Characters for Complex Bundles and their Connections
aa r X i v : . [ m a t h . K T ] M a r Characters for Complex Bundles and their Connections
James Simons † and Dennis Sullivan ‡ Abstract
Given a complex vector bundle E over a base manifold X with connection ∇ we constructinvariants called characters in terms of integrals over manifolds with boundary: ∇ character values C -linear connection C / Z unitary connection R / Z independent of connection Q / Z In logical order: the first result, Theorem AT (Algebraic Topology), shows the Q / Z -charactersderived from C / Z -characters are in a bijective correspondence with complex K -theory. The sec-ond result, Theorem DG (Differential Geometry), shows the C / Z (or R / Z ) characters are in abijective correspondence with differential K -theory defined using complex (or real) valued differ-ential forms. Differential K -theory may be defined as the Grothendieck group of Chern-Simonsequivalence classes of complex bundles with connection (respectively C -linear or unitary). Thethird result, Theorem AN (Analysis): i ) Expresses the unitary bijection in terms of the etainvariants mod one of the spin c Dirac operators with coefficients in ( E, ∇ ) restricted to en-riched closed odd-dimensional stably almost complex manifolds (SACs) in X , and ii )a) Thereis an easy and natural push forward in differential K -theory ( C -linear or unitary) for families ofeven-dimensional SACs defined by pulling back the odd-dimensional SAC cycles in X which areenriched using the direct sum connection, ii )b) in the unitary case the direct sum connectionis CS equivalent to a rescaling (adiabatic) limit of the Levi-Civita connection, and one therebycomputes the push forward (unitary case) as a limit of eta invariants mod one as the rescalingtends to infinity.This adds to the discussion in the literature of the question (communicated by Iz Singer) tohave an index theorem in differential K -theory for families.History and Acknowledgement: The new aspect of Theorem AT is the construction and proofof the bijection using integrals of characteristic forms, whereas the existence of such a bijection isa variant of much earlier understanding [1]. The form of the ˆ K -character definition is a modifiedform of “Differential Characters” [2] motivated by “The Characteristic Variety Theorem” [3].We are indebted to [4] for completing [5] for the non-unitary, C -linear connections describingdifferential K -theory with C -valued differential forms. † Stony Brook University, Simons Foundation ‡ CUNY Graduate Center, Stony Brook University ntroduction Let V be an odd-dimensional Stably Almost Complex (SAC) closed manifold mapping smoothly F : V → X to a target manifold X . Suppose X is the base of a complex vector bundle E togetherwith a C -linear connection ∇ . Suppose V is “enriched” by an independent C -linear connection ∇ /V on its stable tangent bundle. We can form the Chern character, a total even form ch ( E ) on X . We can also form on V the total even Todd form of V (see background in Section 0).One can show [Section 1] that V bounds in the SAC sense a W in such a way that the bundle F ∗ ( E )on V extends to a complex vector bundle E/W on W . One can impose a C -linear connection onthe stabilized tangent bundle of W compatible with that on V in a collar neighborhood of V , andalso a connection on E/W similarly compatible with that on F ∗ ( V ). Using these connections wecan extend Todd( V ) and F ∗ ( ch ( E )) to total forms Todd( W ) and ch ( E/W ) over all of W . Nowform the integral over W of the wedge product Todd( W ) ch ( E/W ) and reduce the value moduloone. We refer to this as an “angle” in the complexified circle C / Z . This “angle” in C / Z may beseen to be independent of the choice of W and the extended connections above. It depends only on F : V → X and the enrichment of V . As the triple ( V, F, ∇ /V ) varies one obtains many “angle”invariants of ( E, ∇ ), the complex bundle with C -linear connection over X . The “angle” invariantsare thought of as a “character” function with values in the complexified circle C / Z defined on thecollection of odd-dimensional enriched SAC closed manifolds or cycles mapping into X .One sees directly the “angle” invariants of ( E, ∇ ) over X are equal as “characters” to the “angle”invariants of ( E, ∇ ′ ) if ∇ and ∇ ′ are CS equivalent. This means the odd-dimensional Chern Simonsforms whose exterior d is the difference between the corresponding Chern character forms are notonly closed so that the Chern character forms of ( E, ∇ ) and ( E, ∇ ′ ) are exactly the same, butthese CS forms are also exact. This statement of equality of “angle invariants” for CS equivalentconnections follows from Stokes’ theorem.One may also change the pair ( E, ∇ ) by direct summing with the trivial connection on the trivialbundle or by changing by a strict isomorphism of bundles with connection, without changing thevalues of the character function associated with ( E, ∇ ).The goal of this paper is twofold. Firstly, to prove the converse of these equalities of “angle” invari-ants. Namely, the “angle” invariant characters actually determine ( E, ∇ ) up to the equivalencesjust mentioned. Secondly, to describe the properties of the character function on enriched SACcycles in X that are necessary and sufficient for such a function to arise as the “angle” invariants ofa complex bundle E with C -linear connection over X . These properties are described / containedin the differential geometry statement Theorem DG in Section 3.First goal: the form of the equivalences was exploited recently [5] to give a geometric description,in the case of the generalized cohomology theory complex K -theory, of the extension introduced in[6] of a contravariant functor called “generalized differential cohomology”. This is a derived fibreproduct of the generalized cohomology functor h ∗ with de Rham differential forms either real orcomplex valued which are labeled by elements in h ∗ ( pt ). The fibre product is over the generalizedde Rham isomorphism canonically identifying h ∗ ( , R or C ) with the de Rham cohomology of forms(over R or C ) with coefficients h ∗ ( pt ). The geometric description of [5] says differential complex2 -theory defined by forms with values in R or C , denoted ˆ K , is represented by the Grothendieckgroup of stable isomorphism classes of complex bundles with C -linear connections in the case of C -valued forms and with unitary connections in the case of R -valued forms, up to CS equivalence(structured bundles) under direct sum. The proof of the odd form lemma in [5] depended on theexistence of inverses up to CS equivalance for the bundles with connection. These inverses wereprovided for unitary connections in [5]. For C -linear connections, the inverses were provided in [4].We are indebted to Leon Takhtajan for explaining how [4] completes the C -linear results of [5]. Inparticular, Corollary 2 of [4] plus Proposition 2.5 of [5] implies Corollary 3 of [4] which producesthe required inverses. There is yet another proof for GL[N,C] of the required inverse property ofbundles with connection in [7].A helpful organizing tool (Section 3) in the proof of the differential