G-Homotopy Invariance of the Analytic Signature of Proper Co-compact G-manifolds and Equivariant Novikov Conjecture
GG -Homotopy Invariance of the Analytic Signature of ProperCo-compact G -manifolds and Equivariant Novikov Conjecture Yoshiyasu Fukumoto ∗ Research Center for Operator Algebras, Department of Mathematics,East China Normal University
Abstract
The main result of this paper is the G -homotopy invariance of the G -index of signature operatorof proper co-compact G -manifolds. If proper co-compact G manifolds X and Y are G -homotopyequivalent, then we prove that the images of their signature operators by the G -index map arethe same in the K -theory of the C ∗ -algebra of the group G . Neither discreteness of the locallycompact group G nor freeness of the action of G on X are required, so this is a generalization ofthe classical case of closed manifolds. Using this result we can deduce the equivariant version ofNovikov conjecture for proper co-compact G -manifolds from the Strong Novikov conjecture for G . Mathematics Subject Classification (2010).
Keywords.
Novikov conjecture, Higher signatures, Almost flat bundles
Contents G -signature 8 G -index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 On proof of Corollary B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 G -index map in KK -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Infinite product of C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Index of the product bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Proof of Theorem C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 On proof of Corollary D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ∗ Research Center for Operator Algebras, Department of Mathematics, East China Normal University3663 North ZhongShan Road, Shanghai, CHINA. [email protected] a r X i v : . [ m a t h . K T ] S e p G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds Introduction
Before discussing on our case of proper G -action, let us review the classical case of closed manifolds.For even dimensional oriented closed manifold M , The ordinary Fredholm index of the signatureoperator ∂ M is equal to the signature of the manifold M which is defined using the cup product ofthe ordinary cohomology of M . In particular it follows that ind( ∂ M ) is invariant under orientationpreserving homotopy. We have the following classical and important result; Theorem 0.1 [Kas75], [KM], [HiSk]
Let M and N be even dimensional oriented closed manifoldswith fundamental group Γ = π ( M ) = π ( N ) . Assume that M and N are orientation preservinghomotopy equivalent to each other. Then ind Γ ( ∂ M ) = ind Γ ( ∂ N ) ∈ K (Γ) . Notice that we can deduce the Novikov conjecture from the Strong Novikov conjecture by using thistheorem. Moreover, we also have a more generalized result;
Theorem 0.2 [RW, 3.3. PROPOSITION and 3.6. THEOREM]
Let a finite group Γ acts on M and N and let Γ = π ( M ) = π ( N ) . Let ind G Γ be the G -equivariant Γ -index map with value in K G ( C ∗ red (Γ)) (cid:39) K ( C ∗ red ( G Γ )) , where G Γ denotes the group extension { } → Γ → G Γ → G →{ } . Assume that M and N are orientation preserving Γ -equivariantly homotopy equivalent. Then ind G Γ ( ∂ M ) = ind G Γ ( ∂ N ) ∈ K ( C ∗ red ( G Γ )) . Our main theorem is a generalization of them. Let us fix the settings. Let X and Y be orientedeven-dimensional complete Riemnannian manifolds and let G be a second countable locally compactHausdorff group acting on X and Y isometrically, properly and co-compactly. Theorem A
Let X and Y be oriented even-dimensional complete Riemnannian manifolds and let G be a second countable locally compact Hausdorff group acting on X and Y isometrically, properlyand co-compactly. Let ∂ X and ∂ Y be the signature operators. Assume the we have a G -equivariantorientation preserving homotopy equivalent map f : Y → X .Then ind G ( ∂ X ) = ind G ( ∂ Y ) ∈ K ( C ∗ ( G )) . This claim is also stated in [BCH] without proofs and here we will give a proof for it to obtain CorollaryB. The method we use in this paper is based on [HiSk], so we will construct a map that sends ind G ( ∂ X )to ind G ( ∂ Y ). Our group C ∗ -algebras can be either maximal one or reduced one.Theorem 0.1 is the case when X and Y are the universal covering of closed manifolds M and N .Thus, analogously to the case of closed manifolds, the equivariant version of the Novikov conjecturecan be deduced from the Strong Novikov conjecture for the acting group G . In particular, by usingthis theorem and the result discussed in [F], we obtain the following equivariant version of Novikovconjecture for low dimensional cohomologies: Corollary B
Let X , Y and G as above and let L be a G -hermitian line bundles over X which isinduced from a G -line bundle over E G , or more generally, G -hermitian line bundle L over X satisfying c ( L ) = 0 ∈ H ( X ; R ) . Suppose, in addition, that G is unimodular and H ( X ; R ) = H ( Y ; R ) = { } .Then, (cid:90) X c X ( x ) L ( T X ) ∧ ch( L ) = (cid:90) Y c Y ( y ) L ( T Y ) ∧ ch( f ∗ L ) , where c X denotes the cut-off function, that is, c X is a R ≥ -valued compactly supported function on X satisfying (cid:82) G c ( γ − x ) d γ = 1 for any x ∈ X . In the case of the closed manifold, that is, when X isobtained as the universal covering of a closed manifold M , and the acting group is the fundamentalgroup, the above value is equal to the ordinary, so called, higher signature (cid:104)L ( T X ) ∪ ch( L ) , [ M ] (cid:105) .The same result in this case of closed manifolds was obtained in [Ma] and [HaSc]. . Fukumoto G -homotopy invariance of the analytic signature twistedby almost flat bundles as in [HiSk, Section 4.]. However we will use a different method from [HiSk] todeal with general G -invariant elliptic operators. To be specific, we will prove the following TheoremC to obtain Corollary D. Theorem C
Let X be a complete oriented Riemannian manifold and let G be a locally compactHausdorff group acting on X isometrically, properly and co-compactly. Moreover we assume that X is simply connected. Let D be a G -invariant properly supported elliptic operator of order on G -Hermitian vector bundle over X .Then there exists ε > satisfying the following: for any finitely generated projective Hilbert B -module G -bundle E over X equipped with a G -invariant Hermitian connection such that (cid:13)(cid:13) R E (cid:13)(cid:13) < ε ,we have ind G (cid:16) [ E ] (cid:98) ⊗ C ( X ) [ D ] (cid:17) = 0 ∈ K ( C ∗ Max ( G ) ⊗ Max B ) if ind G ([ D ]) = 0 ∈ K ( C ∗ Max ( G )) . If we only consider commutative C ∗ -algebras for B , then the sameconclusion is also valid for C ∗ red ( G ) . Corollary D
Consider the same conditions as Theorem A on X , Y and G and assume additionallythat X and Y are simply connected.Then there exists ε > satisfying the following: for any finitely generated projective Hilbert B -module G -bundle E over X equipped with a G -invariant Hermitian connection such that (cid:13)(cid:13) R E (cid:13)(cid:13) < ε ,we have ind G ([ E ] (cid:98) ⊗ [ ∂ X ]) = ind G ([ f ∗ E ] (cid:98) ⊗ [ ∂ Y ]) ∈ K ( C ∗ Max ( G ) (cid:98) ⊗ Max B ) . If we only consider commutative C ∗ -algebras for B , then the same conclusion is also valid for C ∗ red ( G ) . Definition 1.1
Let G be a second countable locally compact Hausdorff group. Let X be a completeRiemannian manifold. • X is called a G -Riemannian manifold if G acts on X isometrically. • The action of G on X is said to be proper or X is called a proper G -space if the followingcontinuous map is proper: X × G → X × X, ( x, γ ) (cid:55)→ ( x, γx ) . • The action of G on X is said to be co-compact or X is called G -compact space if the quotientspace X/G is compact.
