not on the group part, the same is true for ¯ ι .Thus d G = ( ¯ d + ¯ ι ) + ( − p ( ¯ d + ¯ ι )( d + ι ) + ( − p ± ( d + ι )( ¯ d + ¯ ι ) + ( d + ι ) = ¯ d ¯ ι + ¯ ι ¯ d + ( dι + ιd )= − L + L = 0 . (cid:3) Remark 2.21
The reader, who compares this with the original paper of Getzlerwill note that we changed some signs. It just seems more natural to us in this way.Furthermore Getzler uses some reduced subcomplex, which is, by standard argumentson simplicial modules (compare Proposition 1.6.5 in [23]), quasi-isomorphic to thefull complex, which we have taken.One can check that M p + q = nq + r = k C p ( G, S q ( g ∨ ) ⊗ Ω r ( M )) n,k , ¯ d + ( − p ι, ( − p d + ¯ ι is a double complex. But this point of view will not fit to the construction, whichwe want to do with this bigraded module later: We want to turn the groups C p ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) into simplicial sheaves on G • × M . Definition 2.22 A simplicial homotopy cochain complex of modules is a triple( M • , ∗ , f, s ), where M • , ∗ is a Z -graded simplicial module, f is a map of simplicialmodules, which increases the degree by one and s is a simplicial zero homotopy of f which commutes with f and squares to zero, i.e., s∂ + ∂s = − f , sf = f s, and s = 0 . Example 2.23
Observe that( C • ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) , d + ι, ¯ ι )is a simplicial homotopy cochain complex. Definition + Proposition 2.24
The total complex of a simplicial homotopycochain complex ( M • , ∗ , f, s ) is the chain complex M p + q = n M p,q ! n , ∂ + s + ( − p f . Proof.
We have to check that ∂ + s + ( − p f defines a boundary map. Thereforecalculate( ∂ + s + ( − p f ) = ∂ + s + ∂s + s∂ + ( − p ( ∂ + s ) f + ( − p − f ( ∂ + s ) + f = s∂ + ∂s + f = 0 . (cid:3) Observe that the total complexes of the interpretation of C • ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ))as double complex and as simplicial homotopy cochain complex coincide. Moreover,note for our applications later, that in the first column, of the double complexinterpretation and the degree zero part of the interpretation as simplicial homotopycochain complex are equal. In formulas this means M p + q = nq + r = k C p ( G, S q ( g ∨ ) ⊗ Ω r ( M )) n, = C n (cid:0) G, S ( g ∨ ) ⊗ Ω ( M ) (cid:1) . A quasi-isomorphism.
In this section, we will discuss a map defined in [16,Section 2.2.]. It will relate the complex C ∗ ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) from the last sectionto the double complex Ω ∗ ( G • × M ), which consists in degree ( p, q ) of q -forms on G p × M with horizontal boundary map d = d G p + d M and vertical boundary map ∂ from the simplicial manifold structure. Thus we have an explicit identificationsof chains in the one complex with chains in the other complex. This will allow usto compare our definition of equivariant differential cohomology (Section 3.3) withdefinitions given before. Definition 2.25 (Def. 2.2.1. of [16]) The map J : Ω ∗ ( G p × M ) → p M l =0 C l ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ))is defined by the formula J ( ω )( g , . . . , g l | X ) := X π ∈ S ( l,p − l ) sgn( π ) ( i π ) ∗ (cid:16) ι π ( l +1) ( X ( π ) l +1 ) . . . ι π ( p ) ( X ( π ) p ) ω (cid:17) . Here S ( l, p − l ) is the set of shuffles, i.e., permutations π of { , . . . , p } , satisfying π (1) < · · · < π ( l ) and π ( l + 1) < · · · < π ( p ) , QUIVARIANT DIFFERENTIAL COHOMOLOGY 15 X ( π ) j = Ad( g m . . . g l ) X , where m is the least integers less than l , such that π ( j ) <π ( m ), ι j means, that the Lie algebra element should be a tangent vector at the j -thcopy of G , and i π : G l × M → G p × M is the inclusion x ( h , . . . , h p , x ) with h j = ( g m if j = π ( m ) , ≤ m ≤ le ∈ G otherwise,which is covered by the bundle inclusion T M → T ( G p × M ).Observe that the image of ω under J does only depend on the zero form partand, in direction of any copy of G , on the one form part at the identity e ∈ G .The next Lemma – which is mainly a citation of [16, Lemma 2.2.2.], but withsigns corrected – shows, that the map J can be interpreted as a map of doublecomplex. Lemma 2.26
The map J respects the boundaries with the correct sign, i.e., J ◦ ∂ = (cid:0) ¯ d + ( − p ι (cid:1) ◦ J and, after decomposing d = d G + d M with respect to the Cartesian product G p × M J ◦ (( − p d M ) = ( − p d ◦ J and J ◦ ( − p ) d G = ¯ ι ◦ J , where p is the simplicial degree before and p the simplicial degree after applicationof the map J ,Proof. The proof is given in [21, Lemma 2.13]. The idea is, to checks the terms typeby type. (cid:3)
Moreover, the map J induces an isomorphism in the cohomology of the associatedtotal complexes. Theorem 2.27 (Theorem 2.2.3. of [16]) J is a quasi-isomorphism. The definition of equivariant differential cohomology
The are several attempts to a definition [24, 18, 28]. The most elaborated oneis given by Kiyonori Gomi in [18], where he defines equivariant smooth Delignecohomology of a smooth manifold M acted on by a Lie group G . His investigations(for G a compact group) can be summarized in the following diagram with exactdiagonals(3) Ω n − ( M, C ) G (cid:30) im d Ω n cl ( M, C ) G ˆ H n ( M, Z ) H nG ( M, C ) H n − ( G ∗ × M, π − C / Z ) H nG ( M, Z ) . → a → R → I → , → → The subscript G stands for equivariant cohomology and the superscript G forequivariant forms. Gomi defines the maps and shows that the diagonals are exactin the middle.From our point of view, this diagram is not satisfactory: On the one hand, onedoes not have the Bockstein sequence. On the other hand, closed equivariant formsis not, what one expects in the upper right corner, as there indeed exists a mapΩ n cl ( M, C ) G → H nG ( M, C ) . But this map is in general not surjective, as not every n -class in equivariant coho-mology is represented by a closed equivariant n -form: There are classes representedby (non-zero degree) polynomials g → Ω ∗ ( M ). As we have seen these are related to the moment map, which plays an important role when discussing equivariant char-acteristic classes and forms. Thus this information is neglected in Gomis curvaturemap.We will start with the case of finite groups, as there is no moment map if thegroup is discrete and thus everything is much more similar to the non-equivariantcase. In particular, the reader who is not used to differential cohomology hopefullywill have an easier access in this way. Afterwards we will discuss the definition ofGomi and show how to define a better curvature map R , such that one obtains ahexagon with Gomi’s definition of equivariant Deligne cohomology in the middle.By this we are motivated to give another definition, which incorporates additionalgeometric data. The difficulty is in general not to show that there is a hexagon,as this follows directly from the way of the definition by ideas of [7]. What thediscussion is about, is which groups sit on the corners of the hexagon.At the end of this section we will give some remarks on the definitions of [24, 28]for equivariant differential cohomology. Notice that we always work with complex valued differential forms for simplicity.
All statements also hold for real forms and real cohomology.3.1.
The case of finite groups.
Let, in this subsection, G denote a finite group.Thus a de Rham-type complex for equivariant cohomology is given byΩ ( M ) G d → Ω ( M ) G d → Ω ( M ) G d → . . . . To prepare for the case of Lie groups, we will, nevertheless, work on the simplicialmanifold G • × M . For definition of the Deligne complex, we will use a coneconstruction similar to [6]. Definition 3.1
Let M be a G -manifold. The equivariant Deligne complex is definedas D ( n ) G • × M = Cone (cid:0) Z ⊕ σ ≥ n Ω • , ∗ → Ω • , ∗ , ( z, ω ) ω − z (cid:1) [ − . Here Z denotes the locally constant simplicial sheaf of Z and Ω • , ∗ the cochaincomplex of simplicial sheaves of complex valued differential forms on the simplicialmanifold G • × M . Definition 3.2
Let G be a finite group acting on a smooth manifold M. The G -equivariant differential cohomology of M is defined to be the hypercohomologyˆ H nG ( M ) := H n ( G • × M, D ( n ) G p × M ) . Note that D ( n ) G • × M is a cochain complex of simplicial sheaves, i.e., it induces adouble complex of sheaves so that a resolution is given by triple complex. We givehere the general idea of how to define the cohomology of such an object. For detailssee Section 2.1. QUIVARIANT DIFFERENTIAL COHOMOLOGY 17
Denote the boundary map of D ( n ) by d and pick an injective resolution I • , ∗ , ∗ ,i.e., for any p ∈ N there is the following double complex of sheaves on G p × M Z C ∞ Ω . . . Ω n − Ω n ⊕ Ω n − . . .I p, , I p, , I p, , . . . I p,n − , I p,n, . . .I p, , I p, , I p, , . . . I p,n − , I p,n, . . .I p, , I p, , I p, , . . . I p,n − , I p,n, . . . ... ... ... ... ... ... → ι → δ → − d → − δ → − d → δ → − d → − d → ( − n − δ → ( d,ι ⊕− d ) → ( − n δ → d → δ → d → − δ → d → δ → d → d → ( − n − δ → d → ( − n δ → d → δ → d → − δ → d → δ → d → d → ( − n − δ → d → ( − n δ → d → → d → → d → → d → d → → d → The triple complex which calculates the cohomology is K p,q,r = (cid:0) Γ( G p × M, I p,q,r ) , ∂ + ( − p d + ( − p + q δ (cid:1) . And in the same spirit as [6] and [18] one investigates differential cohomology bythe following two short exact sequences0 → Cone( σ ≥ n Ω • , ∗ ι → Ω • , ∗ )[ − a →D ( n ) I → Z → → Cone( Z − ι → Ω • , ∗ )[ − →D ( n ) R → σ ≥ n Ω • , ∗ C → D ( n ) G • × M → Z ⊕ σ ≥ n Ω • , ∗ C → Ω • , ∗ C → D ( n ) G • × M [1] , which has the following interesting part in its long exact cohomology sequence(6) H n − ( G • × M, Z ) → H n − ( G • × M, Ω • , ∗ C ) → H n ( G • × M, D ( n )) ( I,R ) → H n ( G • × M, Z ) ⊕ H n ( G • × M, σ ≥ n Ω • , ∗ ) ( − ι,ι ) → H n ( G • × M, Ω • , ∗ ) . Recall from Section 2.1 that H n ( G • × M, Z ) = H n ( k G • × M k , Z ) = H nG ( M, Z )and H n ( G • × M, Ω • , ∗ ) = H nG ( M, C ) . Moreover, we have:
Lemma 3.3 H n ( G • × M, σ ≥ n Ω • , ∗ ) = Ω n cl ( M ) G . Proof. H n ( G • × M, σ ≥ n Ω • , ∗ ) = { ω ∈ Ω n ( M ) | ∂ω = 0 , dω = 0 } = { ω ∈ Ω n ( M ) | ∂ ∗ ω = ∂ ∗ ω, dω = 0 } = Ω n cl ( M ) G , as ∂ : G × M → M is the projection to M and ∂ is the action of G on M . Thuswe obtain closed equivariant differential n -forms. (cid:3) As Ω • , ∗ is a resolution of C δ , the locally constant sheaf of continuous maps tothe field C with discrete topology, we get a quasi isomorphismCone( Z − ι → Ω • , ∗ ) ’ Cone( Z − ι → C δ ) ’ C δ (cid:30)Z . The long exact cohomology sequence of the exact triangleCone( Z − ι → C δ )[ − → Z − ι → C δ → Cone( Z − ι → C δ )is the Bockstein sequence, up to a minus sign. Comparing this with (6) via theinclusion of (4), results in the follwing commutative diagram: H n − G ( M, Z ) H n − G ( M, C ) H n − G ( M, C / Z ) H nG ( M, Z ) H nG ( M, C ) H n − G ( M, Z ) H n − G ( M, C ) ˆ H nG ( M, Z ) H nG ( M, Z ) ⊕ Ω ∗ cl ( M ) G H nG ( M, C ) . → → → → − β → id ⊕ → − ι → → → I ⊕ R → ( − ι,ι ) For the next steps, recall that one can define, similar to Lemma 2.18, a verticalcontraction of the double complex Ω • , ∗ ( G • × M ). Lemma 3.4 ‘Integration’ over the group, defines a map Z G : Ω i C ( G k × M ) → Ω i C ( G k − × M )(7) (cid:18)Z G ω (cid:19) ( g , . . . , g k − , m ) = ( − i | G | X G ω ( g, g , . . . , g k − , m ) such that ∂ R G ( ω ) = ω if ∂ω = 0 .Proof. The proof is given by the same calculation as for Lemma 2.18. (cid:3)
One has Cone( σ ≥ n Ω • , ∗ ι → Ω • , ∗ )[ − ’ σ Proof. Let η ∈ (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) . Recall that the quasi-isomorphism σ 1] is in degree n given by η ( dη, η ).Thus one calculates R ( a ( η )) = R ( dη, η ) = dη, since R comes from the projection to the first summand. (cid:3) Collecting these statements, we have proven the following theorem. Theorem 3.6 Let G be a finite group acting on the smooth manifold M , then thereis a commutative diagram Ω n − ( M ) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) Ω n cl ( M ) G H n − G ( M, C ) ˆ H nG ( M, Z ) H nG ( M, C ) H n − G ( M, C / Z ) H nG ( M, Z ) → a → d → → → → R (cid:16) I , → → → − β → whose top line, bottom line and diagonals are exact. Remark 3.7 If G is the trivial group this is exactly the hexagon of the non-equivariant case. Proposition 3.8 If G acts freely on M , then ˆ H nG ( M, Z ) = ˆ H n (cid:16) M (cid:30) G, Z (cid:17) Proof. Let Q denote the quotient manifold, q : M → Q the quotient map and { e } the trivial group. The simplicial manifold { e } • × Q equals Q in any level and allface and degeneracy maps are the identity. This implies thatˆ H n { e } ( Q, Z ) = H n ( { e } • × Q, D ( n )) = ˆ H n ( Q, Z ) , since ∂ alternately equals id or the zero map.Moreover, q and G → { e } induce a smooth simplicial map G • × M → { e } • × Q ,whose geometric realization is a fattened, homotopy equivalent, version of EG × G M → Q . This map induces homomorphism between the exact lines H n − ( Q, Z ) H n − ( Q, C ) ˆ H n ( Q, Z ) H n ( Q, Z ) ⊕ Ω n cl ( Q ) H n ( Q, C ) H n − G ( M, Z ) H n − G ( M, C ) ˆ H nG ( M, Z ) H nG ( M, Z ) ⊕ Ω n cl ( M ) G H nG ( M, C ) → → → → → → → → →→ → → → As the two maps on the left and the right-hand side are isomorphisms, the same istrue in the middle by the five lemma. (cid:3) How about homotopy invariance in equivariant differential cohomology? It is thesame as in the non-equivariant case: It is not, but one can measure the deviationfrom being so. Proposition 3.9 Let i t : M → [0 , × M be the inclusion at t ∈ [0 , , let G acttrivially on the interval and let ˆ x ∈ ˆ H nG ([0 , × M, Z ) , then i ∗ ˆ x − i ∗ ˆ x = a Z [0 , × M/M R (ˆ x ) ! . Proof. This is almost verbatim [6, Prop. 3.28]. As integral cohomology is homotopyinvariant, there is y ∈ H nG ( M, Z ), such that I ( x ) = pr ∗ M y . By surjectivity of I there is a lift ˆ y ∈ ˆ H nG ( M, Z ), with I (ˆ y ) = y . Hence I (ˆ x − pr ∗ M ˆ y ) = 0 and thusˆ x = pr ∗ M ˆ y + a ( ω ) for some ω ∈ (Ω n − ([0 , × M )) G . Note that R (ˆ x ) = R (pr ∗ M ˆ y )+ dω . i ∗ ˆ x − i ∗ ˆ x = i ∗ a ( ω ) − i ∗ a ( ω ) = a Z [0 , × M/M dω ! = a Z [0 , × M/M R (ˆ x ) ! . Where the second equality is Stokes theorem, and the third equality follows as thefiber integral of pullback forms is zero: Z [0 , × M/M R (pr ∗ M ˆ y ) = Z [0 , × M/M pr ∗ M ( R (ˆ y )) = 0 . (cid:3) A version for Lie groups by Gomi. From now on, let G denote a Lie group.In this section we restate the definition for equivariant smooth Deligne cohomologygiven in [18] and show how one can define a ‘curvature’ map, which does lead to adifferential cohomology hexagon. Gomi notes the lack of his definition himself (see[18, Lemma 5.9]) and this lemma is the starting point for our alteration.Combining the ideas of Gomi with the cone construction, we reformulate thedefinition for the equivariant Deligne complex for Lie groups by Gomi: Definition 3.10 Let M be a G -manifold for a Lie group G . The equivariant Delignecomplex in degree n is defined as D Gomi ( n ) G • × M = Cone( Z ⊕ F n Ω • , ∗ → Ω • , ∗ , ( z, ω ) ω − z )[ − . Here F n Ω ∗ C is the simplicial sub-sheaf achieved from the simplicial sheaf of differentialforms on G • × M by imposing the following conditions: in simplicial level zero, i.e.,on M , forms shall have at least degree n and on any other level the differential formdegree on the G -part is at least 1, if the total form degree is less then n .In particular, if G is discrete, this is the same complex as in the last section,because on discrete groups G there are no positive degree differential forms. Definition 3.11 Let G be a Lie group acting on a smooth manifold M. The G - equivariant differential cohomology of M is defined to be the hypercohomologyˆ H nG ( M ) := H n ( G • × M, D Gomi ( n ) G p × M ) . We investigate equivariant differential cohomology for Lie groups with the samemethods as for finite groups namely with the following two short exact sequences0 → Cone( F n Ω • , ∗ ι → Ω • , ∗ )[ − a →D Gomi ( n ) I → Z → → Cone( Z − ι → Ω • , ∗ )[ − →D Gomi ( n ) R → F n Ω • , ∗ → D Gomi ( n ) G • × M → Z ⊕ F n Ω • , ∗ → Ω • , ∗ → D Gomi ( n ) G • × M [1] , which has the following interesting part in its long exact cohomology sequence H n − ( G • × M, Z ) → H n − ( G • × M, Ω • , ∗ ) → H n ( G • × M, D Gomi ( n )) ( I,R ) → H n ( G • × M, Z ) ⊕ H n ( G • × M, F n Ω • , ∗ ) ( − ι,ι ) → H n ( G • × M, Ω • , ∗ ) . As before one has ([14, Prop. 5.15 and Prop. 6.1]) H n ( G • × M, Z ) = H nG ( M, Z ) , H n ( G • × M, Ω • , ∗ ) = H nG ( M, C ) QUIVARIANT DIFFERENTIAL COHOMOLOGY 21 and Cone( Z − ι → Ω • , ∗ ) ’ C δ (cid:30)Z . Thus, the only things left to discuss, are the differential form sheaves on the leftin (8) and on the right in (9). Lemma 3.12 (Lemma 4.5 of [18]) Let G be a compact Lie group, then H n ( G • × M, Cone( F n Ω • , ∗ ι → Ω • , ∗ )[ − n − ( M ) G (cid:30) d (Ω n − ( M ) G ) and H n +1 ( G • × M, Cone( F n Ω • , ∗ ι → Ω • , ∗ )[ − . Proof. Since all sheaves of differential forms occuring are fine, we can turn to theglobal sections. Moreover notice, that if π : G • × M → { e } • × M is the map of simplicial manifolds induced by the unique map G → { e } , then thereis a quasi-isomorphism Cone( F n Ω • , ∗ ι → Ω • , ∗ ) → π ∗ Ω • , Let η ∈ (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) . The quasi-isomorphism applied in theproof of Lemma 3.12 together with the inclusion of the invariant forms into thesimplical chain complex is defined, such that a ( η ) = (0 , ( . . . , , ∂η, dη ) , ( . . . , η )).Moreover R is the projection to the tuple of forms in the middle. Thus we obtainthe assertion. (cid:3) Collecting the statements, shown above, yields to the following theorem. Theorem 3.14 Let G be a compact Lie group acting from the left on the smoothmanifold M . Then there is the following commutative diagram (10) (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) H n ( G • × M, F n Ω ∗ ) H n − G ( M, C ) ˆ H nG ( M, Z ) H nG ( M, C ) ,H n − G ( M, C / Z ) H nG ( M, Z ) → a → d + ∂ (cid:16) → → → R (cid:16) I → → − β → where the top line, the bottom line and the diagonals are exact. Remark 3.15 Parts of this diagram are due to Gomi ([18]), but, as he – partially– defined maps to different groups in the corners, he did not achieve the entirehexagon. The curvature map of Gomi can be recovered by combining the curvaturemap R , given above in the hexagon, with the map H n ( G • × M, F n Ω ∗ ) → Ω n cl ( M ) G , induced from projecting a cocycle L ni =0 Ω n − ( G i × M ) ( ω i ) ω to the invariantform part.If G is a discrete group, this reduces to the diagram of Theorem 3.6. If G isnon-discrete and acting freely on M , such that the quotient space is a manifold,one would like to compare equivariant differential cohomology with differentialcohomology of the quotient. In general, one can not expect, that ˆ H nG ( M, Z ) =ˆ H n ( M/G, Z ) as (cid:0) Ω n − ( M ) (cid:1) G is different from Ω n − ( M/G ). To see this in a veryexplicit example, take M = G , then, in degree n = 2, Ω n − ( M ) G = Ω ( G ) G = g ∨ ,but Ω n − ( M/G ) = Ω ( pt ) = 0.Moreover, one can not expect, that the map H n − G ( M, C / Z ) → ˆ H nG ( M, Z ) isinjective as in the discrete case, because H n − ( G • × M, F n Ω ∗ ) will not vanish ingeneral. To see this, take the following example for any positive dimensional Liegroup G : H ( G • × M, F Ω ∗ ) = ker (cid:0) d + ∂ : F Ω ( G × M ) → Ω ( G × M ) ⊕ Ω ( G × G × M ) (cid:1) If ω ∈ Ω ( G × M ) has form degree one on G , then ∂ω = 0 means that for any g , g ∈ G , m ∈ M and any vector field X = X + X + X M , decomposed into thetangent direction of the first copy of G , the second copy of G and M , one has0 = ( ∂ω )( g , g , m )[ X ]= ω ( g , m )[ X ] − ω ( g g , m )[ X g + g X ] + ω ( g , g m )[ X ](11)Taking X = 0 this implies, that actually ω = f ∈ C ∞ ( M, g ∨ ⊗ C ). Moreover,taking X = 0 in (11), we obtain Ad g ◦ f = L ∗ g f for any g ∈ G . Finally, since dω = 0, one has d M f = 0. Hence H ( G • × M, F Ω ∗ ) = map( π ( M ) , g ∨ ) = ∅ . Example 3.16 When constructing characteristic classes, the cohomology of theclassifying space is highly interesting. Let G be a group and EG → BG the universalbundle. Then for