geometry statement TheoremDG is the “character” diagram of groups placing ˆ K in the hexagons between the interlocking sinewave Bockstein long exact sequence and the de Rham long exact sequence. Verifying the analogousdiagram for ˆ K characters is the rest of the work in the proof of Theorem DG beyond Theorem ATmentioned above and described more fully below.Second goal: the pattern of the necessary and sufficient conditions on “angle” invariants to comefrom a bundle with C -linear connection modifies the notion of “differential character” introducedin [2]. These were functions C with values in R / Z on smooth cycles in X in the sense of algebraictopology. These functions were additive under union of cycles. They were not homology invariantbut their values in R / Z changed under a homological deformation of a cycle by integrating a closedreal differential form with integral periods on X over the homology. Namely if z and z ′ are cyclesin X then C ( z ) − C ( z ′ ) in R / Z is given by the integral of w ( C ) over any homology in X between z and z ′ reduced [mod one]. In particular a small smooth deformation produces a small change inthe value, thus the name differential character. This definition was motivated by the attempt todefine objects in the base related to the Chern-Simons forms in the bundle.The modification of the notion of differential character to the main notion of differential ˆ K characteremployed here to describe the “angle” invariants goes as follows: replace algebraic topology cyclein X as in [2] by smooth enriched odd-dimensional SAC cycle in X , and replace the variation k -form w ( C ) of the differential character by a total even complex valued form W ( C ) representingthe Chern character of some complex vector bundle over X . Replace the variation formula fordifferential characters by C ( V ) − C ( V ′ ) equals the integral mod Z of W ( C ) ∧ Todd M over M ,where M is an enriched SAC mapped into X whose boundary is the formal difference of enrichedcycles V − V ′ in X . A new “product relation” beyond [2] appears because of the multiplicativeproperties of the Todd form. This modification follows the pattern of the Characteristic VarietyTheorem [3].Theorem DG gives a second geometric interpretation added to that of [5] for differential K -theoryof X in terms of differential ˆ K characters defined on enriched SAC cycles in X . In the real caseusing unitary connections, thanks to the APS theorem [8], this bijection is elegantly described byforming the eta invariants mod one of the spin c Dirac operator of the odd-dimensional SAC cyclewith coefficients in the bundle E restricted to V to build the ˆ K character of the bundle with unitaryconnection. 3he proof scheme for Theorem DG uses the machinery of algebraic topology to arrive first ata corresponding result, Theorem AT (Algebraic Topology), giving a complete theory of rational“angles” associated to just the bundle E independent of the choice of connection. The rational“angles” of order k in C / Z are associated with pairs ( V, W ) where V is an odd SAC cycle in X with a given way, W , to bound k copies of V in X . These rational angles are derived in Section 2,in terms of the general “angles” associated with enriched SAC cycles in X probing bundles withconnections. As the enrichment connections vary continuously, the rational “angles” cannot varycontinuously without staying constant and thus become topological invariants of the bundle. Theyare complete invariants and their precise necessary and sufficient conditions are specified usingRational Hom( h ( , Q / Z ) , Q / Z ). Here h is the remarkable homology theory discovered by Connerand Floyd [9] using SAC cycles and homologies (i.e. SAC bordism classes) in X taken also modulothe purely algebraic “product relation” mentioned above. The fact that this algebraic quotient doesnot destroy the exactness property of a homology theory is the crucial point of the argumentationhere. This is the Algebraic Topology theorem, Theorem AT, described in Section 2 and used inSection 3.One interesting corollary of the Topology discussion is Corollary : A cohomology class c in H even ( X, Q ) is the Chern character of a complex bundle over X if and only for every closed even-dimensional SAC mapping to X , V f −→ X , R f ∗ c Todd V is aninteger. A similar statement holds for the transgressed ch in U , odd-dimensional closed SACs in X , elements in H odd ( X, Q ) and maps X → U .The first application of Theorem DG is a very easy explicit construction of a wrong way map indifferential K C -theory for a fibration with C -linear connection over a base X with fibres closedeven-dimensional SAC manifolds enriched by C -linear connections. If T denotes the total spacethere is a map of enriched SAC cycles from X to those in T by taking the pullback SAC cycle.There is a nuance here, but then applying ˆ K characters reverses the direction to give a wrongway map from the differential K -theory of T to the differential K -theory of X (all over C ). See[14]. The nuance here is the enrichment of the pullback cycle. Since the fibration is enriched witha connection on the vertical subbundle, one may use the direct sum connection (see Section 4)to enrich the pullback cycles. Then a differential ˆ K character on T will induce a differential ˆ K character on the base using the multiplicative nature of Todd forms and Chern character forms todefine the required variation form. (See Section 4.)There is also the statement: Corollary 1 of Proof of Theorem DG :kernel( ˆ KX ch −→ ∧ evenintegrality ) is isomorphic to Hom( ¯Ω C odd , C / Z ), a complex torus of dimension thesum of the odd Betti numbers of X .The final discussion brings in analysis and the eta invariants of the Atiyah-Patodi-Singer [8] theoryrelating Topology, Geometry, and Analysis of a SAC manifold with boundary, but now restrictedto using unitary connections on the bundles and Levi-Civita connections on manifolds.4s mentioned above, the “angle invariant” in R / Z for a unitary connection constructed using thefilling W is just the spectral invariant on V defined using the spin c Dirac operator reduced moduloone. Here it is important though that the connection on V be the Levi-Civita connection so thatthe asymptotic heat kernel analysis of the APS Theorem is valid.Fortunately but not obviously, there is an extension of this calculation relating real “angles” toeta invariants mod one giving an analytic calculation of the wrong way map in differential K -theory in the case of unitary connections. There is a serious stumbling block though. For the APStheory one again needs the asymptotic analysis based on using the Levi-Civita connection on thetotal space T . For the wrong way map one needs the multiplicative property of the characteristicforms associated to the direct sum connection on T in order to define the variation form in thedefinition of ˆ K -character. Even though the metric on T can be taken to be the direct sum metricthe Levi-Civita connection is not the direct sum connection. See the Appendix to Section 5 forthe detailed discussion of the interesting difference. However in the Appendix one sees that byscaling in such a way that the base becomes infinitely large compared to the fibre the limit of theLevi-Civita connections upstairs exists (the adiabatic limit) and is fortunately CS equivalent tothe direct sum connection. This equivalence enables the analytic calculation of the wrong way mapin differential K -theory (unitary case) as a limit of eta invariants of the rescaled metrics. This isthe analytic theorem, Theorem AN, in Section 5.Theorem AN is our response to Iz Singer’s question (on a flight with the authors seven years ago)about having an analytic version of the index theorem in differential K -theory for families. Therehave been other responses which use infinite-dimensional analysis [10] and [references in [11]]. §