Definition 1.2
The action of G on X induces actions on T X and T ∗ X given by γ : T x X → T γx Xv (cid:55)→ γ ( v ) := γ ∗ v and γ : T ∗ x X → T ∗ γx Xξ (cid:55)→ γ ( ξ ) := ( γ − ) ∗ ξ. The action on X ( X ) and Ω ∗ ( X ) is given by γ [ V ] := γ ∗ V and γ [ ω ] := ( γ − ) ∗ ω for γ ∈ G , V ∈ X ( X ) and ω ∈ Ω ∗ ( X ). Obviously, γ [ ω ∧ η ] = γ [ ω ] ∧ γ [ η ] and d( γ [ ω ]) = γ [d ω ]. Proposition 1.3 (Slice theorem)
Let G be a second countable locally compact Hausdorff group andact properly and isometrically on X . Then for any neighborhood O of any point x ∈ X there exists acompact subgroup K ⊂ G including the stabilizer at x , K ⊃ G x := { γ ∈ G | γx = x } and there existsa K -slice { x } ⊂ S ⊂ O . G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds Here S ⊂ X is called K -slice if the followings are satisfied; • S is K -invariant; K ( S ) = S , • the tubular subset G ( S ) ⊂ X is open, • there exists a G -equivariant map ψ : G ( S ) → G/K satisfying ψ − ([ e ]) = S , called a slice map. Corollary 1.4
We additionally assume that X is G -compact. Then for any open covering X = (cid:83) x ∈ X O x , there exists a sub-family of finitely many open subsets { O x i , . . . O x N } such that (cid:91) γ ∈ G N (cid:91) i =1 γ ( O x i ) = X. In particular, X is of bounded geometry, namely, the injective radius is bounded below and thenorm of Riemannian curvature is bounded. Lemma 1.5
Let X and Y be manifolds on which G acts properly. Suppose that the action on Y isco-compact. Let f : Y → X be a G -equivariant continuous map. Then f is a proper map.Proof. Since the action on Y is co-compact, there exists a compact subset F ⊂ Y satisfying G ( F ) = Y .Fix a compact subset C ⊂ X and assume that the closed set f − C ⊂ Y is not compact. Then thereexists a sequence { y j } ⊂ f − C tending to the infinity, that is, any compact subset in Y containsonly finitely many points of { y j } . Since the action on Y is proper, there exists a sequence { γ j } ⊂ G tending to the infinity satisfying y j ∈ γ j F . Then it follows that f ( y j ) ∈ f ( γ j F ) = γ j f ( F ). Due tothe compactness of f ( F ) ⊂ X and the properness of the action on X , the sequence { f ( y j ) } ⊂ X tends to the infinity. However, the compact subset C cannot contain such a sequence. So, f − C iscompact. In this section, we will discuss on some technical method introduced in [HiSk, Section 1 and 2]. Fornow, we will forget about the manifolds and group actions. Let A be a C ∗ -algebra, which may not beunital. Especially we will consider A = C ∗ ( G ). Let E be a Hilbert A -module equipped with A -valuedscalar product (cid:104)· , ·(cid:105) . Let us fix some notations: • L ( E , E ) denotes a space consisting of adjointable A -linear operators, and we also use L ( E ) := L ( E , E ). • K ( E , E ) denotes a sub-space of L ( E , E ) consisting of compact A -linear operators, namely, thenorm closure of the space of operators whose A -rank are finite. We also use K ( E ) := K ( E , E ). Definition 2.1 (Regular quadratic forms) Q : E × E → A is called a quadratic form on E if itsatisfies Q ( ξ, ν ) = Q ( ν, ξ ) ∗ and Q ( ν, ξa ) = Q ( ν, ξ ) a for ν, ξ ∈ E , a ∈ A. (2.1)A quadratic form Q is said to be regular if there exists a invertible operator B ∈ L ( E ) satisfying that Q ( ξ, Bν ) = (cid:104) ξ, ν (cid:105) .For an operator T ∈ L ( E ), let T (cid:48) denote the adjoint with respect to Q , that is, an operatorsatisfying that Q ( T ξ, ν ) = Q ( ξ, T (cid:48) ν ). Using B , it is written as T (cid:48) = BT ∗ B − . . Fukumoto Definition 2.2 (Compatible scalar product)
Another scalar product (cid:104)· , ·(cid:105) : E × E → A is calledcompatible with (cid:104)· , ·(cid:105) if there exists a linear bijection P : E → E satisfying that (cid:104) ν, ξ (cid:105) = (cid:104) ν, P ξ (cid:105) .Note that P is a positive operator with respect to both of the scalar product and √ P : ( E , (cid:104)· , ·(cid:105) ) → ( E , (cid:104)· , ·(cid:105) ) is a unitary isomorphism. In particular, neither the spaces L ( E ) nor K ( E ) depends on thechoice of compatible scalar product. Lemma 2.3
Let Q be a regular quadratic form on E . then there exist a compatible scalar product (cid:104)· , ·(cid:105) Q with the initial scalar product of E and U ∈ L ( E ) satisfying that Q ( ξ, U ν ) = (cid:104) ξ, ν (cid:105) Q and U = 1 .Moreover they are unique.Proof. With respect to the initial scalar product (cid:104)· , ·(cid:105) , we have that (cid:10) ν, B − ξ (cid:11) = Q ( ν, ξ ) = Q ( ξ, ν ) ∗ = (cid:10) ξ, B − ν (cid:11) ∗ = (cid:10) B − ν, ξ (cid:11) = (cid:10) ν, ( B − ) ∗ ξ (cid:11) , which implies that B − is an invertible self-adjoint operator. Thus, it has the polar decomposition B − = U P in which B − , U and P commute one another, here U is unitary and P is positive. To bespecific, U and P are given by the continuous functional calculus. Let f and g be continuous functionsgiven by f ( x ) := x | x | and g ( x ) := | x | on the spectrum of B − , which is contained in R \ { } , and set U := f ( B − ) and P := g ( B − ). Note that U = P − B − = P − ( B − ) ∗ = P − P U ∗ = U ∗ , so it follows that U = U ∗ U = 1. Let us set (cid:104) ν, ξ (cid:105) Q := (cid:104) ν, P ξ (cid:105) . Then, Q ( ν, U ξ ) = Q ( ν, U − ξ ) = Q ( ν, BP ξ ) = (cid:104) ν, P ξ (cid:105) = (cid:104) ν, ξ (cid:105) Q . If there is another such operator U satisfying that U = 1 and that Q ( ν, U ξ ) is another scalarproduct, then U − U is a positive unitary operator, which implies that U − U = 1. Thus we obtainedthe uniqueness. Remark 2.4