cohomology with any coefficient group one has H ∗ ( BG ) = H ∗ ( EG/G ) = H ∗ G ( EG ) = H ∗ G ( pt ) , where pt is the manifold consisting of a single point. Hence our question is: Whatis ˆ H ∗ G ( pt, Z )?In this case the hexagon becomes(12) Ω n − ( pt ) H n ( G • × pt, F n Ω ∗ ) H n − ( BG, C ) ˆ H nG ( pt, Z ) H n ( BG, C ) .H n − ( BG, C / Z ) H n ( BG, Z ) → a → d + ∂ → → → → R (cid:16) I → → − β → Hence ˆ H nG ( pt, Z ) = H nG ( pt, Z ) if n = 1 and, as H ( G • × pt, F Ω ∗ ) = 0, we getˆ H G ( pt ) = H G ( pt, C / Z ) = C / Z if G is connected. Maybe one wonders whether this C / Z yields some characteristic class like information. The answer is: Pulling backan element of C / Z via the classifying map of some principal G bundle, just gives aconstant function on the base space. QUIVARIANT DIFFERENTIAL COHOMOLOGY 23 In Section 2.2 we defined the Cartan complex( d + ι ) n : Ω nG ( M ) → Ω n +1 G ( M )which calculates equivariant cohomology, where Ω nG ( M ) = (cid:16) ( S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) G (cid:17) n .We want to compare the group H n ( G • × M, F n Ω ∗ ) in the upper right corner of (10)with the Cartan model. Proposition 3.17 There is a natural isomorphism H n ( G • × M, F n Ω ∗ ) → ker( d + ι ) n (cid:30) ( d + i ) n/ L k =1 (cid:0) S k ( g ∨ ) ⊗ Ω n − − k ( M ) (cid:1) G ! . Proof. In Section 2.4 we defined a quasi-isomorphism J : Ω ∗ ( G p × M ) → p M l =0 C l ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) . Let X l,k,m = ( k = 0 and m < n,C l ( G, S k ( g ∨ ) ⊗ Ω m ( M ) otherwise.The double complex ( X • , (2 ∗ + ∗ ) , d + ι + ¯ d + ¯ ι ) is a subcomplex of C • ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )): One has to check, that the inclusion commutes with boundaries. By theway X is defined, the only reason for which it is maybe not a subcomplex, couldarise from the maps which are turned into zero maps, as they map to the zero space.Thus, the problem can only come from maps lowering indices, namely ι and ¯ ι , butthese two raise the second index, hence there image does not lie in one the spaces X l, ,m , with m < n .From the definition of J one checks that J ( F n Ω ∗ ( G • × M )) ⊂ X • , ∗ , ∗ . Moreover, J is the identity on those forms, which have vanishing degree on thegroup part and H n − (cid:16) C • ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )) (cid:30) X • , ∗ , ∗ (cid:17) = Ω n − ( M ) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) by integration over the first copy of G (compare Lemma 3.4). Hence, J andthe inclusion of the Cartan complex into Getzler’s resolution induce the followingcommutative diagram with exact rows H n − G ( M, C ) (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) H n ( G • × M, F n Ω ∗ ) H nG ( M, C ) 0 H n − G ( M, C ) (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) H n ( X ∗ , (2 ∗ + ∗ ) ) H nG ( M, C ) 0 H n − G ( M, C ) (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) ker( d + ι ) n (cid:30) ∼ H nG ( M, C ) 0 →→ J ∗ → id → d + ∂ →→ J ∗ →→→ → d + ι → →→→ → id → d + ι → → → → where ker( d + ι ) n (cid:30) ∼ should denote the right-hand side of the assertion. By the fivelemma this diagram shows that there is the isomorphism as claimed. (cid:3) The discussion of this section thus manifests in the following alteration of Theorem3.14. Theorem 3.18 For any compact Lie group acting on a smooth manifold M , thereis the commutative diagram (13) (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) ker( d + ι ) n (cid:30) ( d + i ) (cid:16)L n/ k =1 S k ( g ∨ ) ⊗ Ω n − − k ( M ) (cid:17) G H n − G ( M, C ) ˆ H nG ( M, Z ) H nG ( M, C ) H n − G ( M, C / Z ) H nG ( M, Z ) → a → d + ι (cid:16) → → → R (cid:16) I → → − β → whose top line, bottom line and diagonals are exact. The new version for Lie groups. In the last section, the geometric refine-ment was done only with respect to the manifold. In this section, we will giveanother solution, where one enriches the equivariant cohomology by all closed Cartanforms.Therefore we want to use the model for equivariant cohomology defined by Getzler,which we introduced in Section 2.3. As noted there, this model is not a cochaincomplex of simplicial modules, but only a simplicial homotopy cochain complex. Toproceed as in the previous Section and do a similar cone-construction, we first haveto investigate the algebraic structure of simplicial homotopy cochain complexes inmore detail. Definition 3.19 A simplicial sheaf homotopy cochain complex of modules on asimplicial manifold M • is a triple ( F • , ∗ , f, s ), where F • , ∗ is a Z -graded simplicialsheaf of modules on M • , which is bounded from below , f is a map of simplicialsheaves, which increases the Z -grading by one and s is a simplicial zero homotopyof f , i.e., in simplicial degree p , s = ( s i ) i =0 ,...,p − , where s i : σ − i F p,q → F p − ,q +1 , i = 0 , . . . , p − s commutes with f and s p ◦ ˜ ∂ p +1 = − f : (cid:0) σ − p (cid:0) ∂ − p +1 F p,q (cid:1)(cid:1) = F p,q → F p,q +1 s i ◦ ˜ ∂ j = ( ˜ ∂ j ◦ s i − if i < j ˜ ∂ j − ◦ s i if i > j + 1 s j ◦ ˜ ∂ j = s j ◦ ˜ ∂ j +1 s ◦ ˜ ∂ = 0 . A morphism of simplicial sheaf homotopy cochain complex is a map of the simplicialsheaves, which respects the grading and commutes with both the ‘boundary map’ f and the zero homotopy. Definition + Proposition 3.20 Let w : ( F • , ∗ , f, s ) → ( e F • , ∗ , ˜ f , ˜ s ) be a morphismof simplicial homotopy cochain complex. The cone of w is the simplicial sheafhomotopy cochain complex Cone( w ) := (cid:18)(cid:16) F • ,k +1 ⊕ e F • ,k (cid:17) k ∈ N , (cid:18) − f − w f (cid:19) , (cid:18) s 00 ˜ s (cid:19)(cid:19) . This means, there is an Integer k such that each F p,q = 0 if q < k . QUIVARIANT DIFFERENTIAL COHOMOLOGY 25 Proof. The only point, which is worth to check, is the relation between the ‘boundarymap’ and the homotopy: − (cid:18) − f − w f (cid:19) = − (cid:18) f f w − w ˜ f f (cid:19) = (cid:18) − f − ˜ f (cid:19) = (cid:18) s∂ + ∂s 00 ˜ s∂ + ∂ ˜ s (cid:19) = (cid:18) s 00 ˜ s (cid:19) ∂ + ∂ (cid:18) s 00 ˜ s (cid:19) . (cid:3) We are now going to define the cohomology of a simplicial sheaf homotopycochain complex ( F • , ∗ , f, s ) using a Čech model. Let U • be a simplicial cover ofthe simplicial manifold M • . This defines for each q a resolution of the simplicialsheaf F • ,q (compare Section 2.1.2)ˇ C • ,q, ∗ ( U • , F • ,k )with Čech boundary map δ . The properties of the simplicial cover imply, that ∂ and s restrict to the Čech groups. Hence, on the total complex of this triple gradedcollection of modules, we have a boundary map M p + q + r = n ˇ C p,q,r , ∂ + s + ( − p f + ( − p + q δ ! , where ∂ and s are the alternating sums over the maps ˜ ∂ i and s i respectively.Thus we can define ˇ H ( U • , ( F • , ∗ , f, s )) to be the cohomology of this cochaincomplex. As for classical Čech cohomology, refinements of the simplicial coverinduce homomorphisms of the associated cohomology theories. Thus we defineˇ H ( M • , ( F • , ∗ , f, s )) = lim U • ˇ H ( U • , ( F • , ∗ , f, s ))to be the limit over all refinements of open covers.If the simplicial sheaf homotopy cochain complex ( F • , ∗ , f, s ) = ( F • , ∗ , d, 0) ac-tually is a cochain complex of simplicial sheaves, the total complex of the Čechresolution of both types (compare Section 2.1.2) coincides, and hence the cohomol-ogy defined here, coincides with the simplicial sheaf cohomology. Moreover, if thesheaves of ( F • , ∗ , f, s ) are fine, then the Čech direction contracts by the standardargument and the cohomology of ( F • , ∗ , f, s ) is the cohomology of the total complex( ⊕ p + q = n F p,q ( G p × M ) , ( − p f + s + ∂ ).Now, turn to our specific case, i.e., we would like to find a simplicial sheaf homotopycochain complex C • = C • , ∗ consisting of fine sheaves, such that its global sectionsare given by C • ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )).The map π : G • × M → { e } • × M, ( g , . . . , g p , m ) ( g . . . g p m ) is a morphismof simplicial manifolds. S ∗ ( g ∨ ) ⊗ Λ ∗ T ∨ M is a bundle over M , with left actionof G on M , the induced action on the cotangent bundle and coadjoint action onthe polynomial, whose global sections are S ∗ ( g ∨ ) ⊗ Ω ∗ ( M ). We can interpret thisbundle as simplicial bundle on the simplicial manifold { e } • × M , with all face anddegeneracy maps being the identity. The global sections of the pullback bundle π ∗ ( S ∗ ( g ∨ ) ⊗ Λ ∗ T ∨ M ) in simplicial level p are C p ( G, S ∗ ( g ∨ ) ⊗ Ω ∗ ( M )), thus takefor U ⊂ G p × M open C p ( U ) := Γ( U, ( π ∗ ( S ∗ ( g ∨ ) ⊗ Λ ∗ T ∨ M ) p ) . This is a sheaf of C ∞ ( G p × M )-modules, hence fine. The morphism between thesimplicial levels ˜ ∂ i : ∂ − i C p → C p +1 and ˜ σ i : σ − i C p → C p − are given by pullbackalong the simplicial bundle maps. The map d + ι : C • ,l → C • ,l +1 increases the second grading and is clearly a map ofsheaves, as booth operations are local. The maps ¯ d and ¯ ι operate between differentsimplicial levels: On global sections ¯ d is the alternating sum of the maps ˜ ∂ i , while ¯ ι ¯ ι : C k ( G, S l ( g ∨ ) ⊗ Ω m ( M )) → C k − ( G, S l +1 ( g ∨ ) ⊗ Ω m ( M ))is given by the formula ¯ ι = P k − i =0 ( − i ¯ ι i , where each ¯ ι i is the map of sheaves¯ ι i : σ − i C k → C k − (¯ ι i f )( g , . . . , g k − | X ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( g , . . . , g i , exp( tX i ) , g i +1 , . . . , g k − | X ) , with X i = Ad( g i +1 . . . g k − ) X .From the discussion of the maps d + ι , ¯ ι and ¯ d in subsection 2.3 one obtains that( C • , ∗ , d + ι, ¯ ι )is a simplicial sheaf homotopy cochain complex. C • , is the simplicial sheaf of smooth functions, in which the simplicial sheaf Z injects. This induces a map of simplicial sheaf homotopy cochain complexes( Z , , → ( C • , ∗ , d + ι, ¯ ι ) , where Z is located in degree zero. With respect to this injection, we define D C ( n ) G • × M = Cone( Z ⊕ C • , ≥ n → C • , ∗ , ( z, ω ) ω − z )[ − . Definition 3.21 Let G be a Lie group acting on a smooth manifold M. The full G -equivariant differential cohomology of M is defined to