0. Background about connections, Chern-Weil characteristic forms,and Chern-Simons forms
Let W Π −→ X be a real n -dim vector bundle over a smooth manifold. Let ∇ be a connection, and R ∈ ∧ ( X, End( W )) the curvature tensor.A real valued polynomial on the Lie algebra of GL ( n, R ) is called invariant if it is fixed underthe adjoint action. If P l ( B ) = t r ( B l ), it is well known that the ring of invariant polynomials isgenerated by P , · · · , P n . For a degree l invariant polynomial P , the Chern-Weil homomorphismyields P ( l z }| { R ∧ R ∧ · · · ∧ R ) ∈ ∧ l ( X ) , a closed form.This map is a ring homomorphism, and the associated cohomology class of an element in its imageis independent of the choice of connection. This is made apparent by 1.1) and 1.2) below. We learned of the adiabatic limit connection from the work of Cheeger [12] and Freed [17]. The equality of the Chern character forms was known before [communication of Dan Freed] but CS equivalenceis a stronger condition. ∇ and ∇ are two connections on W with curvature forms R and R , and ∇ t is a smooth curveof connections joining ∇ to ∇ , and R t its curvature, set B t = ddt ( ∇ t ) ∈ ∧ ( X, End( W )) . For P invariant of deg l ,1 . P ( R ∧ · · · ∧ R ) − P ( R ∧ · · · ∧ R ) = d ( TP ( ∇ , ∇ )) where1 . TP ( ∇ , ∇ ) = l Z P ( B t ∧ l − z }| { R t ∧ · · · ∧ R t ) mod ∧ l − . (Chern-Simons forms)It may be shown that TP is independent of the curve joining ∇ to ∇ , and thus TP ( ∇ , ∇ ) iswell defined. We also have1 . PQ ( R ∧ · · · ∧ R ) = P ( R ∧ · · · ∧ R ) ∧ Q ( R ∧ · · · ∧ R )1 . T ( PQ )( ∇ , ∇ ) = TP ( ∇ , ∇ ) ∧ Q ( R ∧ · · · ∧ R ) + TQ ( ∇ , ∇ ) ∧ P ( R ∧ · · · ∧ R ) . The first, because the Chern-Weil map is a ring homomorphism, and the second by calculation(recall TP is defined mod exact). Definition : ∇ and ∇ are called CS equivalent if TP ( ∇ , ∇ ) is exact for all invariant P . Thisis easily shown to be an equivalence relation.Since { P l } generates the ring of invariant polynomials, 1.3) and 1.4) show Proposition 1.5 : ∇ and ∇ are equivalent if and only if TP l ( ∇ , ∇ ) is exact for all l ≤ n . Remark : The discussion for C -linear connections and complex-valued Chern-Weil and Chern-Simons is similar. §
1. Construction I for Stably Almost Complex Manifolds (SACs)
Proposition 1 : Any complex vector bundle E over Σ, a closed odd-dimensional SAC, can be filledin. Namely, there is an even-dimensional SAC, W with ∂W = Σ, and a complex bundle E W over W extending E . Proof : The SAC bordism of a point, Ω C ∗ ( pt ) is torsion free and is concentrated in even degrees [acelebrated result of the 60’s]. So is the homology of B U n the classifying space of isomorphism classesof complex bundles of rank n . The SAC bordism of B U n , Ω C ∗ ( BU n ), is the limit of the Atiyah-Hirzebruch spectral sequence which begins with H ∗ ( BU n , Ω C ∗ ( pt )) and which for any homologytheory collapses when tensored with Q . Thus it already collapses in this case and Ω C ∗ BU n is torsionfree and concentrated in even degrees. This proves Proposition 1.6onstruction I below defines a pairing < ∇ Σ ; Σ f −→ X | E → X ; ∇ E > in C / Z where Σ f −→ X is aclosed odd-dimensional SAC in X , E → X is a complex vector bundle over X , ∇ Σ is a C -linearconnection on the stable tangent bundle of Σ, and ∇ E is a C -linear connection on E over X . Construction I : By Proposition 1 we can fill in Σ by W and extend E Σ = E/ Σ to E W over W .Similarly we can extend ∇ Σ to ∇ W on the stable tangent bundle of W and ∇ E / Σ to the extendedbundle E W over W . We suppose there is a product neighborhood near ∂W where the extendedconnections are product-like. On W there are two even-dimensional differential forms ch ( E W , ∇ ),the Chern character form defined by ( E W , ∇ ), and Todd W , the characteristic form associated tothe universal total Todd class which is constructed from ∇ W . (To be precise, we use the Toddform, as defined in the Remark below, associated with the inverse of the stable tangent bundle.)Define the pairing < ∇ Σ ; Σ f −→ X | E → X ; ∇ E > in C / Z by R W ch ( E W ) · Todd W reduced modulo 1. Proposition 2 : The value of the integral of W mod one only depends on the SAC cycle in X ,Σ f −→ X “enriched” by the connection ∇ Σ and the complex vector bundle E → X “enriched” by itsconnection ∇ E . Proof : If we had chosen a different filling ¯
W , E ¯ W so that ∂ ¯ W = Σ and E ¯ W / Σ = E Σ and different“enrichment” ∇ ¯ W and ∇ E ¯ W extending the enrichments on Σ we can form the union of these twochoices along Σ, namely W ∪ Σ ¯ W and E W ∪ E Σ E ¯ W and also glue the enrichments. The differenceof the two integrals which are the two definitions of the pairing is the entire integral over the closedmanifold W ∪ Σ ¯ W . This integral is well known to be an integer. (See the next remark for somehistory.) This proves Proposition 2. Remark 1 : Here we fix the definition of the Todd class and recall how the integrality of R V ch ( E )Todd V was understood for closed even-dimensional SACs V .Imagine V embedded in a large sphere S with a complex structure on its normal bundle V providedwith a unitary structure. Pull back V to the normal disk bundle N to obtain a complex bundle E on N . For each point v of N not in the zero section of ν there is an isomorphism between the twohalves of the exterior algebra bundle associated to E , ∧ even E ↔ ∧ odd E defined by (wedging with v ) plus (contracting with v ).This defines an element in the complex K -theory of the pair K even C (disk bundle, sphere bundle)which we pull back to the big sphere S by the collapsing map S ↔ disk bundle / sphere bundle.The Chern character of this pullback element in the sphere is an integer [Bott, Milnor, Adams].One calculates in the universal example over B U that ch ( ∧ even − ∧ odd ) in the cohomology of theuniversal Thom space M U satisfies ch ( ∧ even − ∧ odd ) = U · Todd , where U is the Thom class, for some universal class Todd (defined by this equation). This revealsthe integrality above. 7reviously, in his celebrated treatise, Hirzebruch calculated Todd in terms of the multiplicativeseries x/e x −
1, motivated by Todd’s work on the arithmetic genus (or holomorphic Euler charac-teristics of algebraic varieties). This series shows the multiplicative property of the Todd formula.The integrality made precise by Adams, Bott, and Milnor circa 1960 inspired Atiyah and Singer tobuild Dirac operators and to develop the index theorem. (Recounted to one of the authors by IzSinger (late 60’s)) We revisit this later.