A regular quadratic form Q on a Hilbert A -module E determines the renewed compatiblescalar product (cid:104)· , ·(cid:105) Q associated to Q and the ( Z / Z )-grading given by the ± U .Conversely, if a Hilbert A -module E is equipped with a ( Z / Z )-grading, then it determines a regularquadratic form Q given by Q ( ν, ξ ) = (cid:10) ν, ( − deg( ξ ) ξ (cid:11) for homogeneous elements. Definition 2.5
Let A be a C ∗ -algebra. J ( A ) denotes the space consisting of unitary equivalentclasses of triples ( E , Q, δ ), where E is a Hilbert A -module, Q is a regular quadratic form on E and δ : dom( δ ) → E is a densely defined closed operator satisfying the following conditions;(1) δ (cid:48) = − δ , namely, Q ( − δ ( ν ) , ξ ) = Q ( ν, δ ( ξ )) for ν, ξ ∈ dom( δ ).(2) Im( δ ) ⊂ dom( δ ) and δ = 0.(3) There exists σ, τ ∈ K ( E ) satisfying σδ + δτ − ∈ K ( E ).The typical example, which we will use for dealing with the signature, is given by Definition 3.7.Roughly speaking, E is a completion of the space of compactly supported differential forms Ω ∗ c , Q isgiven by the Hodge ∗ -operation and δ is the exterior derivative. Remark 2.6
This definition is slightly different from L nb ( A ) in [HiSk, 1.5 D´efinition] and our J ( A )is smaller. However it is sufficient for our purpose. G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds Lemma 2.7
If a closed operator δ satisfies the condition (3), then both operators ( δ + δ ∗ ± i ) − canbe defined and they belong to K ( E ) . Here, δ ∗ denotes the adjoint of δ with respect to a certain scalarproduct on E .Proof. Since δ is a closed operator, δ + δ ∗ is self-adjoint. Thus Im( δ + δ ∗ ± i ) are equal to E andboth operators δ + δ ∗ ± i are invertible. We now claim that both ( δ + δ ∗ ± i ) − ∈ L ( E ) are compactoperators. Since Im(( δ + δ ∗ ± i ) − ) = dom( δ + δ ∗ ± i ) = dom( δ ) ∩ dom( δ ∗ ) and δ and δ ∗ are closedoperators, the following operators α ± := δ ( δ + δ ∗ ± i ) − and β ± := δ ∗ ( δ + δ ∗ ± i ) − are closed operator defined on entire E , which implies that they are bounded; α, β ∈ L ( E ).On the other hand, note that ( σδ ) = ( σδ )(1 − δτ ) = σδ and ( δτ ) = δτ modulo K ( E ). Let p bethe orthogonal projection onto Im( δτ ) and let q = 1 − p . Then we have that p ( δτ ) = δτ and ( δτ ) p = p modulo K ( E ). Moreover, ( σδ ) q = (1 − δτ )(1 − p ) = 1 − δτ − p + ( δτ ) p = 1 − δτ = σδ,q ( σδ ) = (1 − p )(1 − δτ ) = 1 − p − δτ + p ( δτ ) = 1 − p = q, − ( δ ∗ σ ∗ ) q − ( δτ ) p = 1 − ( qσδ ) ∗ − p = 1 − q ∗ − p = 1 − q − p = 0 modulo K ( E ) . Then, set (cid:96) := 1 − ( δ ∗ σ ∗ ) q − ( δτ ) p ∈ K ( E ). Now we conclude that1 = (cid:96) + ( δ ∗ σ ∗ q − δτ p ) , ( δ + δ ∗ ± i ) − = ( δ + δ ∗ ± i ) − (cid:96) + ( α ∗∓ σ ∗ q − β ∗∓ τ p ) ∈ K ( E )because (cid:96) , σ and τ belong to K ( E ) and α ± and β ± belong to L ( E ). Definition 2.8
For ( E , Q, δ ) ∈ J ( A ), we define the K -homology class Ψ( E , Q, δ ) ∈ K ( A ) as follows.As in Lemma 2.3, let E be equipped with the compatible scalar product (cid:104)· , ·(cid:105) Q and ( Z / Z )-gradingassociated to Q . Next, put F δ := ( δ + δ ∗ ) (cid:0) δ + δ ∗ ) (cid:1) − ∈ L ( E ) , where δ ∗ is the adjoint of δ with respect to the scalar product (cid:104)· , ·(cid:105) Q . Obviously F δ is self-adjoint and F δ is an odd operator since U δU = δ (cid:48) = − δ . Moreover it follows that1 − F δ = (cid:0) δ + δ ∗ ) (cid:1) − ∈ K ( E )by the previous lemma. Then we define Ψ( E , Q, δ ) := ( E , F δ ) ∈ KK ( C , A ) ∼ = K ( A ). The action of C on E is the natural multiplication. Lemma 2.9
For ( E , Q, δ ) ∈ J ( A ) satisfying Im( δ ) = Ker( δ ) , Ψ( E , Q, δ ) = 0 ∈ K ( A ) .Proof. First, remark that Im( δ ) and Ker( δ ∗ ) are orthogonal to each other, and hence, Im( δ ) ∩ Ker( δ ∗ ) = { } . Indeed, for δ ( η ) ∈ Im( δ ) and ν ∈ Ker( δ ∗ ), it follows that (cid:104) δ ( η ) , ν (cid:105) = (cid:104) η, δ ∗ ( ν ) (cid:105) = 0. Now let ξ ∈ Ker( δ + δ ∗ ). Then0 = (cid:10) ξ, ( δ + δ ∗ ) ( ξ ) (cid:11) = (cid:104) ξ, δ ∗ δ ( ξ ) + δδ ∗ ( ξ ) (cid:105) = (cid:104) δ ( ξ ) , δ ( ξ ) (cid:105) + (cid:104) δ ∗ ( ξ ) , δ ∗ ( ξ ) (cid:105) , which implies that ξ ∈ Ker( δ ) ∩ Ker( δ ∗ ) = Im( δ ) ∩ Ker( δ ∗ ) = { } . Therefore, Ker( F δ ) = { } . Since F δ is a bounded self-adjoint operator, it is invertible. To conclude, ( E , F δ ) = 0 ∈ KK ( C , A ). . Fukumoto Lemma 2.10 [HiSk, 2.1. Lemme]
Let ( E X , Q X , δ X ) , ( E Y , Q Y , δ Y ) ∈ J ( A ) . Suppose that we have...(1) T ∈ L ( E X , E Y ) satisfying T (dom( δ X )) ⊂ dom( δ Y ) , T δ X = δ Y T and T induces an isomorphism [ T ] : Ker( δ X ) / Im( δ X ) → Ker( δ Y ) / Im( δ Y ) ;(2) φ ∈ L ( E X ) satisfying φ (dom( δ X )) ⊂ dom( δ X ) and − T (cid:48) T = δ X φ + φδ X ;(3) ε ∈ L ( E X ) satisfying ε = 1 , ε (cid:48) = ε , εδ X = − δ X ε and ε (1 − T (cid:48) T ) = (1 − T (cid:48) T ) ε .Then, Ψ( E X , Q X , δ X ) = Ψ( E Y , Q Y , δ Y ) ∈ K ( A ) .Proof. First, we may assume that φ (cid:48) = − φ . Indeed, since 1 − T (cid:48) T = (1 − T (cid:48) T ) (cid:48) = ( δ X φ + φδ X ) (cid:48) = − ( δ X φ (cid:48) + φ (cid:48) δ X ), we may replace φ by ( φ − φ (cid:48) ) which satisfies the same assumption.Set E := E X ⊕E Y , Q := Q X ⊕ ( − Q Y ) and ∇ := (cid:20) δ X − δ Y (cid:21) . Note that the replacing of Q Y by − Q Y means the reversing of the grading of E Y . Then it is easy to see that Ψ( E , Q, ∇ ) = Ψ( E X , Q X , δ X ) − Ψ( E Y , Q Y , δ Y ). Therefore it is sufficient to verify that Ψ( E , Q, ∇ ) = 0.Let us introduce invertible operators R t ∈ L ( E ) and a quadratic form B t on E given by the formula: R t := (cid:20) itT ε (cid:21) and B t ( ν, ξ ) := Q ( R t ν, R t ξ ) = Q ( R (cid:48) t R t ν, ξ )for t ∈ [0 , E , B t , ∇ ) ∈ J ( A ).It is easy to see that ∇ R t = R t ∇ , and hence, B t ( ν, ∇ ξ ) = B t ( −∇ ν, ξ ) . Clearly the scalar productsassociated to B t and Q are compatible with each other, also the condition (2) and (3) in the definitionof J ( A ) are satisfied. Therefore ( E , B t , ∇ ) ∈ J ( A ) and Ψ( E , B t , ∇ ) = Ψ( E , Q, ∇ ).Next let us introduce L t := (cid:20) − T (cid:48) T ( iε + tφ ) T (cid:48) T ( iε + tφ ) 1 (cid:21) and C t ( ν, ξ ) := Q ( L t ν, ξ ) . Notice that since Q = Q X ⊕ ( − Q Y ) and that T (cid:48) denotes the adjoint of T with respect to Q X and Q Y , the adjoint of the matrix (cid:20) T (cid:21) with respect to Q is equal to (cid:20) − T (cid:48) (cid:21) . Thus we have that R (cid:48) t = (cid:20) itεT (cid:48) (cid:21) and that R (cid:48) R = (cid:20) − εT (cid:48) T ε iεT (cid:48) iT ε (cid:21) = (cid:20) ε (1 − T (cid:48) T ) ε iεT (cid:48) iT ε (cid:21) = (cid:20) (1 − T (cid:48) T ) ε iεT (cid:48) iT ε (cid:21) = L . In particular, B = C . Since L t is invertible at t = 0, there exists t > L t is invertible for t ∈ [0 , t ]. Besides it is clear that L (cid:48) t = L t , so C t is a regular quadratic form for t ∈ [0 , t ].Moreover consider the operator ∇ t := (cid:20) δ X tT (cid:48) − δ Y (cid:21) , and we claim that ( E , C t , ∇ t ) ∈ J ( A ). for t ∈ [0 , t ]. The adjoint of ∇ t with respect to the quadratic form C t is equal to L − t ∇ (cid:48) t L t so in order tocheck that it is equal to −∇ t , we should check that L t ∇ t = −∇ (cid:48) t L t . L t ∇ t = (cid:20) (1 − T (cid:48) T ) δ X t (1 − T (cid:48) T ) T (cid:48) − ( iε + tφ ) T (cid:48) δ Y T ( iε + tφ ) δ X tT ( iε + tφ ) T (cid:48) − δ Y (cid:21) ∇ (cid:48) t L t = (cid:20) − δ X (1 − T (cid:48) T ) − δ X ( iε + tφ ) T (cid:48) − tT (1 − T (cid:48) T ) − δ Y T ( iε + tφ ) − tT ( iε + tφ ) T (cid:48) + δ Y (cid:21) G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds Obviously the (1,1) and (2,2)-entries are the negative of each other. Besides we can see that[(1,2)-entry of L t ∇ t ] = t ( δ X φ + φδ X ) T (cid:48) − ( iε + tφ ) δ X T (cid:48) = tδ X φT (cid:48) − iεδ X T (cid:48) = δ X ( iε + tφ ) T (cid:48) = − [(1,2)-entry of ∇ (cid:48) t L t ] . Since ( L t ∇ t ) (cid:48) = ∇ (cid:48) t L t , it automatically follows that [(2,1)-entry of L t ∇ t ] = − [(2,1)-entry of ∇ (cid:48) t L t ] aswell, and now we obtained that L t ∇ t = −∇ (cid:48) t L t . It is easy to see that ( ∇ t ) = 0. If σ X , τ X ∈ K ( E X )and σ Y , τ Y ∈ K ( E Y ) satisfy σ X δ X + δ X τ X − ∈ K ( E X ) and σ Y δ Y + δ Y τ Y − ∈ K ( E Y ), then itfollows that (cid:104) σ X − σ Y (cid:105) ∇ t + ∇ t (cid:104) τ X − τ Y (cid:105) − ∈ K ( E ) since T ∈ L ( E X , E Y ). Thus we obtained that( E , C t , ∇ t ) ∈ J ( E ) and Ψ( E , C t , ∇ t ) = Ψ( E , B , ∇ ) = Ψ( E , Q, ∇ ).Finally check that Ker( ∇ t ) = Im( ∇ t ) for any t ∈ (0 , t ]. Ker( ∇ t ) ⊃ Im( ∇ t ) is implied by ( ∇ t ) = 0,so let (cid:20) θ θ (cid:21) ∈ Ker( ∇ t ). Then θ ∈ Ker( δ Y ) and tT (cid:48) θ = − δ X θ ∈ Im( δ X ). Since T (cid:48) induces anisomorphism [ T (cid:48) ] : Ker( δ Y ) / Im( δ Y ) → Ker( δ X ) / Im( δ X ), it follows from the injectivity that θ ∈ Im( δ Y ). There exists η ∈ E such that δ Y η = θ . On the other hand, θ + tT (cid:48) η ∈ Ker( δ X ) andthe surjectivity of [ T (cid:48) ] imply that there exists ζ ∈ Ker( δ Y ) such that T (cid:48) ζ = t ( θ + tT (cid:48) η ). ThereforeIm( ∇ t ) (cid:51) ∇ t (cid:20) ζ − η (cid:21) = (cid:20) tT (cid:48) ( ζ − η ) − δ Y ( η ) (cid:21) = (cid:20) θ θ (cid:21) , which concludes that Ker( ∇ t ) ⊂ Im( ∇ t ).Due to Lemma 2.9, it follows that Ψ( E , C t , ∇ t ) = 0 ∈ KK ( C , A ) and we conclude that Ψ( E X , Q X , δ X ) − Ψ( E Y , Q Y , δ Y ) = Ψ( E , Q, ∇ ) = 0. G -signature G -index Let G be a second countable locally compact Hausdorff group. Let X be a G -compact proper complete G -Riemannian manifold. And let V be a G -Hermitian vector bundle over X . In this section, we willdefine and investigate a C ∗ ( G )-module denoted by E ( V ) obtained by completing C c ( X ; V ). This willbe used for the definition of the index of G -invariant elliptic operators, in particular, the signatureoperator. Definition 3.1 [Kas16, Section 5] First we define on C c ( X ; V ) the structure of a pre-Hilbert moduleover C c ( G ) using the action of G on C c ( X ; V ) given by γ [ s ]( x ) = γ ( s ( γ − x )) for γ ∈ G . • The action of C c ( G ) on C c ( X ; V ) from the right is given by s · b = (cid:90) G γ [ s ] · b ( γ − )∆( γ ) − d γ ∈ C c ( X ; V ) (3.1)for s ∈ C c ( X ; V ) and b ∈ C c ( G ). Here, ∆ denotes the modular function. • The scalar product valued in C c ( G ) is given by (cid:104) s , s (cid:105) E ( γ ) = ∆( γ ) − (cid:104) s , γ [ s ] (cid:105) L ( V ) (3.2)for s i ∈ C c ( V ).Define E ( V ) as the completion of C c ( V ) in the norm (cid:107)(cid:104) s, s (cid:105)(cid:107) C ∗ ( G ) . . Fukumoto Theorem 3.2 [Kas16, Theorem 5.8]
Let G be a second countable locally compact Hausdorff group.Let X be a G -compact proper complete G -Riemannian manifold. Let D : C ∞ c ( X ; V ) → C ∞ c ( X ; V ) be aformally self-adjoint G -invariant first-order elliptic operator on a G -Hermitian vector bundle V . Thenboth operators D ± i have dense range as operators on E ( V ) and ( D ± i ) − belong to K ( E ( V )) . Theoperator D (1 + D ) − / ∈ L ( E ( V )) is a Fredholm and determines an element ind G ( D ) ∈ K ( C ∗ ( G )) . In this paper, mainly we consider V as (cid:86) ∗ T ∗ X equipped with the Z / Z -grading given by the Hodge ∗ -operation and D as a signature operator. Definition 3.3