be the cohomology ofsimplicial sheaf homotopy cochain complexes D C ( n ): b H nG ( M ) := H n ( G • × M, D C ( n ) G • × M ) . Theorem 3.22 If G is a compact group, one has the following hexagon (14) Ω n − G ( M ) (cid:30) ( d + ι ) Ω nG ( M ) cl H n − G ( M, C ) b H nG ( M, Z ) H nG ( M, C ) H n − G ( M, C / Z ) H nG ( M, Z ) → a → d + ι → → → → R (cid:16) I , → → → − β → where the line along top, the one along the bottom and the diagonals are exact.Proof. This follows by the same arguments as in the last section, and the fact, thatfor compact Lie groups, the Getzler resolution contracts to the Cartan complex (seeSection 2.3). (cid:3) Example 3.23 Let M = pt be a point, then the hexagon (14) reduces in evendegrees to S n ( g ∨ )) G b H nG ( pt, Z ) H n ( BG, C ) H n − ( BG, C / Z ) H n ( BG, Z ) → a → d + ι → → → → R (cid:16) I , → → → − β → QUIVARIANT DIFFERENTIAL COHOMOLOGY 27 and in odd degrees to ( S n ( g ∨ )) G H n ( BG, C ) b H n +1 G ( pt, Z ) 0 H n ( BG, C / Z ) H n +1 ( BG, Z ) → a → d + ι → → → → R (cid:16) I , → → → − β → Hence b H nG ( pt, Z ) = ( H n ( BG, Z ) if n is even H n − ( BG, C / Z ) if n is odd . The contravariant functor b H G assigning an abelian group to the G -manifold M is not homotopy invariant, but its deviation from homotopy invariance is measuredby the homotopy formula. Lemma 3.24 Let i t : M → [0 , × M be the inclusion determined by t ∈ [0 , andlet G act trivially on the interval. Let ω ∈ ( S ∗ ( g ∨ ) ⊗ Ω ∗ ([0 , × M )) n ) G ( d M + ι ) Z [0 , × M/M ω ! = i ∗ ω − i ∗ ω + Z [0 , × M/M ( d M + ι ) ω Proof. Going to local coordinates (using a partition of unity), this is the derivativeof the integral by the lower bound, the upper bound and the interior derivative. (cid:3) Proposition 3.25 If ˆ x ∈ b H nG ([0 , × M, Z ) , then i ∗ ˆ x − i ∗ ˆ x = a Z [0 , × M/M R (ˆ x ) ! , where we have kept the notions of the previous lemma.Proof. As equivariant integral cohomology is homotopy invariant, there is a class y ∈ H n ( M, Z ), such that p ∗ M y = I (ˆ x ). As I is surjective, choose a lift ˆ y ∈ b H nG ( × M ; Z )with I (ˆ y ) = y . Thus I ( p ∗ M ˆ y − ˆ x ) = 0 and hence ˆ x = p ∗ M ˆ y + a ( ω ) for some ω ∈ ( S ∗ ( g ∨ ) ⊗ Ω ∗ ([0 , × M )) n − ) G . Therefore ( d + ι ) ω = R ( a ( ω )) = R (ˆ x ) − R ( p ∗ M ˆ y ).We can write ω = dt ∧ α + β , where dt corresponds to the interval and α, β areforms on p ∗ M T M On the one hand i ∗ ˆ x − i ∗ ˆ x = a ( i ∗ ω − i ∗ ω ) = a ( i ∗ β − i ∗ β ) . On the other hand a Z [0 , × M/M R (ˆ x ) ! = a Z [0 , × M/M R (ˆ x ) − p ∗ M R (ˆ y ) ! , and, as fiber integrals over basic forms vanish,= a Z [0 , × M/M ( d + ι ) ω ! = a Z [0 , × M/M ( d M + ι ) ω ! + a Z [0 , × M/M d [0 , ω ! = a Z [0 , × M/M ( d M + ι ) dt ∧ α ! + a Z [0 , × M/M d [0 , β ! = a ( i ∗ − i ∗ ) dt ∧ α + ( d M + ι ) Z [0 , × M/M dt ∧ α !! + a (( i ∗ − i ∗ ) β )= a ( i ∗ β − i ∗ β ) . In the last step we use that a vanishes on exact forms. (cid:3) To compare our definition with the construction in the last section, we define asubsheaf F n C • , ∗ ⊂ C • , ∗ . In the bundle S ∗ ( g ∨ ) ⊗ Λ ∗ ( T ∨ M ), we have the subbundle S ≥ ( g ∨ ) ⊗ Λ The image of F n Ω • , ∗ under the Getzler map J : Ω • , ∗ → C • , ∗ , definedin section 2.4, lies in F n C • , ∗ .Proof. Let U ⊂ G p × M an open set and ω ∈ F n Ω p,k ( U ). If k ≥ n there is nothingto show. Let k < n . The projection of the image of J ( ω ) to C • , ∗ ( U ) / F n C • , ∗ ( U ) isthe part of J ( ω ) whose polynomial degree is zero. This is zero, since the form degreeof ω on the G part is positive (by the condition k < n ) and hence ω is mapped tozero in the quotient and hence to a positive degree polynomial. (cid:3) Let D C (1 , n ) = Cone( Z ⊕ F n C • , ∗ → C • , ∗ , ( z, ω, η ) ω + η − z )[ − Lemma 3.27 The map of chain complexes of simplicial sheaves J ∗ : D Gomi ( n ) G • × M → D C (1 , n ) G • × M induces an isomorphism ˆ H ∗ G ( M, Z ) → H ∗ ( G • × M, D C (1 , n )) Proof. The same arguments as given above show, that H ∗ ( G • × M, D C (1 , n )) sitsin the same hexagon (13), as ˆ H ∗ G ( M, Z ) and the induced maps on all corners is theidentity. (cid:3) We have an inclusion D C ( n ) → D C (1 , n ), which, composted with the isomorphismof Lemma 3.27, induces a map f : b H ∗ G ( M, Z ) → ˆ H ∗ G ( M, Z ) . Theorem 3.28 f is an isomorphism in degree 0,1 and 2 and surjective in higherdegrees. QUIVARIANT DIFFERENTIAL COHOMOLOGY 29 Proof. This again follows from the hexagons, which coincide in degree 0 , , 2. Inhigher degrees, the sequence along the bottom is the same and along the top onehas surjections. (cid:3) Remark 3.29 Michael Luis Ortiz discusses an idea of a definition of equivariantdifferential cohomology in [28, p.7-9]. He gives a recipe what to do for general Liegroups, but does not make things precise. In particular he talks about differentialforms on M × G EG . As you will have noted, giving them a precise meaning, inwhich one can compare them with integral cohomology and the Cartan model isone of the major lines in this thesis and found its final answer in this section.On the other hand, there is a definition of Deligne cohomology for orbifolds byErnesto Lupercio and Bernardo Uribe in [24]. This includes the ‘action orbifold’of G on M with objects M and morphisms G × M , whose nerve is our simplicialmanifold G • × M . Translating his definition to our language, one gets the complexCone (cid:16) Z ⊕ Γ (cid:0) · , ( ∂ ∗ ) • Λ ≥ n T ∨ M (cid:1) ∗ → Γ ( · , ( ∂ ∗ ) • Λ ∗ T ∨ M ) , ( z, ω ) ω − z (cid:17) [ − , of cochain complexes of simplicial sheaves on G • × M , where Γ( · , E ) denotes thesheaf of local sections of the bundle E . This yields (for G compact) the hexagon: (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) Ω n cl ( M, C ) G (Ω n − ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) ˆ H nG ( M, Z ) (Ω n cl ( M )) G (cid:30) d (cid:0) Ω n − ( M ) G (cid:1) H n − G ( M, Cone( Z → Ω ∗ ( · ) G )) H nG ( M, Z ) → a → d → → → → R (cid:16) I , → → → − β → In the case of finite groups, one has H ∗ G ( M, C ) = Ω n cl ( M ) G /d Ω n − ( M ) G , thus thisis the same as we had before. In the case of positive dimensional Lie groups it iseven less satisfactory then the definition of Gomi, as there is not even equivariantcomplex cohomology at the left and the right end.4. Equivariant differential characteristic classes Definitions. Let us restrict to compact groups G acting on the manifold andon vector bundles. As rank n vector bundles admit a hermitian metric, they are inone to one correspondence with principal U ( n )-bundles. Thus any characteristicform for vector bundles corresponds to an invariant polynomial P ∈ I ∗ ( U ( n )) (see,e.g. [21, Corollary 5.13].Let E → M be a G -equivariant vector bundle. Recall that a connection is map ∇ : Ω ( M, E ) → Ω ( M, E ) , which satisfies a Leibniz rule ∇ ( f ϕ ) = df ∧ ϕ + f ∇ ϕ for f ∈ Ω ( M, C ) , ϕ ∈ Ω ( M, E ) . Further, a connection ∇ extends uniquely to a C -linear map ∇ : Ω ∗ ( M, E ) → Ω ∗ +1 ( M, E ) , called exterior connection, by imposing the sign respecting Leibniz rule ∇ ( ω ∧ ϕ ) = dω ∧ ϕ + ( − k ω ∧ ∇ ϕ for ω ∈ Ω k ( M, C ) , ϕ ∈ Ω ∗ ( M, E ) . One observes, that ∇ ◦ ∇ : Ω ( M, E ) → Ω ( M, E ) is C ∞ -linear and hence givenby left multiplication with an endomorphism valued 2-form, which is known as thecurvature operator R ∇ ∈ Ω ( M, End E ). If the connection is G -invariant, then thereis another associated map. Definition 4.1 (Def. 2.23. of [6]) Let ∇ be a G -invariant connection on the G -vectorbundle E . The moment map µ ∇ ∈ Hom( g , ω ( M, End( E ))) G is defined by µ ∇ ( X ) ∧ ϕ := ∇ X ]M ϕ + L E X ϕ, ϕ ∈ ω ( M, E ) . Here L E X denotes the derivative L E X ϕ = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( tX ) ∗ ϕ. From any invariant polynomial we obtain a equivariant differential forms of the G -invariant connection by ω ( ∇ ) = P ( R ∇ + µ ∇ ) ∈ Ω G ( M ) . Moreover, if ω is integral, i.e. has integral periods, then there is an integral equi-variant characteristic class c ω coinciding with the class of ω in complex cohomology. Definition 4.2 A differential refinement of ω associates to every G -equivariantvector bundle with connection ( E, ∇ ) on M a class ˆ ω ( ∇ ) ∈ b H G ( M ; Z ) such that R (ˆ ω ( ∇ )) = ω ( ∇ ) , I ( ω ( ∇ )) = c ω ( E )and for every map f : M → M , we have f ∗ ˆ ω ( ∇ ) = ˆ ω ( f ∗ ∇ ).As the intersection of the kernelsker( R ) ∩ ker( I ) = H n − G ( M, C ) (cid:30) H n − G ( M, Z )is in general non-trivial, the differentially refined class ˆ ω ( ∇ ) can contain finerinformation than the pair ( ω ( ∇ ) , c ω ( E )). Thus it is a priori not clear that fora given equivariant characteristic form, there is only one equivariant differentialcharacteristic class. Theorem 4.3 An integral equivariant characteristic form admits a unique equivari-ant differential extension. The line of arguments proof this assertion is (almost) the following: A simplicialmanifold model of the universal U ( n )-bundle is given by (compare [14, Section 5])the simplicial principal U ( n )-bundle γ : N U ( n ) • → N U ( n ) • , with N U ( n ) p = U ( n ) p +1 ,∂ i removes the i -th coefficient, and σ i doubles the i -th coefficient. N U ( n ) = U ( n ) • × pt and γ ( g , . . . , g p ) = ( g g − , . . . , g p − g − p ). As b H nU ( n ) ( pt, Z ) = H n ( BU ( n ) , Z ), wewould like to define a map of simplicial manifolds G • × M → N U ( n ) classifying ourbundle and pull back the universal class together with a corresponding connection.Now we can compare this connection with the one defined on our bundle and changethe differential characteristic class according to this. Lemma 4.4 Let ∇ and ∇ are two connections on the same bundle, then ˆ ω ( ∇ ) − ˆ ω ( ∇ ) = a (˜ ω ( ∇ , ∇ )) QUIVARIANT DIFFERENTIAL COHOMOLOGY 31 Proof. Let ∇ t denote the convex combination of ∇ and ∇ . Then by Proposition3.25 ˆ ω ( ∇ ) − ˆ ω ( ∇ ) = i ∗ ˆ ω ( ∇ t ) − i ∗ ˆ ω ( ∇ t )= a Z [0 , × M/M R (ˆ ω ( ∇ t )) ! = a Z [0 , × M/M ω ( ∇ t ) ! = a (˜ ω ( ∇ , ∇ )) . (cid:3) This Lemma implies, in particular, that we are done, if we have defined therefined