Remark 2 : There are several properties of the pairing < ∇ Σ , Σ f −→ X | E → X, ∇ E > in C / Z . i ) Since the Todd form is multiplicative for the direct sum of bundles with connections, if Σ → X is multiplied by V → pt then < ∇ , Σ × V → X | E X , ∇ E > = Todd V < ∇ Σ , V → X | E X , ∇ > . ii ) Fixing E X , ∇ E and varying the cycle the function φ ( ∇ , V → X ) = < ∇ , V → X | E X , ∇ > satisfies: there is a closed form C on X so that whenever V f −→ X = ∂ ( W F −→ X ) then φ ( ∇ , V f −→ X ) = R W F ∗ C Todd W (mod one). Remark 3 : It follows that the closed form C of property ii) has integral periods in the sense thatfor every closed W F −→ X Z W F ∗ C Todd W belongs to Z , and that C is unique given the values in C / Z for SAC cycles. (We denote such forms C in Section3 by ∧ evenintegrality .) iii ) If we fix E X but change the connection from ∇ to ∇ ′ , then if CS ( ∇ , ∇ ′ ) denotes the ChernSimons difference form so that dCS ( ∇ , ∇ ′ ) = ch ∇ − ch ∇ ′ , then < ∇ Σ , Σ f −→ X | E X , ∇ > − < ∇ E , Σ f −→ X | E X , ∇ ′ > = Z V f ∗ CS ( ∇ , ∇ ′ )Todd V. In particular, if CS ( ∇ , ∇ ′ ) = exact, the difference is zero.8
2. Algebraic Topology invariants in Q / Z of complex vector bundlesderived from the pairing < ∇ Σ , Σ → X | E → X, ∇ E > in C / Z Recall that the Z / C ∗ ( X ) = Ω C ∗ ( X ) ⊗ Ω C ∗ ( X ) Z where the SAC bordism Z -gradedmodules over SAC bordism of a point are collapsed to a Z / V · x = 0if Todd V = 0. This was introduced by Conner and Floyd and recall their theorem [9] implyingthe unexpected fact that ¯Ω C ∗ ( ∗ ) is a Z / C ∗ ( X ) is a homologytheory we can form ¯Ω C ∗ ( X, Z /n ) which can also be defined by SAC Z /n -manifolds via the formula¯Ω C ∗ ( X, Z n ) = Ω C ∗ ( X, Z /n ⊗ Ω ∗ pt Z (see below). Then we can define the homology theory ¯Ω C ∗ ( X, Q / Z )as thelim → n ¯Ω C ∗ ( X, Z /n ) . The first theorem, (whose terms are explained more fully in the Remark) is
Theorem AT (Algebraic Topology): Complex K -theory is isomorphic based on C / Z charactersto Rational Hom( ¯Ω C ∗ ( X, Q / Z ) , Q / Z ), whose elements are called Q / Z characters. Remark : We will use the pairing of Section 1 < ∇ Σ , Σ → X | E → X, ∇ E > ∈ C / Z in the proof, to define Z /n pairings for Σ a Z /n SAC bordism class of Z /n -manifolds in X . < Σ → X | E → X > ∈ Z /n will form an inverse limit. This limit is uncountable, but the rationality condition will characterizethe image we seek. Remark : For a homology theory h ∗ C ∈ Rational Hom ( h ∗ ( , Q / Z ) , Q / Z )means by definition the boxed commutative diagram: β −→ h ∗ ( X, Z ) → h ∗ ( X, Q ) → h ∗ ( X, Q / Z ) β −→↓ C Z ↓ C Q ↓ C → Z → Q → Q / Z → h ∗ . C determines C Q uniquely when C Q exists(see below). 9 V, β V) β V − + − +−+ +− a) b) c) β W W + d) − Figure 1: Z /n manifold in X , n = 3Now we turn to the proof of Theorem AT using the pairing < ∇ Σ ; Σ → X | E → X ; ∇ E > ∈ C / Z ,and the Z /n -manifold definition of ¯Ω C ∗ ( X, Z /n ). (See Remark below for an explanation of the Z /n -manifold definition of Ω ∗ C ( X, Z /n ).) Definition 1 : A Z /n -manifold is a pair ( V, βV ) where boundary V is the disjoint union of n copies of a closed manifold βV , (read “Bockstein of V ”). We say the Z /n -manifold ( V, βV ) is theboundary of (
W, βW ) if βV is the boundary of βW , and V union n -copies of βW glued on the n boundary components βV is a closed manifold which is the boundary of W . Word picture: a Z /n -manifold looks like a book with n -pages attached along a binding βV but whose edges are allglued together to form V . Construction II : Given an even-dimensional SAC- Z /n manifold in X , ( V, βV ) → X and a com-plex bundle E → X , we construct an element in C / Z of order n as follows: write βV = ∂Q andextend E/V = E V to EQ over Q using Proposition 1 of Section 1. Enrich these objects withconnections as in Section 1. For dim V even, consider the expression ∗ ) n R ch E V Todd V − R Q ch E Q Todd Q defining an element in C / Z after reducing mod Z .10 roposition 2 :a) The element defined in ∗ ), denoted by < V f −→ X | E → X > is an element of order n in C / Z andis independent of the choices ∇ V , ∇ E , and Q, ∇ Q .b) If the Z /n -manifold V → X in X bounds as a Z /n manifold in X then ∗ ) is zero in C / Z .c) < U → pt × V f −→ X | E → X > = Todd
U < V f −→ X | E → X > .
Proof : For part a), multiply ∗ ) by n and look at the boundary R of Figure 1c. This expression isthe integral of ( ch E Todd R ) where R is the closed manifold ( V union n -copies of Q ) = R . Thisis an integer. So n · ( ∗ ) = 0 in C / Z .Changing Q to another filling of βV changes the second term in ∗ ) by an integer as in Construction I.Changing the connection can be done continuously. Elements of order n cannot move continuously.This proves a).To prove b), note the integer defined by [ ∗ ) times n ] is actually zero since R = ∂W . Thus dividingby n , it is still zero as a real number. Thus its reduction mod Z is zero.c) follows from the definitions, and the multiplicative properties of the Todd form. (cid:4) Corollary : For every n we have character invariants defined using the C / Z characters, K even C ( X ) < | > −−−→ Hom( ¯Ω C even ( X, Z /n ) , Z /n )where ¯Ω C ( ∗ ) is the Conner-Floyd homology theory ( Z / C ( ∗ ) ( X ) = Ω C ∗ ( X ) ⊗ Ω C ∗ pt Z. Proposition 3 : Elements on ¯Ω ( ∗ ) ( X, Z /n ) have order n . ¯Ω ( ∗ ) ( X, Z /n ) is a multiplicative theory,(which means one can multiply cycles in X and in Y to get cycles in X × Y ). Proof : We prove the second statement first for Ω C ( ∗ ) ( X, Z /n ) (without the “bar”). The cartesianproduct of two Z /n -manifolds (like the product of two smooth manifolds with boundary) has acodimension two locus L that needs attention. The neighborhood of the locus L has the form L × c n × c n