Let X and Y be proper and co-compact Riemannian G -manifolds and let V and W be G -Hermitian vector bundles over X and Y respectively. Let T : C ∞ c ( X ; V ) → C ( Y ; W ) be a linearoperator. The support of the distributional kernel of T is given by the closure of the complement ofthe following union of all subsets K X × K Y ⊂ X × Y ; (cid:91) (cid:104) T s , s (cid:105) =0 for any sections s ∈ C c ( X ; V ) and s ∈ C c ( Y ; W ) satisfyingsupp( s ) ⊂ K X , supp( s ) ⊂ K Y K X × K Y .T is said to be properly supported if bothsupp( k T ) ∪ ( K X × Y ) and supp( k T ) ∪ ( X × K Y ) ⊂ X × Y are compact for any compact subset K X ⊂ X and K Y ⊂ Y . T is said to be compactly supported if supp( k T ) ⊂ X × Y is compact.The following proposition is used for the construction of the bounded operators on E ( V ). Proposition 3.4 [Kas16, Proposition 5.4]
Let G , X , Y , V and W be as above. Let T : C c ( X ; V ) → C c ( Y ; W ) be a properly supported G -invariant operator which is L -bounded. Then T defines anelement of L ( E ( V ) , E ( W )) . For the proof, we will use the following Lemma 3.5 and Lemma 3.6.
Lemma 3.5
Let P ∈ L ( L ( X ; V ) , L ( Y ; W )) be a compactly supported bounded operator. Then theoperator (cid:101) P := (cid:90) G γ [ P ] d γ is well defined as a bounded operator in L ( L ( X ; V ) , L ( Y ; W )) and the inequation (cid:13)(cid:13)(cid:13) (cid:101) P (cid:13)(cid:13)(cid:13) op ≤ C (cid:107) P (cid:107) op holds, where C is a constant depending on its support.Proof. Assume that the support of the distributional kernel of P is contained in K X × K Y for somecompact subsets K X ⊂ X and K Y ⊂ Y . We will follow the proof of [CM, Lemma 1.4–1.5]. Fix an arbi-trary smooth section with compact support s ∈ C ∞ c ( X ; V ) and let us consider F s ∈ L (cid:0) G ; L ( Y ; W ) (cid:1) given by F s ( γ ) := γ [ P ] s. Note that for any γ ∈ G the support of the distributional kernel of γ [ P ] is contained in γ ( K X ) × γ ( K Y ).This is because for any s ∈ C ∞ c ( X ; V ), it follows that supp( γ [ P ] s ) ⊂ γ ( K Y ) and γ [ P ] s = 0 wheneversupp( s ) ∩ γ ( K X ) = ∅ . In particular, since the actions are proper, F s has compact support in G . In0 G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds addition, again since the actions are proper, γ ( K Y ) ∩ η ( K Y ) = γ ( K Y ∩ γ − η ( K Y )) = ∅ if γ − η ∈ G isoutside some compact neighborhood Z ⊂ G in particular, (cid:107) F s ( γ ) (cid:107) L ( Y ; W ) · (cid:107) F s ( η ) (cid:107) L ( Y ; W ) = 0for such γ and η ∈ G . Remind that Z is determined only by K Y so independent of s . Then, (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) G F s ( γ ) d γ (cid:13)(cid:13)(cid:13)(cid:13) L ( Y ; W ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) G F s ( γ ) d γ (cid:13)(cid:13)(cid:13)(cid:13) L ( Y ; W ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) G F s ( η ) d η (cid:13)(cid:13)(cid:13)(cid:13) L ( Y ; W ) ≤ (cid:90) G (cid:90) G (cid:107) F s ( γ ) (cid:107) L ( Y ; W ) (cid:107) F s ( η ) (cid:107) L ( Y ; W ) d γ d η ≤ (cid:90) G (cid:107) F s ( γ ) (cid:107) L ( Y ; W ) (cid:18)(cid:90) G χ Z ( γ − η ) (cid:107) F s ( η ) (cid:107) L ( Y ; W ) d η (cid:19) d γ ≤ (cid:107) F s (cid:107) L ( G ) (cid:107) χ Z (cid:107) L ( G ) (cid:107) F s (cid:107) L ( G ) ≤ | Z | (cid:107) F s (cid:107) L ( G ) , where χ Z : G → [0 ,
1] is the characteristic function of C , that is χ Z ( γ ) = 1 for γ ∈ Z and χ Z ( γ ) = 0for γ / ∈ Z .Next, take a compactly supported smooth function c ∈ C ∞ c ( X ; [0 , c = 1 on K X .Noting that P = P c , we obtain (cid:107) F s (cid:107) L ( G ) = (cid:90) G (cid:107) F s ( γ ) (cid:107) L ( Y ; W ) d γ = (cid:90) G (cid:13)(cid:13) γP c γ − s (cid:13)(cid:13) L ( Y ; W ) d γ ≤ (cid:90) G (cid:107) P (cid:107) (cid:13)(cid:13) c γ − s (cid:13)(cid:13) L ( X ; V ) d γ ≤ (cid:107) P (cid:107) (cid:90) G (cid:90) X | c ( x ) | (cid:13)(cid:13) γ − s ( x ) (cid:13)(cid:13) V d x d γ ≤ (cid:107) P (cid:107) (cid:90) G (cid:90) X | c ( γ − x ) | (cid:107) s ( x ) (cid:107) V d x d γ ≤ (cid:107) P (cid:107) sup x ∈ X (cid:18)(cid:90) G | c ( γ − x ) | d γ (cid:19) (cid:107) s (cid:107) L ( X ; V ) . Since the action of G is proper, (cid:8) γ ∈ G (cid:12)(cid:12) γ − x ∈ supp( c ) (cid:9) ⊂ G is compact so the value (cid:82) G | c ( γ − x ) | d γ is always finite for any fixed x ∈ X . Besides, since X/G is compact, this value is uniformly bounded; C := sup x ∈ X (cid:18)(cid:90) G | c ( γ − x ) | d γ (cid:19) = sup [ x ] ∈ X/G (cid:18)(cid:90) G | c ( γ − x ) | d γ (cid:19) < ∞ . Remind that C depends only on K X , not on s . We conclude that (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) G γ [ P ] s d γ (cid:13)(cid:13)(cid:13)(cid:13) L ( Y ; W ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) G F s ( γ ) d γ (cid:13)(cid:13)(cid:13)(cid:13) L ( Y ; W ) ≤ | Z | (cid:107) F s (cid:107) L ( G ) ≤ | Z | C · (cid:107) P (cid:107) (cid:107) s (cid:107) L ( X ; V ) . Lemma 3.6 [Kas16, Lemma 5.3]
Let P be a bounded positive operator on L ( X ; V ) with a compactlysupported distributional kernel. Then the scalar product ( s , s ) (cid:55)→ (cid:28) s , (cid:18)(cid:90) G γ [ P ] d γ (cid:19) s (cid:29) E ( V ) ∈ C ∗ ( G ) is well defined and positive for any s = s ∈ C c ( X ; V ) . . Fukumoto Proof.