for hermitian bundles with hermitian connection, since any connection canby symmetrized (compare [6, Section 2.5]).To construct the classifying map we will need an intermediate bundle, for whichone can easily write down pullback maps to the given bundle and to the universalbundle. Therefor we need to recall the following construction from [21, Section 4].Let U = { U α | α ∈ A } be an open cover of some G -manifold M . This induces asimplicial cover of G • × M : Define the simplicial index set A ( p ) = A p +1 with faceand degeneracy maps given by removing respective doubling of the i -th element.Then define the simplicial cover U ( p ) = { U ( p ) α } α ∈ A ( p ) inductively by U ( p ) α = p \ i =0 ∂ − i (cid:16) U ( p − ∂ i ( α ) (cid:17) , where U (0) α = U α for any α ∈ A (0) = A .From this simplicial cover one obtains the the simplicial manifold ( G • × M ) U as(( G • × M ) U ) p := a ( α ,...,α p ) U ( p ) α ∩ · · · ∩ U ( p ) α p , where the disjoint union is taken over all ( p + 1)-tuples ( α , . . . , α p ) ∈ ( A ( p ) ) p +1 with U ( p ) α ∩ · · · ∩ U ( p ) α p = ∅ . The face and degeneracy maps are given on the indexsets ( A ( p ) ) p +1 by removing, respective doubling of the i -th index and on the opensets by the corresponding inclusions composed with the i -th face and degeneracymap of G • × M .Let π : E → M be a G -equivariant hermitian vector bundle with hermitianconnection ∇ and B be the associated principal U ( n )-bundle furnished with theassociated principal connection ϑ . From an open cover U of M we obtain the cover π − U of B and thus the construction above yields a simplicial bundle π : ( G • × B ) π − U → ( G • × M ) U . and the commutative diagram( G • × B ) π − U G • × B ( G • × M ) U G • × M → π U → →→ induced by the inclusions of the covering sets is a pullback, since the cover we takeon G • × B is induced by π and U • .Suppose the cover U = { U α } α ∈ A of M trivializes B with trivialization ϕ α : V α = π − ( U α ) → U α × U ( n ) and transition functions g αβ : U α ∩ U β → U ( n ). Then there is an induced map ψ : ( G • × B ) π − U → N U ( n ) , which is given on the intersection of p + 1 covering sets of G p × BV = p \ j =0 V ( p ) α j ,...,α jp by ( g , . . . , g p , x ) ( ϕ α ( g . . . g p x ) , ϕ α ( g . . . g p x ) , . . . , ϕ α pp ( x )) ∈ U ( n ) p +1 , where, on the right-hand side, the maps ϕ α are understood to be composed withthe projection to U ( n ).Next, we want to define ψ : ( G • × M ) U → N U ( n ), such that ψ covers ψ . Thereforewe need some additional transition functions of the bundle. Define h αβ : G × M ⊃ ∂ − U α ∩ ∂ − U β → U ( n )( g, m ) ( π ◦ ϕ α ( gx ))( π ◦ ϕ β ( x )) − , for any x ∈ π − ( m ). Define ψ on U = q \ j =0 U ( p ) α j ,...,α jp by(15) ( g , . . . , g p , m ) ( h α α ( g , g . . . g p m ) ,h α α ( g , g . . . g p m ) , . . . , h α p − p − α pp ( g p , m ) , ∗ ) . These maps combine to the following commutative diagram of simplicial manifolds: G • × B ( G • × B ) π − U N ¯ U ( n ) G • × M ( G • × M ) U N U ( n ) . → π → ¯ i → ¯ ϕ → →→ i → ϕ Proposition 4.5 The map i induces an isomorphism i ∗ : b H nG ( M, Z ) → H n (( G • × M ) U , i ∗ D C ( n )) and isomorphisms between all corners of the hexagons (14) with the correspondingcorners of H n (( G • × M ) U , i ∗ Cone( C • , ≥ n → C • , ∗ )[ − H n (( G • × M ) U , i ∗ C • , ≥ n ) H n − G ( M, C ) H n (( G • × M ) U , i ∗ D C ( n )) H nG ( M, C ) .H n − G ( M, C / Z ) H nG ( M, Z ) → a → d + ι → → → → R (cid:16) I , → → → − β → Proof. Recall that k i k : k ( G • × M ) U k → k G • × M k is a homotopy equivalence. Theshort exact sequence of simplicial sheaves0 → Cone( Z → C • , ∗ ) → D C ( n ) G • × M → C • , ≥ n → i induce the following diagram with exact rows H n − G ( M, C / Z ) b H nG ( M, Z ) Ω nG ( M ) cl H n − G ( M, C / Z )0 H n − G ( M, C / Z ) H n (( G • × M ) U , i ∗ D C ( n )) H n (( G • × M ) U , i ∗ C ∗ , ≥ n ) H n − G ( M, C / Z ) . →→ = →→ = →→ i ∗ →→ i ∗ → = → → → → Thus, by the five lemma, it is sufficient to show, that i ∗ : Ω nG ( M ) cl → H n (( N G M ) U , i ∗ C • , ≥ n ) QUIVARIANT DIFFERENTIAL COHOMOLOGY 33 is an isomorphism. Observe that H n (( N G M ) U , i ∗ C • , ≥ n ) = ker (cid:16) d + ι : C ,n (cid:16)a U α (cid:17) → C ,n +1 (cid:16)a U α (cid:17)(cid:17) ∩ ker ∂ : C ,n (cid:16)a U α (cid:17) → C ,n a α ,α ,β ,β U (1) α α ∩ U (1) β β . Let ( ω α ) ∈ C ,n ( ‘ U α ). The definition of the map ∂ m ∈ U β U (1) α α ∩ U (1) β β ( g, m ) gm ∈ U α → → ∂ → → ∂ implies that ∂ ( ω α ) = 0 is equivalent to ∂ ∗ ω β | U (1) αβ = ∂ ∗ ω α | U (1) αβ . Moreover, since e × ( U α ∩ U β ) ⊂ ∂ ∗ U α ∩ ∂ ∗ U β = U (1) αβ , this equation implies that ( ω α )is the restriction of a global section ω ∈ C ,n ( M ), which is by the same equation G -invariant. Hence ω ∈ ker( d + ι ) = Ω nG ( M ) cl . This proves the first claim.The claim about the hexagon follows by the same argument, because the ‘deRham’ sequence along the top is exact. (cid:3) One defines (compare [14, p.94]) a connection ¯ ϑ on N U ( n ) → N U ( n ): Let ϑ ∈ Ω ( K, k ) denote the unique connection of the trivial bundle K → pt, i.e., ϑ ( k ) = L k − : T k K → T e K = k . Let π i : ∆ p × K p +1 → K denote the projection to the i -th coefficient, i = 0 , . . . , p and ϑ i = π ∗ i ϑ . Then wedefine ¯ ϑ on ∆ p × ( N K ) p by ¯ ϑ = X i t i ϑ i , where ( t , . . . , t p ) are barycentric coordinates on the simplex. ¯ ϑ | ∆ p × ( NK ) p is aconnection on ∆ p × ( N K ) p , as it is a convex combination of connections. It canbe seen easily from the definition, that ¯ ϑ is a simplicial Dupont 1-form. For moredetails see also [21].Let P ∈ I ∗ ( U ( n )) denote the polynomial and c P ∈ H n ( BU ( n ) , Z ) = b H nU ( n ) ( pt, Z )the universal characteristic class corresponding to the integral characteristic form ω P . Definition + Proposition 4.6 The differential refinement is given by the formula ˆ ω ( ∇ ) = ( i ∗ ) − ( ϕ ∗ c P + a ( f ω P ( i ∗ ϑ, ¯ ϕ ∗ ϑ ))) . This definition is independent of the chosen cover and trivializations and defines thedifferential refinement of the integral characteristic form ω P .Proof. We will prove the independence of the cover in three steps: Step 1: Let U = { U β } be a refinement of the cover U , i.e., for any β , there is some α ( β ), such that U β ⊂ U α ( β ) ; let ϕ β = ϕ α ( β ) | U β . The inclusion of the refinement yields a commutative diagram G • × B ( G • × B ) π − U N ¯ U ( n ) G • × B ( G • × B ) π − U N ¯ U ( n ) G • × M ( G • × M ) U N U ( n ) G • × M ( G • × M ) U N U ( n ) → → ¯ i → ¯ ϕ → → →→ ¯ i → ¯ ϕ →→ i → ϕ → → → i → ϕ → from which the independence of the cover follows, because the direct pullback issame as the one factorized over the coarser cover. Step 2: Take one cover U = { U α } , with two different families of trivializationmaps ϕ α , ϕ α : π − U α → U α × G .Then there is a family of maps ψ α : U α → G , such that ψ α ( π ( b )) · ϕ α ( b ) = ϕ α ( b )for any b ∈ π − U α and any α .The difference of the two definitions is ϕ ∗ c P + a ( f ω P ( i ∗ ϑ, ¯ ϕ ∗ ϑ )) − ϕ c P − a ( f ω P ( i ∗ ϑ, ¯ ϕ ϑ ))= ϕ ∗ c P − ϕ c P − a ( f ω P ( ¯ ϕ ∗ ϑ , ¯ ϕ ϑ ))First assume each U α is contractible, then there is a homotopy e ψ α : [0 , × U α → G such that i ∗ e ψ α = ψ α and i ∗ e ψ α maps any point to e ∈ G . These homotopies inducea homotopy e ϕ : [0 , × ( G • × B ) π − U → N U ( n )between ˜ ϕ = ϕ and ˜ ϕ = ϕ and one can calculate ϕ ∗ c P − ϕ c P = i ∗ ˜ ϕ ∗ c P − i ∗ ˜ ϕ ∗ c P = a Z [0 , R ( ˜ ϕ ∗ c P ) ! = a Z [0 , ˜ ϕ ∗ R ( c P ) ! = a Z [0 , ˜ ϕ ∗ Z ∆ P ( ϑ ) ! = a Z [0 , Z ∆ P ( ˜ ϕ ∗ ϑ ) ! = a ( ˜ ω P ( ¯ ϕ ∗ ϑ , ¯ ϕ ϑ )) . In the last step, we use that e ω P is independent of the path between the connections.The case of non contractible U α follows by Step 1. Step 3: Let ( U , ( ϕ α )) , ( U , ( ϕ β )) be two different covers with trivializations. Let˜ U = { U α ∩ U β | α, β } be the common refinement on which there are two differentfamilies of trivializations are introduced by ϕ and ϕ . Now the statement followsfrom the previous steps.Next, we check the properties of the differential refinement: I (ˆ ω ( ∇ )) = I (( i ∗ ) − ( k ϕ k ∗ c P )) = c ω ( B ) QUIVARIANT DIFFERENTIAL COHOMOLOGY 35 and R (ˆ ω ( ∇ )) = R (( i ∗ ) − ( k ϕ k ∗ c P ) + a (˜ ω ( i ∗ ϑ, ¯ ϕ ∗ ϑ ))= R (( i ∗ ) − ( k ϕ k ∗ c P )) + ( d + ι )˜ ω ( i ∗ ϑ, ¯ ϕ ∗ ϑ )= ( i ∗ ) − ( ω ( ¯ ϕ ∗ ϑ ) + ω ( i ∗ ϑ ) − ω ( ¯ ϕ ∗ ϑ ))= ω ( ∇ ) . Let ( F, f ) : ( B, M ) → ( B , M ) be a pullback. As a trivialization of ( B , M )induces a trivialization of ( B, M ), one has a commutative diagram G • × B ( G • × B ) π − f − U N ¯ U ( n ) G • × B ( G • × B ) π − U N ¯ U ( n ) G • × M ( G • × M ) f − U N U ( n ) G • × M ( G • × M ) U N U ( n ) → → → →→ → →→ → → → → → → → → →→ which clearly implies the pullback property.The refinement is unique, since we used for our definition only properties thedifferential refinement necessarily has, namely the pullback-property and Lemma4.4. (cid:3) Multiplicative structures.Definition 4.7 (compare [6, Definition 3.94]) Let G be a compact Lie group. Aproduct on equivariant Deligne cohomology is the datum of a graded commutativering structure (denoted by ∪ ) on b H ∗ G ( M, Z ) for every G -manifold M such that(1) f ∗ : b H ∗ G ( M, Z ) → b H ∗ G ( M , Z )is a homomorphism of rings for every smoothmap f : M → M ,(2) R : b H ∗ G ( M, Z ) → Ω ∗ G ( M ) cl is multiplicative for all M ,(3) I : b H ∗ G ( M, Z ) → H ∗ G ( M, Z ) is multiplicative for all M , and(4) a ( α ) ∪ x = a ( α ∧ R ( x )) for all α ∈ ω ∗ G ( M ; C ) / im( d + ι ) and x ∈ b H ∗ G ( M, Z ) Proposition 4.8 There exists a unique product on equivariant Deligne cohomology.Proof. Uniqueness follows by (almost) verbatim the same arguments as given by [6,p. 60]: The difference between two products B = ∪ − ∪ : b H pG ( M, Z ) ⊗ b H qG ( M, Z ) → b H p + qG ( M, Z )factorizes over a bilinear map˜ B : H pG ( M, Z ) ⊗ H qG ( M, Z ) → H p + qG ( M, C / Z ) , by the hexagon (14), since R ◦ B = 0 and B ◦ ( a × id) = 0. The bilinear map ˜ B corresponds to a map of Eilenberg-Maclane spaces K ( Z, p ) ∧ K ( Z, q ) → K ( C/Z, p + q − , which is homotop to a constant map, as the smash product on the left hand side is p + q − Existence: We will leave this to the reader. The idea is to copy the argumentsof [6, Section 3.4], but replace the de Rham d in the definition of the map on thelevel of chain complexes [6, Equation (29)] by the boundary map ¯ d + ¯ ι + d + ι ofGetzler. (cid:3) Recall that the total equivariant differential Chern class is the sum of the equi-variant differential Chern classesˆ c ( ∇ ) = 1 + ˆ c ( ∇ ) + ˆ c ( ∇ ) + · · · ∈ M n even b H nG ( M, Z ) . Proposition 4.9 The total equivariant differential Chern class satisfies a Whitneysum formula, i.e., given two G -equivariant vector bundles ( E, ∇ ) , ( E , ∇ ) withequivariant connection over the G -manifold M and let ∇ ⊕ ∇ be the Whitney sumconnection on E ⊕ E , then ˆ c ( ∇ ⊕ ∇ ) = ˆ c ( ∇ ) ∪ ˆ c ( ∇ ) . Proof. The proof consists of two steps: First we will prove the formula for theclassifying space and afterwards, we will show that the difference terms fit.Since the U ( n )-equivariant differential cohomology of a point equals in evendimension the U ( n )-equivariant integral cohomology of a point, the formular followsfrom the non-differential Whitney sum formula and the compatibility of the cupproducts.Thus by construction of the equivariant differential characteristic classes, we doonly have to check, that the difference terms fit, i.e., the classifying maps of E, E and E ⊕ E induce connections ∇ , ∇ and ∇ ⊕ ∇ , for whichˆ c ( ∇ ⊕ ∇ ) = ˆ c ( ∇ ) ∪ ˆ c ( ∇ )holds by the pullback property of ˆ c and the first step for the universal bundles.Denote the characteristic form of c by ω and the transgression form by e ω . Nowcalculate by applying the properties of the cup productˆ c ( ∇ ) ∪ ˆ c ( ∇ )= (ˆ c ( ∇ ) + a ( e ω ( ∇ , ∇ ))) ∪ (ˆ c ( ∇ ) + a ( e ω ( ∇ , ∇ )))= ˆ c ( ∇ ⊕ ∇ ) + a ( e ω ( ∇ , ∇ )) ∪ ˆ c ( ∇ ))+ ˆ c ( ∇ ) ∪ a ( e ω ( ∇ , ∇ )) + a ( e ω ( ∇ , ∇ )) ∪ a ( e ω ( ∇ , ∇ ))= ˆ c ( ∇ ⊕ ∇ ) + a ( e ω ( ∇ , ∇ ) ∧ R (ˆ c ( ∇ )))+ a ( R (ˆ c ( ∇ )) ∧ ( e ω ( ∇ , ∇ )) + a ( e ω ( ∇ , ∇ ) ∧ R ◦ a ( e ω ( ∇ , ∇ )))= ˆ c ( ∇ ⊕ ∇ ) + a ( e ω ( ∇ , ∇ ) ∧ ω ( ∇ ))+ a ( ω ( ∇ ) ∧ ( e ω ( ∇ , ∇ )) + a ( e ω ( ∇ , ∇ ) ∧ ( ω ( ∇ ) − ω ( ∇ )))= ˆ c ( ∇ ⊕ ∇ ) + a ( e ω ( ∇ , ∇ ) ∧ ω ( ∇ )) + a ( ω ( ∇ ) ∧ ( e ω ( ∇ , ∇ ))= ˆ c ( ∇ ⊕ ∇ ) + a ( e ω ( ∇ ⊕ ∇ , ∇ ⊕ ∇ ) + a (( e ω ( ∇ ⊕ ∇ , ∇ ⊕ ∇ ))= ˆ c ( ∇ ⊕ ∇ ) . (cid:3) Examples for equivariant differential cohomology Free actions. Let the Lie group G act freely on the manifold M from the left.Does equivariant differential cohomology groups make a difference between the G manifolds M and G × M/G ? As equivariant cohomology does not make one, thequestion reduces to differential forms.To discuss this, we collect the following statements: Definition 5.1 (Def. 13.5. of [30]) The action is proper , if the action map G × M → M × M, ( g, m ) ( gm, m )is proper, i.e., the pre-image of any compact set is compact. QUIVARIANT DIFFERENTIAL COHOMOLOGY 37 Theorem 5.2 (Th. 13.8. of [30]) Suppose G acts properly on M . Then each orbit G · m is an embedded closed submanifold of M , with T m ( G · m ) = { X ]M ( m ) | X ∈ g } = g ]m . Theorem 5.3 (Th. 13.10. of [30]) Suppose that G acts properly and freely on M ,then the orbit space M/G is a manifold and the quotient map π : M → M/G is asubmersion. Suppose the action is free and proper, thus M/G is a manifold. The quotientmap always induce injections q ∗ : Ω n ( M/G ) → Ω n ( M ) G and pr ∗ : Ω n ( M/G ) → Ω n ( G × M/G ) G . These lead to two resolutions of Ω ∗ ( M/G ): The first one is given as the doublecomplex... ... ... ...Ω n ( M/G ) Ω n ( M ) G (cid:0) g ∨ ⊗ Ω n − ( M ) (cid:1) G (cid:0) S ( g ∨ ) ⊗ Ω n − ( M ) (cid:1) G . . . Ω n +1 ( M/G ) Ω n +1 ( M ) G ( g ∨ ⊗ Ω n ( M )) G (cid:0) S ( g ∨ ) ⊗ Ω n − ( M ) (cid:1) G . . . ... ... ... ... → → → →→ q ∗ → d → ι → d → ι → d →→ d → q ∗ → d → ι → d → ι → d →→ d whose total complex is the Cartan complex Ω ∗ G ( M ), while the total complex of thesecond resolution is Ω ∗ G ( G × M/G ). The question now is: Are these two complexesequivalent on the level of cycles? This is clearly true for zero forms as the two maps C ∞ ( M ) G q ∗ ←− C ∞ ( M/G ) pr ∗ −→ C ∞ ( G × M/G ) G are isomorphisms. For higher degrees let h be a G -invariant Riemannian metric on M . Then the tangent bundle T M = g ] ⊕ (cid:0) g ] (cid:1) ⊥ splits with respect to h . Moreover dq m : (cid:0) g ]m (cid:1) ⊥ → T q ( m ) ( M/G ) is an isomorphismfor any m ∈ M . Thus we have the following lemma, what shows the equivalence indegree one Lemma 5.4 Let G act properly and freely on M , then → Ω ( M/G ) q ∗ → Ω ( M ) G ι → (cid:0) g ∨ ⊗ Ω ( M ) (cid:1) G → splits.Proof. Restriction to (cid:0) g ] (cid:1) ⊥ ⊂ T M defines a map Ω ( M ) G → Ω ( M/G ) which isleft inverse of q ∗ . Thus it is a split. (cid:3) For the higher degrees, recall the following relation between exterior algebras. Proposition 5.5 (Prop. 10 of [3, Ch.III, §7.7]) Let V, W be vector spaces. Thenthere is a natural isomorphism of algebras Λ ∗ ( V ) ⊗ Λ ∗ ( W ) → Λ ∗ ( V ⊕ W ) , from the graded tensor product of the exterior algebras to the exterior algebra of thedirect sum. We will now restrict to the case, where the adjoint action of G on g is trivial.This includes, in particular, the case of abelian Lie groups.An element of Ω ∗ G ( G × ( M/G )) is an invariant section of S ∗ ( g ∨ ) ⊗ Λ ∗ (cid:16) T ∨ (cid:16) G × M (cid:30) G (cid:17)(cid:17) → G × M (cid:30) G, what by the splitting of the cotangent space and Proposition 5.5 is a G -invariantsection of S ∗ ( g ∨ ) ⊗ Λ ∗ (pr ∗ T ∨ G ) ⊗ Λ ∗ (cid:16) pr ∗ T ∨ M (cid:30) G (cid:17) → G × M (cid:30) G. This is the same as a section of S ∗ ( g ∨ ) ⊗ Λ ∗ ( g ∨ ) ⊗ Λ ∗ (cid:16) pr ∗ T ∨ M (cid:30) G (cid:17) → M (cid:30) G, since the action of G on S ∗ ( g ∨ ) is trivial. Pulling this section back to M along thequotient map yields a G -invariant section of S ∗ ( g ∨ ) ⊗ Λ ∗ ( g ∨ ) ⊗ q ∗ Λ ∗ (cid:16) T ∨ M (cid:30) G (cid:17) → M. Composition with id ⊗ ] ⊗ (cid:16) dq (cid:12)(cid:12)(cid:12) ( g ] ) ⊥ (cid:17) − turns this section to an G -invariant sectionof S ∗ ( g ∨ ) ⊗ Λ ∗ T ∨ M → M and thus an element of Ω ∗ G ( M ), because X ]gm = g ( g − Xg ) ]m = g · X ]m . As any ofthese steps may be gone in the opposite direction, we have an isomorphism betweenΩ ∗ G ( M ) and Ω ∗ G ( G × ( M/G )).Thus for free proper actions of abelian groups, there is no difference between M and G × ( M/G ) in equivariant differential cohomology. The most easy example for afree proper action of a non-abelian Lie group on a manifold is the left multiplicationof S ⊂ H on S ⊂ H . We will leave this discussion to future research.Let E → M be a G -equivariant vector bundle with free and proper G -action onthe base and the total space. Given a connection on ∇ on E , there is the question,whether this connection is a pullback from the quotient bundle E E := E (cid:30) GM M := M (cid:30) G. → ¯ q → →→ q Clearly, if the connection is a pullback, then every equivariant differential character-istic class ˆ c ( ∇ ) must lie in the image of˜ q ∗ : ˆ H ( M , Z ) → b H G ( M, Z ) , where ˜ q is the projection of the simplicial manifolds G • × M → { e } • × ¯ M . Inparticular, the connection must be G -invariant and the moment map must vanish(compare also [6, Section 2.2]).Now turn the question the other way around: Assume, the there is some collectionof equivariant differential characteristic classes for a connection on E → M , whichall lie in the image of ˜ q ∗ . Does this imply that connection descends to the quotientbundle?We want to remark to following observations according to an answer of thisquestion: Let ∇ be a connection on the equivariant complex vector bundle E → M of rank n . Then total equivariant Chern form is given by R (ˆ c ( ∇ ) = det (cid:18) πi R ∇ + µ ∇ (cid:19) . QUIVARIANT DIFFERENTIAL COHOMOLOGY 39 For any X ∈ g , this form induces a polynomial P X ( t ) = det (cid:18) πi R ∇ + µ ∇ ( tX ) (cid:19) = det (cid:18) πi R ∇ + tµ ∇ ( X ) (cid:19) in t . If the total equivariant Chern form lies in the image of the quotient map, thenthe degree of polynomial in t is zero.In the case of R ∇ = 0, t n P X ( t ) is exactly the characteristic polynomial of µ ∇ ( X )and hence all eigenvalues of µ ∇ ( X ) are zero, if total equivariant Chern form lies inthe image of the quotient map. In general, this does not imply that µ ∇ ( X ) is zero,but if there is a metric on E , we can say more.Let h be a hermitian metric on E and ∇ be compatible with h . Then E is incorrespondence to a principal U ( n )-bundle and, as the Lie algebra u ( n ) consists ofanti-hermitian matrices, the image of µ ∇ ( X ) at any point of M is anti-hermitian.The Jordan normal form of an anti-hermitian matrix is diagonal, because theconjugate of an anti-hermitian matrix by an unitary one is anti-hermitian,( U ∗ AU ) ∗ = U ∗ A ∗ U = − U ∗ AU and hence all 1’s’ in the first the upper diagonal must vanish. Since an invariantconnection descends, if and only if the moment map vanishes (compare [6, Problem2.24]), we have proven the following Proposition. Proposition 5.6 Let ( E, h ) → M be a G -equivariant hermitian vector bundle, suchthat the G -action is free and proper, and let ∇ be a G -invariant hermitian connectionon E , such that the curvature R ∇ vanishes, then ∇ descends to a connection on E (cid:30) G → M (cid:30) G, if and only if the total Chern form vanishes. Conjugation action on S . The manifold S ⊂ R has a group structure.Recall that one defines on the vector space R a real (non-commutative) divisionalgebra, the quaternions, with three imaginary units i, j, k squaring to − ij = − ji = k . Now the space of unit quaternions is S and has an inducedmultiplication. On the other hand, there is another description of the 3-sphere bythe special unitary group of complex 2 × SU (2) = (cid:26)(cid:18) a − ¯ bb ¯ a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a, b ∈ C , | a | + | b | = 1 (cid:27) . The map (cid:18) a − ¯ bb ¯ a (cid:19) a + jb ∈ S ⊂ H defines a group isomorphism between the two descriptions.We want to investigate the conjugation action of S on itself. Therefore note thefollowing well-known fact (for a proof see e.g. [20, Lemma 4.44]): Lemma 5.7 Half the trace or the real part of the quaternion is an invariantsurjective mapping 12 tr : S → [ − , , which induces an isomorphism of the quotient S (cid:30) SU (2) → [ − , . The isotropygroup of any point besides and − is isomorphic to S . Another helpful picture of S is obtained from stereographic projection withprojection point − 1. In formulas this is expressed as H ⊃ S x = x + ix + jx + kx 11 + x ( x , x , x ) ∈ R ∪ {∞} , where 1 ∈ H is mapped to the 0 ∈ R and − ∞ . Taking subsets S ⊂ H of fixedreal value x , then these are mapped to 2-sphere of Radius q − x x . The conjugationaction acts transitive on each of these 2-spheres and leaves the midpoint and ∞ fixed. 