where c n is the cone on n points (denoted ( n )). Now c n × c n is the cone on the join( n ) ∗ ( n ). But ( n ) ∗ ( n ) defines an element in Ω C ( pt, Z /n ) which is zero as seen by the exact sequence(Ω C ( pt ) n −→ Ω C ( pt ) → Ω C ( pt, Z /n ) → Ω C n −→ Ω C ) =(0 → → Ω C ( pt, Z /n ) → Z n −→ Z ) . Remark : For a general theory, elements in h ( pt ) can create a difficulty at this point of theargument for Proposition 3, e.g. n = 2 and KO ∗ . [Communication of Luke Hodgkin].11ontinuing, choose a two dimensional Z /n SAC manifold C n with boundary, whose boundary is n ∗ n . Use it to repair the neighborhood of the locus L as follows: remove L × c n × c n and replaceit by L × C n glued along the boundary = L × (( n ) ∗ ( n )).Map the repaired cartesian product of cycles to the cartesian product of cycles by projecting C n tothe cone on its boundary. This defines (by repairing bordisms likewise) the multiplicative structure:a map, Ω C ∗ ( X, Z /n ) ⊗ Ω C ∗ ( Y, Z /n ) → Ω C ∗ ( X × Y, Z /n ).Since the zero manifold [ n points] bounds in Z /n bordism of a point, that Ω C ∗ ( X, Z /n ) is a Z /n module follows from the map defining the multiplicative structure. This completes the first partof the proof of Proposition 3. The rest of the proof identifying the Z /n Conner-Floyd theory withthat defined by Z /n -manifolds is in the Appendix to Section 2. §
2. Appendix (Homology Theory)
Continuing the proof of Proposition 3:Let M n denote the Z /n -Moore space, the circle with one two cell attached by degree n . If h ∗ is ahomology theory, then h k ( X, Z /n ) for X connected may be defined as h k +1 ( X ∧ M n , Z ) where X ∧ M n ≡ X × M n /X ∨ M n . Note the Bockstein exact sequence β −→ h ∗ X n −→ h ∗ X → h ∗ ( X, Z /n ) β −→ follows by applying h ∗ to the cofibration X ∧ S ∧ n −−→ X ∧ S → X ∧ M n obtaining the long exact sequence of a cofibration.That Ω C ∗ ( X, Z /n ) defined by SAC Z /n -manifolds agrees with this definition is proved by construct-ing this exact sequence directly for Z /n -manifolds getting a map and using the 5-lemma. Thiscompletes the discussion for Proposition 3.Now we can prove Proposition 4: Proposition 4 : For the homology theory,¯Ω C ∗ ( X ) ≡ Ω C ∗ ( X ) ⊗ Ω C ∗ ( pt ) Z, we have¯Ω C ∗ ( X, Z /n ) = Ω C ∗ ( X, Z /n ) ⊗ Ω C ∗ ( pt ) Z, where Ω C ∗ ( X, Z /n ) is defined by SAC Z /n -manifolds.12 roof of Proposition 4 :¯Ω C ∗ ( X, Z /n ) ≡ ¯Ω C ∗ +1 ( X ∧ M n ) ≡ Ω C ∗ +1 ( X ∧ M n ) ⊗ Ω C ∗ ( pt ) Z = Ω C ∗ +1 ( X, Z /n ) ⊗ Ω C ∗ ( pt ) Z by the above. This is one crucial place where ¯Ω being homology theory is used. Continuing Proof of Theorem AT : We combine the mod n character invariants of Construc-tion II and the fact that ¯Ω C ∗ ( X, Z /n ) are Z /n modules, i.e. each element has order n . K ∗ C ( X, Z ) pairing −−−−→ lim ← n Hom( ¯Ω C ∗ ( X, Z /n ) , Z n ) , from the Corollary to Proposition 2= lim ← n Hom( ¯Ω C ∗ ( X, Z /n ) , Q / Z ) , using the Z /n -module property= Hom(lim → n ¯Ω C ∗ ( X, Z /n ) , Q / Z )= Hom(Ω C ∗ ( X, Q / Z ) , Q / Z ) , because we have Z /n -modules, which implies by the universal properties of the finite completionfunctor ∧ , a commutative diagram K ∗ C ( X, Z ) ∧ −→ K ∗ C ( X, ˆ Z ) < | > ∧ −−−→ Hom( ¯Ω C ∗ X, Q / Z , Q / Z ) Id l l Id K ∗ C ( X, Z ) < | > −−→ Hom( ¯Ω C ∗ X, Q / Z , Q / Z ) .< | > ∧ is a map of cohomology theories which on a point maps ˆ Z → Hom( Q / Z , Q / Z ) by anisomorphism. Thus < | > ∧ is an isomorphism. Remark : Hom( ¯Ω ∗ C ( X, Z /n ) , Z n ) is a cohomology theory since Z n is an injective Z /n module.The inverse limit of finite cohomology theories is also a cohomology theory. Putting these to-gether yields Hom( ¯Ω ∗ C ( X, Q / Z ) , Q / Z ), is a cohomology theory. A second way is Q / Z is divisibleso Hom(homology theory, Q / Z ) satisfies exactness and thus all the axioms to be a cohomologytheory.We also have K ∗ C X < | > Z −−−→ Hom( ¯Ω C ∗ , Z ) defined by: for any closed even-dimensional SAC V f −→ X ,form the integer R V ch E Todd V where E is any complex bundle over X . One knows < | > Z ⊗ Q is an isomorphism of ( K ∗ C X ) ⊗ Q with Hom( ¯Ω C ∗ ( X, Q ) , Q ).Furthermore, for any complex bundle E , the diagram ∧ replaces an Abelian group by the inverse limit of its finite quotients. We referred to this construction as the “Pontryagin dual” cohomology theory [13] which we learned about fromDon Anderson who never published it to my knowledge. It is now referred to by specialists as the “Anderson dual”.It’s a great construction. C ∗ ( X, Q ) → ¯Ω C ∗ ( X, Q / Z ) < | E > Q ↓ ↓ < | E > Q / Z = diagram Q → Q / Z commutes.The map K ∗ C → { diagram } is a map of cohomology theories (see next Remark) which for a point is Q mod1 −−−→ Q / Z n ∈ Z n ↓ n , Q mod1 −−−→ Q / Z an isomorphism. Thus it is an isomorphism of functors. This proves Theorem AT. (cid:4) Remark : Diagrams are easily identified (using iso < | > ∧ ) with the kernel of( K ∗ C ( X ) ⊗ Q ) ⊕ K ∗ C ( X, ˆ Z ) ∆ −→ K ∗ C ( X, ˆ Z ) ⊗ Q where ∆ is the difference of the natural maps. By the fibre construction and the exactness of0 → kernel → ( ) ⊕ ( ) → K ∗ C ( X, ˆ Z ) ⊗ Q → §