Note that (cid:104) γ [ s ] , P ( γ [ s ]) (cid:105) L ( X ; V ) = (cid:68) √ P ( γ [ s ]) , √ P ( γ [ s ]) (cid:69) L ( X ; V ) for γ ∈ G and s ∈ C c ( X ; V ). Regarding the each side of the above equation as a function in γ ∈ G , itis clear that the left hand side vanishes outside some compact subset in G depending on the support of s and P . This implies √ P ( γ [ s ]) has a compact support in G . Take any unitary representation space H of G and h ∈ H . By the above observation of the compact support, v := (cid:90) G ∆( γ ) − √ P ( γ [ s ]) ⊗ γ [ h ] d γ ∈ L ( X ; V ) ⊗ H is well-defined. Then we obtain that0 ≤ (cid:107) v (cid:107) = (cid:90) G (cid:90) G ∆( γ ) − ∆( η ) − (cid:68) √ P ( γ [ s ]) , √ P ( η [ s ]) (cid:69) L ( X ; V ) (cid:104) γ [ h ] , η [ h ] (cid:105) H d γ d η = (cid:90) G (cid:90) G ∆( γ ) − ∆( η ) − (cid:10) s, γ − [ P ( η [ s ])] (cid:11) L ( X ; V ) (cid:10) h, γ − η [ h ] (cid:11) H d γ d η = (cid:90) G (cid:90) G ∆( γ ) − ∆( γ − η ) − (cid:10) s, γ − [ P ]( γ − η [ s ]) (cid:11) L ( X ; V ) (cid:10) h, γ − η [ h ] (cid:11) H d γ d( γ − η )= (cid:90) G (cid:90) G ∆( ζ ) − (cid:10) s, γ − [ P ]( ζ [ s ]) (cid:11) L ( X ; V ) (cid:104) h, ζ [ h ] (cid:105) H d( γ − ) d ζ = (cid:90) G ∆( ζ ) − (cid:28) s, (cid:18)(cid:90) G γ [ P ] d γ (cid:19) ( ζ [ s ]) (cid:29) L ( X ; V ) (cid:104) h, ζ [ h ] (cid:105) H d ζ = (cid:90) G (cid:28) s, (cid:18)(cid:90) G γ [ P ] d γ (cid:19) ( s ) (cid:29) E ( V ) ( ζ ) · (cid:104) h, ζ [ h ] (cid:105) H d ζ Recall that the action of f := (cid:10) s, (cid:0)(cid:82) G γ [ P ] d γ (cid:1) ( s ) (cid:11) E ( V ) ∈ C c ( G ) on H is given by f [ h ] = (cid:82) G f ( ζ ) ζ [ h ] d ζ for h ∈ H . Thus, by rewriting the above inequality, we have (cid:104) h, f [ h ] (cid:105) H ≥ h , which meansthat this f is a positive operator on any unitary representation space H . To conclude, f is positive in C ∗ ( G ) for any s ∈ C c ( E ( V )). Proof of Proposition 3.4 . Let T := ( cT ∗ T + T ∗ T c ), which is bounded self-adjoint operator L ( X ; V ) → L ( X ; V ). Moreover the distributional kernel of T is contained in K × K for some compact subset K ⊂ X . By Lemma 3.5, (cid:82) G γ [ T ] is well-defined in L ( L ( X ; V )) and (cid:90) G γ [ T ] = (cid:90) G
12 ( γ [ c ] T ∗ T + T ∗ T γ [ c ]) = T ∗ T. Consider a compactly supported continuous function f ∈ C c ( X ; [0 , c = 1 on K so that c T c = T holds. Consider the following self-adjoint operator; P := c (cid:16) (cid:107) T (cid:107) (cid:107) c (cid:107) − T (cid:17) c = c (cid:107) T (cid:107) (cid:107) c (cid:107) − T ∈ L ( L ( X ; V )) . Obviously P is compactly supported and since T ≤ (cid:107) T (cid:107) ≤ (cid:107) T (cid:107) (cid:107) c (cid:107) , P is positive. Using Lemma3.6, for any s ∈ C c ( V ), the following value is positive;0 ≤ (cid:28) s, (cid:18)(cid:90) G γ [ P ] d γ (cid:19) s (cid:29) E ( V ) ≤ C (cid:107) T (cid:107) (cid:107) c (cid:107) (cid:104) s, s (cid:105) E ( V ) − (cid:28) s, (cid:18)(cid:90) G γ [ T ] (cid:19) s (cid:29) E ( V ) ∈ C ∗ ( G ) , G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds where C is the maximum of a G -invariant bounded function (cid:82) G γ [ c ], which is independent of s . Toconclude, (cid:104) T ( s ) , T ( s ) (cid:105) E ( W ) = (cid:104) s, T ∗ T ( s ) (cid:105) E ( V ) = (cid:28) s, (cid:18)(cid:90) G γ [ T ] (cid:19) s (cid:29) E ( V ) ≤ C (cid:107) T (cid:107) (cid:107) c (cid:107) (cid:104) s, s (cid:105) E ( V ) . The theorem we will discuss is the following;
Theorem A
Let X and Y be oriented even-dimensional complete Riemnannian manifolds and let alocally compact Hausdorff group G acts on X and Y isometrically, properly and co-compactly. f : Y → X be a G -equivariant orientation preserving homotopy equivalent map. Let ∂ X and ∂ Y be the signatureoperators. Then ind G ( ∂ X ) = ind G ( ∂ Y ) ∈ K ( C ∗ ( G )) . From now on we will slightly change the notation for simplicity. We will only consider V for thecotangent bundle (cid:86) ∗ T ∗ X ⊗ C . Let us use E X for E ( (cid:86) ∗ T ∗ X ⊗ C ). Let Ω ∗ c ( X ) be the space consistingof compactly supported smooth differential forms on X , namely, C ∞ c ( X ; V ). We will prove TheoremA using Lemma 2.10. Definition 3.7