1 ijk This figure shows the stereo-graphic projection of the 3-sphere S \ {− } to R , filled with 2-spheres. i, j and k are the imag-inary units of the quaternions,which span the tangent space at0 ∈ R .The vector field in real direction,discussed in the text, points out-ward like the spines of a hedgehog,perpendicular to the correspond-ing 2-sphere and its length is theradius of this 2-sphere.Let f ∈ C ∞ ( S ) S . It is clear that the map does only depend on the real value or,in the other picture, not on the point itself, but only on the 2-sphere, on which thepoint is located. To be smooth, the function must depend smoothly on the realvalue and the different direction must fit at 1 and − 1. As the function has the samevalue in any direction of 1, fitting smoothly means that all odd derivatives mustvanish. Thus C ∞ ( S ) S ∼ = (cid:26) f ∈ C ∞ ([ − , (cid:12)(cid:12)(cid:12)(cid:12) d k fdt k ( − 1) = d k fdt k (1) = 0 , for all odd k > (cid:27) ⊂ C ∞ ([ − , , C )Now, we are going to understand invariant differential forms of the conjugationaction on S .Let ω ∈ Ω ( S ) S . Let v be a tangent vector on one of the two fixed points. Thenthere exists g ∈ S , s.t. g − vg = − v , hence an invariant one form must be zero onthe fixed points. As the real part of the quaternion is invariant under conjugation,the vector field pointing in this direction, projects to a invariant tangent field on S ,which vanishes only at 1 and − 1. In the R picture, this is the radial vector fieldpointing outward everywhere. Let X now denote the normalization of this vectorfield on S \ { , − } , and ω the one form dual to X . Let ω = ω − ( ι ( X ) ω ) ω ,where ι is the contraction of the form by the field. A priori this forms are onlydefined on S \ { , − } , but as ω is zero at 1 and − 1, we can extend ( ι ( X ) ω ) ω and ω by zero to obtain a smooth form on all of S . Taking any slice of S withfixed real part in ( − , S and ω actually is a one form oneach of these 2-spheres. The S -isotropy found above, acts non-trivially on tangentvectors. Hence with the same argument as above (Rotating the tangent vector tominus itself) one sees, that ω actually is zero. Thus ω = ( ι ( X ) ω ) ω . Let f be theintegral of ι ( X ) ω ∈ C ∞ ( S ) S ⊂ C ∞ ([ − , ω = df and QUIVARIANT DIFFERENTIAL COHOMOLOGY 41 f (1) = f ( − 1) = 0 as ω vanishes at the fixed points. Thus we have shownΩ ( S ) S (cid:30) dC ∞ ( S ) S = 0 . Let ω ∈ Ω ( S ) S . Contracting with the radial field X as defined in the lastparagraph yields ι ( X ) ω = f ω , for some function f . As ι = 0, f = 0. Thus,restricting ω to each of the level of fixed real part in the open interval, one obtainsa multiple of the volume form on S . At the fixed points one gets a SO (3)-invariant2-form on R , since the adjoint action on the Lie algebra of SU (2) is how one definesthe double cover of SU (2) → SO (3). But there is no non-zero skew-symmetricmatrix commuting with the whole SO (3). Thus ω must vanish on the fixed points.Moreover, as any invariant 1-form is exact,Ω ( S ) S (cid:30) d Ω ( S ) S = Ω ( S ) S ∼ = (cid:26) f ∈ C ∞ ([ − , (cid:12)(cid:12)(cid:12)(cid:12) f ( − 1) = f (1) = 0 , d k fdt k ( ± 1) = 0 , k odd (cid:27) . A volume form on the manifold induces an isomorphism Ω ( S ) ∼ = C ∞ ( S ). Sincethe standard volume is invariant, we get an isomorphism for invariant forms andfunctions. Let X ∈ s ⊂ H . Then X ] ( m ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (1 + tX ) m (1 − tX ) = Xm − mX. Thus for ω ∈ Ω ( S ) S (16) ι ( X ] ) ω ( m ) = ι ( Xm − mX ) ω ( m ) ω =Ad ∗ ω = ι ( Xm ) ω ( m ) − ι ( mX ) Ad ∗ m ω ( m )= ι ( Xm ) ω ( m ) − ι ( m − mXm ) ω ( m ) = 0 . Moreover, d vanishes on top forms, hence the Cartan differential on Ω ( S ) S iszero. As S has empty boundary Z S : d Ω ( S ) S → C is the zero map by Stokes theorem. ThusΩ ( S ) S (cid:30) d Ω ( S ) S → C , ω Z S ω. is a well defined injective homomorphism. From the calculation of the cohomologybelow, we see, that it is surjective.What is the classical equivariant cohomology of the conjugation action of S withvalues in R ∈ { Z , C , C / Z } ? Taking the simplicial manifold model for ES × S S anda cellular resolution with cell structure on S given by one zero cell, correspondingto the neutral element of S , and one three cell, then all simplicial maps are cellularand we obtain the following double complex with the cellular resolution horizontallyto the right and the simplicial complex in vertical direction downwards (compare page 10). S R R . . .S × S R R . . . ( S ) × S R R . . . ( S ) × S R R . . . ( S ) × S R R . . . ... ... ... ... ... ... →→ →→ →→ →→ ∂ (0) →→→→ →→ →→ →→ ∂ (1) →→→→ →→ →→ →→ ∂ (2) →→→→ →→ →→ →→ ∂ (3) →→→→ →→ →→ →→ →→ The R in the 0-column corresponds to the zero cell and the R k in the 3-columncorresponds to the k S ) × k . The 3-cells in S × S are S × { e } and { e }× S and in S × S × S are S ×{ e }×{ e } , { e }× S ×{ e } and { e }×{ e }× S . Onecalculates directly for the conjugation action, that ∂ (0) = 0 and ∂ (1) ( a, b ) = (0 , , b ),where the i -th entry corresponds to the i -th cell. Hence we obtain H kS ( S , R ) = ( R k = 0 , , k = 1 , S and the fourth cohomology is generating by the ‘acting’ 3-cell S × { e } ⊂ S × S .Now the next proposition follows, in the main, by applying the hexagons (13)and (14). Proposition 5.8 For the conjugation action of the 3-sphere S = SU (2) on itself,we have ˆ H nS ( S , Z ) = Z n = 0 C ∞ ( S ) S / Z n = 10 n = 2Ω ( S ) S ⊕ Z d vol S ⊂ Ω ( S ) S n = 3 C / Z ⊕ Z n = 4 H nS ( S , Z ) n ≥ and b H nS ( S , Z ) = Z n = 0 C ∞ ( S ) S / Z n = 10 n = 2Ω ( S ) S ⊕ Z d vol S ⊂ Ω ( S ) S n = 3 C / Z ⊕ Z ⊕ Ω ( S ) S /C ∞ ( S ) S n = 4 . Proof. For ˆ H nS ( S , Z ), the only open question is the case n = 4. There one obtainsa short exact sequence 0 → C / Z → ˆ H S ( S , Z ) → Z → C / Z is an injective abelian group. QUIVARIANT DIFFERENTIAL COHOMOLOGY 43 In the case of b H S ( S , Z ) one has the following hexagon from (14):(17) C ⊕ ( s ) ∨ ⊗ Ω ( S ))) S (cid:30) d (( s ) ∨ ⊗ Ω ( S ))) S ) (( s ) ∨ ⊗ Ω ( S ))) S ⊕ S (( s ) ∨ ) ⊗ Ω ( S ))) S ) cl C b H S ( S , Z ) CC / Z Z → a → ⊕ ( d + ι ) → → → → R (cid:16) I , → → → → As discussed above s = R i + R j + R k ⊂ H and S acts transitive on the unitsphere of this space. Moreover, the subgroup of S , which leaves i ∈ s invariant, isexactly S ⊂ C ⊂ H . Hence (cid:16)(cid:0) s (cid:1) ∨ ⊗ Ω k (cid:0) S (cid:1)(cid:17) S ∼ = Ω k (cid:0) S (cid:1) S (cid:0) ω : s → Ω k (cid:0) S (cid:1)(cid:1) ω ( i )and, since the first and second de Rham cohomology of S vanish, averaging overthe S implies that d : Ω ( S ) S /dC ∞ ( S ) S → Ω ( S ) S is an isomorphism.Further, let( ω, f ) ∈ (cid:16)(cid:0) ( s ) ∨ ⊗ Ω ( S ) (cid:1) S ⊕ (cid:0) S (( s ) ∨ ) ⊗ Ω ( S ) (cid:1) S (cid:17) cl , i.e., dω = 0 and df = − ιω . Then ω = dη for one and only one η ∈ ( s ) ∨ ⊗ Ω ( S ))) S (cid:30) d (cid:18)(cid:16)(cid:0) s (cid:1) ∨ ⊗ Ω (cid:0) S (cid:1)(cid:17) S (cid:19) and df = − ιdη . On the other f is given by a symmetric 3 × S f ii f ij f ik f ji f jj f jk f ki f kj f kk and this matrix is determined, up to a constant matrix denoted by A , by the form η . By the transitive action of S on the Lie algebra, it is clear, that the informationof the matrix is contained in f ii and f ij . The conjugation by the element k √ ∈ S translates the pair ( i, j ) to − ( j, i ). Hence f ij = − Ad ∗ k √ f ij . Thus the off-diagonalterms of the symmetric matrix A must vanish and hence A must be a multiple ofidentity matrix.Thus, we have described an isomorphism C ⊕ Ω ( S ) S (cid:30) C ∞ ( S ) S → (cid:16)(cid:0) ( s ) ∨ ⊗ Ω ( S ) (cid:1) S ⊕ (cid:0) S (( s ) ∨ ) ⊗ Ω ( S ) (cid:1) S (cid:17) cl ( A, η ) ( f, ω ) . Applying this isomorphism, the hexagon (17) changes toΩ ( S ) S (cid:30) C ∞ ( S ) S Ω ( S ) S (cid:30) C ∞ ( S ) S ⊕ C b H S ( S , Z ) C , C / Z Z , → → a , → → id → → R (cid:16) I , → → → → where again the top line, the bottom line and the diagonals are exact. The map a isinjective because the inclusion in the top line factors as R ◦ a . (cid:3) Actions of finite cyclic groups on the circle. Let C p = Z /p Z denote thecyclic group with p elements. There is an action of C p on any odd sphere S n − ⊂ C n ,where a fixed generator acts by multiplication with e p πi . This diagonal action isalso unitary on the infinite-dimensional separable Hilbert space l ( N , C ) and henceinduces an action on the unit sphere S ∞ . The inclusions of C n s as first coefficientsinduce equivariant inclusions S → S → · · · → S ∞ The sum of the tangent bundle and the normal bundle of S ⊂ C is a complex linebundle, T S ⊕ N ∼ = S × C , which we equip with the connection ∇ , whose associatedparallel transport respects the decomposition in tangent and normal space. Hence,the holonomy once around the circle equals 2 π , thus is trivial. The sphere bundle(with respect to the standard metric) of T S ⊕ N is the trivial S bundle on S with the S -invariant connection. Now we have a pullback diagram of bundles withconnection with equivariant maps (cid:0) S × S , ∇ (cid:1) H = S × S S S ∞ S S S ∞ /S . → → → → →→ f → Moreover the first Chern class c ( S ∞ → S ∞ /S ) ∈ H ( S ∞ /S ) = H ( BS ) is agenerator. Now for ˆ H C p ( S , Z ) we have the diagramˆ H C p ( S , Z ) H C p ( S , C / Z ) H C p ( S , Z ) . → I , → → → − β As first and second cohomology are torsion, the Bockstein is an isomorphism, givenby multiplication with p . As the connection on H is flat, ˆ c ( H ) actually is aclass in H C p ( S , C / Z ). Let the cycle au = h , p i ⊂ R / Z ∼ = S be a fundamentaldomain of the C p action on S . Evaluation at f ( au ) induces the isomorphism H C p ( S , C / Z ) → (cid:16) p Z (cid:17) / Z under which c ( H ) is mapped to p . Pulling back theclass along f shows ˆ c ( T S ⊕ N ) = 1 p ∈ C / Z . A finer analysis shows that the bundle S × S → S , where C p acts by multiplicationwith e qp πi on the fiber and e p πi on the base space, has first equivariant differentialChern class qp ∈ C / Z . One may interpret this as a measurement of holonomy alongthe fundamental domain.5.4. G -Representations. In this section, we want to investigate actions of Liegroups on R n . This will lead to some implication to equivariant immersions. Equivari-ant immersions will be subject of further investigation. To generalize the well-knownmethods of characteristic classes applied to immersion, one has, in particular, todefine multiplicative structures in equivariant differential cohomology and generalizethe Whitney-Sum-Formula.An orthogonal representation of the Lie group G on R n (with the standard metric)is given by a map ho : G → O ( n ). This induces an action on the tangent bundle( T R n , ∇ ) = ( R n × R n , d ) with the trivial connection. As d is zero, the curvature EFERENCES 45 vanishes, i.e. R ∇ = 0. But since the trivialization of the tangent bundle is not anequivariant trivialization the moment map will not vanish in general. µ ∇ ( X ) ϕ ( m ) = dϕ m ( X ] ) + ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( ho (exp( tX )) ϕ ( ho (exp( − tX )) m )= dϕ m ( X ] ) + dho ( X ) ϕ ( m ) − dϕ m ( X ] )= dho ( X ) ϕ ( m )Hence for any equivariant differential characteristic class ˆ c with correspondinginvariant polynomial P ∈ I ∗ ( O ( n )), one has R (ˆ c ( G (cid:121) R n )) = P ( µ ∇ + R ∇ ) = P ( dho ) ∈ S ∗ ( g ∨ ) ⊗ Ω ( R n ) = S ∗ ( g ∨ ) . In particular, the characteristic form (and hence the class) will not vanish ingeneral for this flat bundle.5.5. Towards obstruction to immersions? A major application of characteristicclasses in the non-equivariant case is given by obstructions to immersions – moreprecisely, the characteristic classes give lower bounds to the minimal codimensionof an immersion. In the world of classical characteristic classes this can be found,e.g., in [26, Theorem 4.8]). Differential characteristic classes apply for a result, thatconformal immersions have a stronger bound for the minimal codimension, thansmooth immersions (see [25] and [11, §6] for the original work and [27] for a partlystrengthened version).The arguments therefore go as follows: Let M be an (Riemannian) manifold and f : M → R n an (isometric) immersion. Then there is a normal bundle N M → M ,such that T M ⊕ N M = f ∗ T R n . Since the Chern classes on the right hand side vanish, the the total Chern class of N M must be the inverse (with respect to the cup product) of T M . This impliesrestrictions to the values of these classes. Moreover, in the Riemannian case, theLevi-Civita connection on M is compatible with the pullback connection ∇ f of thetrivial connection on R n to T M ⊕ N M . This implies similar statements for thedifferentially refined characteristic classes of the Riemannian connections.More explicitly John Millson calculates the first differential Pontryagin classof some lens spaces and shows, that these do not immerse conformally into R n with certain codimension, where smooth immersions exist. It is, with our theory,straight forward to restate these examples for the lens space action of a finite cyclicgroup on the 3-sphere, which should be immersed into a trivial representation.It is subject of further research to study equivariant conformal immersions intonon-trivial representations. Acknowledgements The first author wants to thank the International Max Planck Research School Mathematics in the Sciences for financial support. This research was supported byERC Starting Grant No. 277728. References [1] Nicole Berline and Michèle Vergne. “Classes caractéristiques équivariantes.Formule de localisation en cohomologie équivariante”. In: C. R. Acad. Sci.Paris Sér. I Math. issn : 0249-6321 (cit. on p. 4).[2] Raoul Bott and Loring W. Tu. “Equivariant characteristic classes in the Cartanmodel”. In: Geometry, analysis and applications (Varanasi, 2000) . World Sci.Publ., River Edge, NJ, 2001, pp. 3–20. arXiv: math/0102001 (cit. on p. 4). [3] Nicolas Bourbaki. Algebra I. Chapters 1–3 . Elements of Mathematics (Berlin).Translated from the French, Reprint of the 1989 English translation. Berlin:Springer-Verlag, 1998, pp. xxiv+709. isbn : 3-540-64243-9 (cit. on p. 37).[4] Jean-Luc Brylinski. Gerbes on complex reductive Lie groups . 2000. arXiv: math/0002158 (cit. on pp. 7–9).[5] Jean-Luc Brylinski. Loop spaces, characteristic classes and geometric quantiza-tion . Vol. 107. Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc.,1993, pp. xvi+300. isbn : 0-8176-3644-7. doi : (cit. on p. 3).[6] Ulrich Bunke. Differential cohomology . Course notes. 2012. arXiv: (cit. on pp. 3, 4, 16, 17, 20, 30, 31, 35, 38, 39).[7] Ulrich Bunke, Thomas Nikolaus, and Michael Völkl. “Differential cohomologytheories as sheaves of spectra”. In: Journal of Homotopy and Related Structures (2014), pp. 1–66. issn : 2193-8407. doi : 10 . 1007 / s40062 - 014 - 0092 - 5 .arXiv: (cit. on pp. 3, 16).[8] Ulrich Bunke and Thomas Schick. Differential K-theory. A survey . 2010.arXiv: (cit. on p. 3).[9] Ulrich Bunke and Thomas Schick. “Uniqueness of smooth extensions of gen-eralized cohomology theories”. In: J. Topol. issn :1753-8416. doi : (cit. on p. 3).[10] Henri Cartan. “La transgression dans un groupe de Lie et dans un espace fibréprincipal”. In: Colloque de topologie (espaces fibrés), Bruxelles, 1950 . GeorgesThone, Liège, 1951, pp. 57–71 (cit. on p. 10).[11] Jeff Cheeger and James Simons. “Differential characters and geometric invari-ants”. In: Geometry and topology (College Park, Md., 1983/84) . Vol. 1167.Lecture Notes in Math. Berlin: Springer, 1985, pp. 50–80. doi : (cit. on pp. 2, 45).[12] Pierre Deligne. “Théorie de Hodge. III”. In: Inst. Hautes Études Sci. Publ.Math. 44 (1974), pp. 5–77 (cit. on pp. 3, 6).[13] Albrecht Dold. Lectures on algebraic topology . Classics in Mathematics. Reprintof the 1972 edition. Berlin: Springer-Verlag, 1995. isbn : 3-540-58660-1. doi : (cit. on p. 10).[14] Johan L. Dupont. Curvature and Characteristic Classes . Lecture Notes inMathematics, Vol. 640. Berlin Heidelberg New York: Springer-Verlag, 1978,pp. viii+175. isbn : 3-540-08663-3 (cit. on pp. 3, 5–9, 20, 30, 33).[15] Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine. Riemanniangeometry . Third. Universitext. Berlin: Springer-Verlag, 2004, pp. xvi+322. isbn : 3-540-20493-8. doi : (cit. on p. 11).[16] Ezra Getzler. “The Equivariant Chern Character for Non-compact Lie Groups”.In: Advances in Mathematics issn : 0001-8708. doi : (cit. on pp. 4, 5, 11, 13–15).[17] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory . Vol. 174.Progress in Mathematics. Basel: Birkhäuser Verlag, 1999, pp. xvi+510. isbn :3-7643-6064-X. doi : (cit. on p. 6).[18] Kiyonori Gomi. “Equivariant smooth Deligne cohomology”. In: Osaka J. Math issn : 0030-6126. url : http://projecteuclid.org/euclid.ojm/1153494380 (cit. on pp. 4, 5, 7–9, 12, 15, 17, 20–22).[19] Robin Hartshorne. Algebraic geometry . Graduate Texts in Mathematics, No.52. New York: Springer-Verlag, 1977, pp. xvi+496. isbn : 978-0-387-90244-9. doi : (cit. on pp. 6, 9). EFERENCES 47 [20] Andreas Kübel. “Equivariant Differential Cohomology”. PhD thesis. Univer-sität Leipzig, 2015. url : http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-185772 (cit. on p. 39).[21] Andreas Kübel and Andreas Thom. Equivariant characteristic forms in theCartan model and Borel equivariant cohomology . 2015. arXiv: (cit. on pp. 4, 15, 29, 31, 33).[22] Matvei Libine. Lecture Notes on Equivariant Cohomology . 2007. arXiv: (cit. on p. 10).[23] Jean-Louis Loday. Cyclic Homology . Springer Berlin Heidelberg, 1998. doi : (cit. on p. 13).[24] Ernesto Lupercio and Bernardo Uribe. “Deligne cohomology for orbifolds,discrete torsion and B -fields”. In: Geometric and topological methods forquantum field theory (Villa de Leyva, 2001) . World Sci. Publ., River Edge,NJ, 2003, pp. 468–482. doi : (cit. on pp. 15,16, 29).[25] John J. Millson. “Examples of nonvanishing Chern-Simons invariants”. In: Journal of Differential Geometry url : http://projecteuclid.org/euclid.jdg/1214433163 (cit. on p. 45).[26] John W. Milnor and James D. Stasheff. Characteristic classes . Annals ofMathematics Studies, No. 76. Princeton, N. J.: Princeton University Press,1974, pp. vii+331. isbn : 978-0-691-08122-9 (cit. on p. 45).[27] John Douglas Moore. “Euler Characters and Submanifolds of Constant PositiveCurvature”. In: Transactions of the American Mathematical Society url : (cit.on p. 45).[28] Michael Luis Ortiz. Differential equivariant K-theory . Thesis (Ph.D.)–TheUniversity of Texas at Austin. 2009, p. 124. eprint: (cit. on pp. 15, 16, 29).[29] Friedhelm Waldhausen. Lecture: Algebraische Topologie . 2006. url : (cit. on p. 10).[30] Zuoqin Wang. Notes on Lie Groups . 2009. url : (cit. on pp. 36, 37).[31] Frank W. Warner. Foundations of differentiable manifolds and Lie groups .Vol. 94. Graduate Texts in Mathematics. Corrected reprint of the 1971 edition.New York: Springer-Verlag, 1983, pp. ix+272. isbn : 978-0-387-90894-6. doi : (cit. on p. 9).[32] Charles A. Weibel. An introduction to homological algebra . Vol. 38. CambridgeStudies in Advanced Mathematics. Cambridge: Cambridge University Press,1994, pp. xiv+450. isbn : 0-521-43500-5. doi : (cit. on p. 7). A.K., Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103Leipzig, Germany E-mail address : [email protected] A.T., Institut für Geometrie, TU Dresden, 01062 Dresden, Germany E-mail address ::