3. The maps from differential K -theories to ˆ K -characters are bi-jections ( C -linear and unitary cases) The C -linear case in detail: Definition : A ˆ K -character on X is an additive over disjoint union function from (enriched closedodd-dimensional SACs mapping to X ) to ( C / Z ) satisfying properties i) and ii) of Remark 2 followingConstruction I in Section 1, using C -valued differential forms. Proposition 3 : The construction of a ˆ K -character for a complex vector bundle ( E, ∇ ) over X with C -linear connection in Section 1 only depends on the equivalence class of ( E, ∇ ) in differential K -theory, ˆ K ( X ) defined using C -valued differential forms. Proof : By Remark 2 property iii ) of Section 1, if CS ( ∇ , ∇ ′ ) is exact, the ˆ K -characters of ( E, ∇ )and ( E, ∇ ′ ) are equal. Since chE is additive, we can pass to the Grothdieck group of ( E, ∇ ) up to CS equivalence, which is the definition of ˆ K ( X ). (cid:4) heorem DG (Differential Geometry) : The natural map produced by Construction I of Sec-tion 1, ˆ KX → ˆ K -characters with values in C / Z , is an isomorphism, where ˆ KX is defined by C -linear connections and C -valued differential forms. Proof : From [5] backstopped by [4] for the odd form lemma in the C -linear case, we have thediagram 0 0 (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ K odd C ( X ) ∧ odd / ∧ oddintegrality ✲ ∧ evenintegrality d (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ i (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ ˆ ch (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ H odd ( X, C ) ˆ K even ( X ) H even ( X, C ) ❅❅❅❘ (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ i δ (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ ch (cid:0)(cid:0)(cid:0)✒ ker ˆ ch ✲ K even C ( X ) ❅❅❅❘ (cid:0)(cid:0)(cid:0)✒ C -linear connections and differential K -theorydefined using C -valued forms. From Construction I of Section 2, we get natural maps of thisdiagram into the diagram (whose exactness will be demonstrated)0 0 (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ Rational Hom( ¯Ω odd ( X, Q / Z ) , Q / Z ) ∧ odd / ∧ oddintegrality ✲ ∧ evenintegrality d (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ i (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ ˆ ch (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ Hom( ¯Ω odd X, C ) ˆ K even - characters Hom( ¯Ω even X, C ) ❅❅❅❘ mod1 (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ i δ (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ ch (cid:0)(cid:0)(cid:0)✒ Hom( ¯Ω C odd X, C / Z ) δ · i ✲ Rational Hom( ¯Ω even ( X, Q / Z ) , Q / Z ) ❅❅❅❘ (cid:0)(cid:0)(cid:0)✒ ∧ evenintegrality and ∧ oddintegrality are those closed forms C whose cohomology classes satisfy inte-grality conditions, namely R M T ( M ) C is an integer dim M even or odd respectively.Notational Diagram:8 ♠✒✑✓✏ ✒✑✓✏ ✲ ✒✑✓✏ (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ (cid:0)(cid:0)(cid:0)✒ ♠✒✑✓✏ ✒✑✓✏ ♠✒✑✓✏ (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ (cid:0)(cid:0)(cid:0)✒ ❅❅❅❘ ✒✑✓✏ ✲ ♠✒✑✓✏ Note to Reader: The double circles mean we enter the proof knowing we have isomorphisms atthese locations. We have to fight hard for positions 1 and 3, but only a little for positions 7 and 2.Here ¯Ω C ∗ denotes the Z / C ∗ ( X, Z ) = Ω C ∗ ( X, Z ) ⊗ Ω C ∗ ( pt ) Z = Ω C ∗ ( X, Z ) / ( V · x − Todd V · x )16here the ring of integers Z is a Ω C ∗ ( pt ) module via the ring homomorphism Ω C ∗ ( pt ) → Z providedby the Todd genus and “Rational Hom” is discussed again momentarily. Note: we also write ¯Ω C ∗ ( X )for ¯Ω C ∗ ( X, Z ), and ¯Ω C ∗ ( X, Q /Z ) for Ω C ∗ ( X, Q /Z ) ⊗ Ω C ∗ ( pt ) Z .The proof of Theorem DG depends on Theorem 2 : For X a finite complex K even C ( X ) is a finitely generated Abelian group whose torsionis identified by Constructions I and II to Hom(torsion ¯Ω C odd ( X, Z ) , C / Z ) and whose quotient bytorsion is identified to Hom( ¯Ω C even ( X, Z ) , Z ) . Also the same statements hold reversing even andodd.
Proof : In the Appendix to Section 2, one shows using
Z/n -manifolds and Construction II that onehas a bijection between K ∗ ( X ) and Rational Hom( ¯Ω ∗ ( X, Q / Z ) , Q / Z ) for ∗ even or odd. Recall anelement C in Rational Hom( , ) is an element C in Hom which is part of a commutative diagram β −→ ¯Ω C even ( X ) ⊗ Q −−→ ¯Ω C even ( X ) ⊗ Q → ¯Ω C even ( X, Q / Z ) β −→ ¯Ω C odd ( X, Z ) ↓ C Z ↓ C Q ↓ C → Z → Q → Q / Z → C Q exists given C , C Q must be unique. This follows since the difference of two would map tozero in Q / Z so it would factor through Z ⊂ Q which is impossible since the domain is a Q vectorspace.Also the unique C Q that fits with C determines C Z . Thus C determines C Q and C Q determines C Z .Conversely given any C Z it determines C Q by C Q = C Z ⊗ C Q , which in turn determines C partiallyon kernel β . Since Q / Z is divisible (and thus an “injective Z -module”) any such partial C extends(non-uniquely) to a full C . This proves the second part of Theorem 2.Note if given C , C Q were zero, then C factors through image β which is the torsion of ¯Ω C odd ( X, C ) . This proves the first part of Theorem 2. (cid:4)
Corollary : A cohomology class c in H even ( X, Q ) is the Chern character of a complex bundle over X if and only for every closed even-dimensional SAC mapping to X , V f −→ X , R f ∗ c Todd V is aninteger. A similar statement holds for the transgressed ch in U , odd-dimensional closed SACs in X , elements in H odd ( X, Q ) and maps X → U . Proof : This is just unraveling the statement of the second part of Theorem 2. The odd case followsusing the suspension isomorphism h ∗ ( X ) = h ∗ +1 (Σ X ) applied to K ∗ C and ¯Ω C ∗ ( X, Q / Z ). (cid:4) roof of Theorem DG : In the upper diagram, the diagonal sequences and the outer sequencesare exact by [5]. By the Corollary to Theorem 2, the maps at positions 7 (cid:13) and 2 (cid:13) are isomorphisms.It follows that 0 → (cid:13) → (cid:13) → (cid:13) → Claim : For the middle diagram, the upper sequence 8 (cid:13) → (cid:13) → (cid:13) → (cid:13) → (cid:13) is exact.Proof of Claim is below. Corollary of Claim : The map is an isomorphism at position 3 (cid:13) . Proof of Corollary of Claim : By the Appendix to Section 2, the map is an isomorphism at 4 (cid:13) and 8 (cid:13) . It is an isomorphism at 6 (cid:13) and 5 (cid:13) by direct inspection. Thus it is an isomorphism at 3 (cid:13) by the 5-lemma.