Let us introduce the following data ( E , Q, δ ) to present the G -index of the signatureoperator; • Let C ∗ ( G )-valued quadratic form Q X be defined by the formula; Q X ( ν, ξ )( γ ) := i k ( n − k ) ∆( γ ) − (cid:90) X ¯ ν ∧ γ [ ξ ] for ν ∈ Ω kc ( X ) , ν ∈ Ω n − kc ( X ) , γ ∈ G, (3.3)here ¯ ν denotes the complex conjugate. If deg( ν ) + deg( ξ ) (cid:54) = dim( X ) then Q X ( ν, ξ ) := 0. Thisdeg means the degree of the differential form. • The grading U X determined by Q X is given by U X ( ξ ) = i − k ( n − k ) ∗ ξ for ξ ∈ Ω kc ( X ) , (3.4)where ∗ denotes the Hodge ∗ -operation.Clearly, U X = 1 and Q X ( ν, U X ( ξ )) = (cid:104) ν, ξ (cid:105) E X hold. • δ X ( ξ ) := i k d X ξ for ξ ∈ Ω kc ( X ), here d X denotes the exterior derivative on X .We will also use the similar notations for Y . Lemma 3.8 ( E X , Q X , δ X ) ∈ J ( C ∗ ( G )) and Ψ( E X , Q X , δ X ) = ind G ( ∂ X ) , where ∂ X is the signatureoperator of X .Proof. First, obviously δ = 0. Applying Theorem 3.2 to the signature operator on X , it follows that δ X − U X δ X U X : Ω ∗ c ( X ) → E X is closable and its closure is self-adjoint. Let us use δ X − U X δ X U X for alsoits closure. Since Im( δ X ) and Im( − U X δ X U X ) are orthogonal to each other with respect to the scalarproduct (cid:104)· , ·(cid:105) E X , it follows that δ X itself is a closed operator on E . Moreover, set σ = τ := δ ∗ X δ ∗ X + δ X ) . . Fukumoto K ( E X ) since δ ∗ X δ ∗ X + δ X ± i ∈ L ( E X ) and δ ∗ X + δ X ± i ∈ K ( E X ). Then from Theorem 3.2, weobtain σδ X + δ X τ − −
11 + ( δ ∗ X + δ X ) ∈ K ( E X ) . Therefore, ( E X , Q X , δ X ) ∈ J ( C ∗ ( G )) and Ψ( E X , Q X , δ X ) = ind G ( ∂ X ) by the definition of Ψ.Let f : Y → X be a G -equivariant proper orientation preserving homotopy equivalent map be-tween n -dimensional proper co-compact Riemannian G -manifolds. In order to construct a map T ∈ L ( E X , E Y ) satisfying the hypothesis of Lemma 2.10, it is sufficient to construct an L -bounded G -invariant operator T : Ω ∗ c ( X ) → Ω ∗ c ( Y ) due to Proposition 3.4. Remark 3.9
Note that f ∗ : Ω ∗ c ( X ) → Ω ∗ c ( Y ) may not be L -bounded unless f : Y → X is submersion.For instance, let Y = X = [ − ,
1] and f ( y ) = y . Consider an L -form ω on X given by ω ( x ) = | x | / .Actually (cid:107) ω (cid:107) L ( X ) = (cid:82) − | x | / d x = 2, however, (cid:107) f ∗ ω (cid:107) L ( Y ) = (cid:82) − | y | / d y = + ∞ . So we need toreplace f ∗ by a suitable operator.Let us construct operator T that we need and investigate its properties in a slightly more generalcondition. • X and Y are Riemannian manifold and G acts on them isometrically and properly. For a while, X and Y may have boundary and the action may not be co-compact if not mentioned. • Let W be an oriented G -invariant fiber bundle over Y whose typical fiber is an even dimen-sional unit open disk B k ⊂ R k . Let q : W (cid:16) Y denote the canonical projection map and q I : Ω ∗ + kc ( W ) → Ω ∗ c ( Y ) be the integration along the fiber. • Let us fix ω ∈ Ω k ( W ) be a G -invariant closed k -form with fiber-wisely compact support suchthat the integral along the fiber is always equal to 1; q I ( ω )( y ) = (cid:82) W y ω = 1 for any y ∈ Y . Let e ω denote the operator given by e ω ( ζ ) = ζ ∧ ω for ζ ∈ Ω ∗ ( W ).We can construct a G -invariant ω as follows; Let τ ∈ Ω k ( W ) be a k -form inducing a Thomclass of W . We may assume that (cid:82) W y τ = 1 for any y ∈ Y . Then ω := (cid:82) G γ [ cτ ]d γ is a desired G -invariant form. • Suppose that we have a G -equivariant submersion p : W → X whose restriction on supp( ω ) ⊂ W is proper. Definition 3.10
For the above data, let us set T p,ω := q I e ω p ∗ : Ω ∗ c ( X ) → Ω ∗ c ( Y ). We may write just T p for simplicity. W q (cid:15) (cid:15) (cid:15) (cid:15) p (cid:32) (cid:32) Y X Ω ∗ c ( X ) → p ∗ Ω ∗ ( W ) → e ω Ω ∗ + kc ( W ) → q I Ω ∗ c ( Y ) . Lemma 3.11
If the actions of G are co-compact, then T p,ω determines an operator in L ( E X , E Y ) .Proof. By Proposition 3.4, it is sufficient to check that T p,ω is L -bounded.Since q I is obviously L -bounded, only the boundedness of e ω p ∗ : Ω ∗ c ( X ) → Ω ∗ c ( W ) is non-trivial.Note that our proper submersion p restricted on supp( ω ) ⊂ W is locally trivial G -invariant fibration.Let p I denotes the integration along this fibration. Then (cid:90) W ζ = (cid:90) X p I ζ G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds holds for any compactly supported differential form ζ ∈ Ω ∗ c ( W ) satisfying supp( ζ ) ⊂ supp( ω ), inparticular, ζ = | ( p ∗ ξ ) ∧ ω | vol W ∈ Ω n + kc ( W ) for ξ ∈ Ω ∗ c ( X ). Let C ω be the maximum of the norm ofbounded G -invariant form p I (cid:0) | ω | vol W (cid:1) ∈ Ω n ( X ). (cid:107) e ω p ∗ ( ξ ) (cid:107) L ( W ) = (cid:90) W | ( p ∗ ξ ) ∧ ω | vol W ( † ) = (cid:90) X | ξ | p I (cid:0) | ω | vol W (cid:1) ≤ C ω (cid:90) X | ξ | vol X = C ω (cid:107) ξ (cid:107) L ( X ) for ξ ∈ Ω ∗ c ( X ) . The equation ( † ) holds because the function p ∗ | ξ | is constant along the fiber p − ( x ). Lemma 3.12