Second Corollary of Claim : The completion of the proof of Theorem DG. Apply the 5-lemmato 0 → (cid:13) → (cid:13) → (cid:13) →
0. QED for Theorem DG. (cid:4)
Proof of Claim : By the second part of Theorem 2 the odd case, the image of 8 (cid:13) → (cid:13) in the middlediagram is Hom( ¯Ω odd X, Z ) ⊂ Hom( ¯Ω odd X, C ). This is the kernel of 6 (cid:13) → (cid:13) . So we have exactnessat 6 (cid:13) . The image of 6 (cid:13) → (cid:13) is the component of the identity of the locally compact Abelian groupHom( ¯Ω C odd X, C / Z ). The quotient by the image is isomorphic to Hom(torsion ¯Ω C odd X, C / Z ). Thisquotient injects into 4 (cid:13) by the first part of Theorem 2. So we have exactness at 3 (cid:13) . The image of3 (cid:13) → (cid:13) is the torsion of 4 (cid:13) again by Theorem 2. This torsion is the kernel of 4 (cid:13) → (cid:13) by the proofof Theorem 2. So we have exactness at 4 (cid:13) . Corollary 1 of Proof of Theorem DG :kernel( ˆ KX ch −→ ∧ evenintegrality ) is isomorphic to Hom( ¯Ω C odd , C / Z ), a complex torus of dimension thesum of the odd Betti numbers of X . Corollary 2 of Proof of Theorem DG : The diagonal and outer sequences of the ˆ K -characterdiagram are exact. Note : Replacing C / Z by R / Z , C -linear connections by unitary connections, and complex valuedforms by real valued forms gives the proof of Theorem DG in that case mutatis mutandis. §
4. The Wrong Way Map on ˆK-characters for a Smooth SAC Fam-ily with Complex Linear Connection on the Vertical Stable TangentSpaces together with a Horizontal Connection
An enriched SAC cycle on the base determines an enriched SAC cycle in the total space by pullback.The stable tangent bundle of the pullback cycle in the total space has a natural direct sum splittingand the direct sum connection, where in the base directions we use the pullback of the connectionon the cycle in the base and in the vertical directions the induced complex connection. Now werestrict attention to even-dimensional SAC fibers over the base of the family.18 efinition of Wrong Way Map on ˆK-characters : Given a function t on enriched SAC cycleson the total space, we get a function b on enriched SAC cycles in the base by the obvious formula: b (base cycle) ≡ t (pulled-back cycle) . Proposition : If t satisfies the properties of a ˆ K -character on the total space then b satisfies theproperties of a ˆ K -character on the base. Proof : The Todd form of the pullback cycle in the total space is the wedge product of (the Toddform of the cycle in the base) with (the Todd form of the vertical).The same will be true for the Todd form of the pullback of a SAC enriched bordism deformationof the cycle in the base.If C ( t ) denotes the variation form of a ˆ K -character on T , define C ( b ) by the integration along thefibres of the product of C ( t ) with the vertical Todd form on the total space.If W fills in the base cycle V and ¯ W is the pullback fill in of the pulled-back cycle ¯ V , then theintegral of Todd ¯ W · C ( t ) over W computed by integrating along the fibres is seen to be the integral C ( b ) over W . This follows since if I denotes integration along the fibres and Π is the projection, I (Todd ¯ W ∧ C ( t )) = I ( Π ∗ Todd W ∧ Todd(vertical) ∧ C ( t ))= Todd W ∧ I (Todd(vertical) ∧ C ( t ))= Todd W ∧ C ( b ) . (cid:4) §
5. The Riemannian and Unitary Case and Eta Invariants of ( X, E, ∇ ) We will use the APS theorem [8] to compute (
V, F ) where F : V → X is an enriched SAC cyclein X . The invariant will be the eta invariant of the spin c Dirac operator on V with coefficients in F ∗ E reduced mod one. Now we assume the connection ∇ is unitary.First, a SAC bundle E has a canonically associated complex line bundle whose first Chern classreduces mod 2 to the second Stiefel-Whitney class of E . Proof : The top exterior power of a complex vector space U is canonically isomorphic to the topexterior power of U ⊕ C . So we have a line bundle (functorially) associated with any SAC bundle.Call this line bundle L . The first Chern class of L is the first Chern class of E which reduces mod2 to the second Stiefel-Whitney class of E . (cid:4) Second, applying this to the actual tangent bundle T ( V ) of a SAC sycle, form T ( V ) ⊗ L and itscomplex Clifford algebra bundle associated to a metric on T ( V ) and a U (1) metric on L . Thereis a complex bundle S which is fibrewise the irreducible complex Cifford module for the Cliffordalgebra on T ( V ) ⊗ L well-defined up to module isomorphism.19 roof : One knows this representation fact is equivalent to having a specific spin c lift of the SO ( n )structure on T ( V ), where spin c is the fibre product of the diagram SO ( n ) “ w ” −−−→ RP ∞ ← ֓ RP = U (1) . We have just seen using L that we have a homotopy commutative diagram V c ( L ) −−−→ K ( Z,
2) = BU (1) T ( V ) ↓ ↓ reduction mod 2 B SO ( n ) w −→ K ( Z , . So given a homotopy class of homotopies making it actually commutative we have a lift V → B spin c which is a fibre product of this diagram. Here B G means classifying space for G .This homotopy class of homotopies comes from the (rigidly) commutative diagram of structures, B SO ( n ) ✲ B SO ✲ w K ( Z/ , tangent ❅❅❅❘ V ✏✏✏✏✏✏✏✏✏✏✶ L ❅❅❅❅❅❅❅❅❘ inclusion ❄ mod 2 SAC (cid:0)(cid:0)(cid:0)✒ B U ✲ c B U (1) ✲ = K ( Z, c Dirac operator on the complex spinors, the sections of S . This operatorcombines the Clifford multiplications with covariant derivatives of the induced unitary connectionon S (see [8]).By the discussion in [8], one has the spectral eta invariants of this operator with coefficients in anyunitary bundle when the dimension of V is odd. By the celebrated theorem in [8], this real numberdefined by eigenvalues of Dirac with coefficients, zeta functions thereof and analytic continuationto zero differs by an integer from the integral of (Todd W · ch E ) over W where W is SAC, ∇ on E is unitary, Dirac has coefficients in E and boundary ( W ) = V . Corollary : (Eta form of Theorem DG)The complex angle invariants in C / Z for the complex bundle E over X with unitary connection liein R / Z and can be defined directly for odd-dimensional SAC cycles F : V → X in X enriched byLevi-Civita connections on T V and a unitary connection on the canonical complex line bundle L over V using the eta invariant reduced mod 1 of the spin c Dirac operator with coefficients in F ∗ E to define a bijectiondifferential K -theory eta ←→ differential ˆ K -characters . (real forms) (values in R / Z )20imilarly, we can give an eta computation of the push forward.Using the result of the Appendix to Section 5 we will be able to work with both connections onthe Riemannian fibration together with a unitary connection on the line bundle L associated to theSAC structure in the vertical tangent bundle.The direct sum connection by definition computes the push forward for bundles with unitaryconnection. The eta invariants of the spin c Dirac operator relative to the rescaled Levi-Civitaconnections on the total space converge to these push forward values as the base becomes infinitelylarge relative to the fibre.This proves the Analytic Theorem:
Theorem AN : The invariants of the push forward of ( E, ∇ ) for E SAC and ∇ unitary arecomputed by the limits of eta invariants mod one of the rescaled Levi-Civita connections as theyconverge to their adiabatic limit. Appendix to § If Π : F → M is a fibration of riemannian manifolds such that Π is a Riemannian submersion,two natural connections are present in T ( F ). The first is the Riemannian connection, ∇ r , andthe second, ∇ ⊕ , is a direct sum connection on the vertical tangent bundle and its orthogonalcomplement.Under a stretching of the base by multiplying its metric by a constant λ , and carrying this throughto the metric on F so that Π remains a Riemannian submersion, ∇ r changes to a connection denotedby ∇ λr . ∇ ⊕ however remains fixed.In this Appendix we show that lim λ →∞ ∇ λr = ˜ ∇ r is a well defined connection, and also show that ˜ ∇ r and ∇ ⊕ are equivalent. This, in the sense that the CS terms relating the characteristic forms ofthe two connections are all exact. We refer to Section 0 for notation.
2. Riemannian Fibrations
Let F Π −→ M be a smooth fibration over a smooth manifold, the fibers of which are Riemannianmanifolds. x ∈ T ( F ) will be called vertical if Π ∗ ( x ) = 0. The collection of vertical vectors forms asub-bundle V ⊆ T ( F ). Clearly V| F m = T ( F m ). V is a Riemannian vector bundle, and we shall use <, > to denote the inner product on its fibers.Now suppose we are given H ⊆ T ( F ) a complementary sub-bundle to V . I.e. T ( F ) ∼ = V ⊕ H .Elements of H will be called horizontal , as will vector fields on F all of whose elements are21orizontal. Clearly a vector field J on M may be lifted to a unique horizontal field Π ∗ ( J ) on F .Such horizontal fields on F will be called special . The following is well known. Lemma 2.1 : Let
H, I be special horizontal fields on F , and X a vertical field. a ) [ H, X ] is vertical b ) [ H, I ] = Π ∗ ([ Π ∗ ( H ) , Π ∗ ( I )]) + vertical Proof : Because H is special, the 1-parameter flow induced by H takes fibers to fibers. a) is theinfinitesimal version of this observation. To see b), let f ∈ C ∞ ( M ). For p ∈ F , the fact that H, I are special shows[
H, I ]( p )( Π ∗ ( f )) = [ Π ∗ ( H ) , Π ∗ ( I )]( Π ( p ))( f )= ⇒ Π ∗ ([ H, I ]) = Π ∗ ( Π ∗ ([ Π ∗ ( H ) , Π ∗ ( I )])) which implies b). (cid:4) The Riemannian connection on the tangent bundles of the fibers of F may be extended to an innerproduct preserving connection, ∇ V , on V over all of F as follows:Let X, Y, Z be vertical vector fields on F , and H a special horizontal field. Let ∇ denote theRiemannian connection on the fibers. Set2 . < ∇ V X Y, Z > = < ∇ X Y, Z >< ∇ V H Y, Z > = { < [ H, Y ] , Z > − < [ H, Z ] , Y > + H ( < Y, Z > ) } . Direct calculation shows that ∇ V is a well defined connection on V , and that ∇ V preserves <, > .Let us now suppose that M itself is a Riemannian manifold. Since H ∼ = Π ∗ ( T ( M )), the metricand the Riemannian connection on T ( M ) induce an inner product and connection on H , denotedrespectively by <, > and ∇ H . Set ∇ ⊕ = ∇ V ⊕ ∇ H . By making V and H orthogonal, <, > becomes a positive definite inner product on T ( F ), on whichit induces the Riemannian connection ∇ r . We wish to compare ∇ r and ∇ ⊕ , two metric preservingconnections on T ( F ). Letting Skew( T ( F )) denote skew symmetric endomorphisms, we set B = ∇ r − ∇ ⊕ ∈ ∧ ( F, Skew( T ( F ))) . Let
X, Y, Z be vertical vector fields and
H, I, J special horizontal fields.22 roposition 2.3 : Assume we are working in a neighborhood where the inner products of all theabove pairs are constant. Then1) < B X Y, Z > = 02) < B H Y, Z > = 03) < B X Y, H > = { < [ H, X ] , Y > + < X, [ H, Y ] > } < B H Y, I > = − < [ H, I ] , Y > < B X I, Z > = − { < [ I, X ] , Z > + < X, [ I, Z ] > } < B H I, Z > = < [ H, I ] , Z > < B X I, J > = − < [ I, J ] , X > < B H I, J > = 0
Proof : We recall the Koszul formula for the Riemannian connection, as applied to the case oftriples of vector fields, the pair-wise inner products of which are constant.2 . < ∇ W W , W > = 12 { < [ W , W ] , W > + < [ W , W ] , W > + < [ W , W ] , W > } To show 1) we note that < B X Y, Z > = < ∇ rX Y, Z > − < ∇ V X Y, Z > .
By definition of ∇ V , it was the extension of the Riemannian connection on the fibers of F to allof T ( F ). Since the Riemannian connection on a submanifold is simply its orthogonal projection tothe sub-tangent bundle, < ∇ rX Y, Z > = < ∇ V X Y, Z > .2) follows immediately by comparing 2.2) to 2.4) and using the fact that inner products of ourfields are constant.3) follows from 2.4) by noting that ∇ ⊕ preserves each of V and H , as does 4) via a) of Lemma 2.1.5) and 6) follow from 3) and 4) respectively, using the skew symmetric action of the values of B .To show 7), we note that ∇ H is the pull-back under Π of the Riemannian connection of T ( M ), andthus, since Π ∗ ( X ) = 0, ∇ H X = 0. The rest follows from 2.4) and a) of Lemma 2.1.To show 8), we use b) of Lemma 2.1, and use 2.4) on both T ( F ) and T ( M ). Together, that showsthat < ∇ H H I, J > = < ∇ rH I, J > . (cid:4) . The Adiabatic Connection We now stretch M by considering the 1-parameter family of metrics <, > λ = λ <, > , where λ ∈ [1 , ∞ ). Lifting this to F we see for X, Y vertical and
H, I horizontal3 . < X, Y > λ = < X, Y >, < X, H > λ = 0 , < H, I > λ = λ < H, I > . Since ∇ V is independent of a metric on M , and since the Riemannian connection on T ( M ) isunchanged under a constant conformal change of metric, ∇ ⊕ = ∇ V ⊕ ∇ H is invariant as λ changes.The Riemannian connection on T ( F ) does change, however, and and we denote this family ofconnections by {∇ λr } . Theorem 3.2 : Set ˜ ∇ r = lim λ →∞ ∇ λr . Then, ˜ ∇ r is well defined, and ˜ ∇ r is equivalent to ∇ ⊕ . Proof : Let B λ = ∇ λr − ∇ ⊕ ∈ ∧ ( F, End( T ( F ))). From 3.1) and Proposition 2.3) we see1) < B λX Y, Z > = 02) < B λH Y, Z > = 03) < B λX Y, H > = λ < B λX Y, H > λ = λ < B X Y, H > < B λH Y, I > = λ < B λH Y, I > λ = λ < B H Y, I > < B λX I, Z > = < B X I, Z > < B λH I, Z > = < B H I, Z > < B λX I, J > = λ < B λX I, J > λ = λ < B X I, J > < B λH I, J > = 0Setting ˜ B = lim λ →∞ B λ , we see from the above3 .
3) ˜ B s |V = 0 and ˜ B s ( H ) = ( B s | H ) V where s is any tangent vector to F , and ( ) V means projection into V . Thus3 .
4) ˜ ∇ r = ∇ ⊕ + ˜ B implying ˜ ∇ r is well defined.Let [ ˜ B, ˜ B ] ∈ ∧ ( F, End( T ( F ))) be defined as usual, i.e. [ ˜ B, ˜ B ]( x, y ) = [ B x , B y ] . By 3.3)3 .
5) [ ˜ B, ˜ B ] = 0 . d denote exterior differentiation with respect to ∇ ⊕ of forms on F taking values in End( T ( F )).Since ∇ ⊕ preserves V and H , 3.3) shows, for any s , u tangent to F . d ˜ B s,u |V = 0 and d ˜ B s,u ( H ) ⊆ V . For t ∈ [0 , γ ( t ) = ∇ ⊕ + t ˜ B , a curve of connections joining ∇ ⊕ to ˜ ∇ r .Let R t denote the curvature tensor of γ ( t ). By the usual formula R t = R + td ˜ B + t [ ˜ B, ˜ B ]and by 3.5)3 . R t = R + td ˜ B. Since ddt ( γ ( t )) = ˜ B , following 1.2) in § TP l ( ∇ ⊕ , ˜ ∇ r ) consists of integrals of terms of the formtr( ˜ B s R ts ,s · · · R ts l − ,s l − ) . Since R s i ,s j preserves V and H , by 3.3), 3.6) and 3.7) we see that the endomorphism inside theparentheses is either 0 or takes H → V and
V →
0. In either case its trace is 0. Thus, TP l ( ∇ ⊕ , ˜ ∇ r ) = 0and thus by Proposition 1.5 in Section 0, ∇ ⊕ ∼ ˜ ∇ r . (cid:4) References