Let us consider proper co-compact G -manifold X , Y and Z and let q : W (cid:16) Y and q : V (cid:16) Z be G -invariant oriented disk bundles over Y and Z with typical fiber B k and B k . Fix G -invariant closed forms ω ∈ Ω k ( W ) and ω ∈ Ω k ( V ) with fiber-wisely compact support satisfying ( q j ) I ( ω j ) = 1 . Let p : W → X p : V → Y be G -equivariant submersions whose restriction on supp( ω j ) are proper.On the other hand, as in the diagram below, let us consider the pull-back bundle p ∗ W = { ( v, w ) ∈ V × W | p ( v ) = q ( w ) } over V and let us regard it as a fiber bundle over Z with projection denoted by q . Let us set ω := (cid:101) p ∗ ω ∧ (cid:101) q ∗ ω ∈ Ω ∗ ( p ∗ W ) , p := p (cid:101) p , where (cid:101) q : p ∗ W → V denotes the projection and (cid:101) p : p ∗ W → W denotes the map induced by p .Then T p T p = T p : E X → E Z . V q (cid:15) (cid:15) (cid:15) (cid:15) p (cid:32) (cid:32) W q (cid:15) (cid:15) (cid:15) (cid:15) p (cid:32) (cid:32) Z Y X p ∗ W (cid:101) q (cid:15) (cid:15) (cid:15) (cid:15) p (cid:24) (cid:24) (cid:101) p (cid:34) (cid:34) q (cid:19) (cid:19) (cid:19) (cid:19) V (cid:15) (cid:15) (cid:15) (cid:15) (cid:34) (cid:34) W (cid:15) (cid:15) (cid:15) (cid:15) (cid:32) (cid:32) Z Y XT p T p : E X → E Y → E Z , T p : E X → E Z . Proof.
First we can see that for ξ ∈ Ω ∗ c ( X ), T p ( ξ ) = ( q ) I ◦ e ω p ∗ ( ξ )= ( q ) I ( (cid:101) q ) I (cid:8) (cid:101) p ∗ p ∗ ξ ∧ ( (cid:101) p ∗ ω ∧ (cid:101) q ∗ ω ) (cid:9) = ( q ) I ( (cid:101) q ) I (cid:8) (cid:101) p ∗ ( p ∗ ξ ∧ ω ) ∧ (cid:101) q ∗ ω (cid:9) = ( q ) I (cid:8) ( (cid:101) q ) I (cid:0) (cid:101) p ∗ ( p ∗ ξ ∧ ω ) (cid:1) ∧ ω (cid:9) = ( q ) I e ω ( (cid:101) q ) I (cid:101) p ∗ e ω p ∗ ( ξ ) ,T p T p ( ξ ) = ( q ) I e ω p ∗ ◦ ( q ) I e ω p ∗ ( ξ ) . Note that ( (cid:101) q ) I in the second bottom row is well defined because the differential form (cid:101) p ∗ e ω p ∗ ( ξ ) iscompactly supported along each fiber of (cid:101) q : p ∗ W (cid:16) V . We need to prove the commutativity of thefollowing diagram; Ω ∗ ( p ∗ W ) ( (cid:101) q ) I (cid:15) (cid:15) Ω ∗ ( V ) Ω ∗ c ( W ) ( q ) I (cid:15) (cid:15) (cid:101) p ∗ (cid:101) (cid:101) Ω ∗ ( Y ) p ∗ (cid:101) (cid:101) (3.5) . Fukumoto W (cid:16) Y is trivialized on U ⊂ Y .Then p ∗ W is trivialized on p − U ⊂ V . We write these trivialization as W | U (cid:39) U × B k and p ∗ W | U (cid:39) p − U × B k . Then for ζ ( y, w ) = f ( y, w )d y ∧ d w ∈ Ω ∗ c ( W | U ), (cid:0) ( (cid:101) q ) I (cid:101) p ∗ ζ (cid:1) ( v ) = (cid:90) B k ( f ( p ( v ) , w ) p ∗ (d y )) d w = ( p ∗ ( q ) I ζ ) ( v ) for v ∈ p − U ⊂ V. We will use the following proposition repeatedly.
Proposition 3.13
Let W and W be oriented G -invariant disk bundles over Y with typical fiber B k and B k , and let q j : W j (cid:16) Y be the projection. Let ω j ∈ Ω k j be closed forms with fiber-wisely compactsupport satisfying ( q j ) I ( ω ) ω j = 1 .Suppose that there exist G -equivariant submersions p j : W j → X whose restriction on the -sections p j ( · ,
0) : Y → X are G -equivariant homotopic to each other.Then, there exists a properly supported G -equivariant L -bounded operator ψ : Ω ∗ c ( X ) → Ω ∗ c ( Y ) satisfying that T p ,ω − T p ,ω = d X ψ + ψ d Y . First, let us prove the following lemma;
Lemma 3.14
Let Q : (cid:102) W (cid:16) Y × [0 , be a G -invariant disk bundle over Y × [0 , and let ω ∈ Ω k ( (cid:102) W ) be a closed form with fiber-wisely compact support satisfying Q I ( ω ) = 1 . Suppose that thereexists a G -equivariant submersion P : (cid:102) W → X whose restriction on supp( ω ) is proper. Then thereexists a properly supported G -equivariant L -bounded operator ψ : Ω ∗ c ( X ) → Ω ∗ c ( Y ) satisfying that T P ( · , ,ω ( · , − T P ( · , ,ω ( · , = d X ψ + ψ d Y .Proof. Let ξ ∈ Ω ∗ c ( X ) and θ := Q I ( P ∗ ξ ∧ ω ) ∈ Ω ∗ c ( Y × [0 , (cid:90) [0 , d θ = − d (cid:32)(cid:90) [0 , θ (cid:33) + ( i ∗ θ − i ∗ θ ) , where i t : Y × { t } (cid:44) → Y × [0 ,
3] denotes the inclusion map. Note that i ∗ t θ = T P ( · ,t ) ,ω ( · ,t ) ξ .Now, set ψ : Ω ∗ c ( X ) → Ω ∗ c ( Y ) by the formula; ψ ( ξ ) := (cid:82) [0 , Q I ( P ∗ ξ ∧ ω ) for ξ ∈ Ω ∗ c ( X ). Note thatthe identity map L ([0 , → L ([0 , , (cid:82) [0 , : Ω ∗ c ( Y × [0 , → Ω ∗ c ( Y ) is L -bounded. Moreover, since P ∗ ξ ∧ ω vanishes at theboundary of each fiber of (cid:102) W , the integration along the fiber commutes with taking exterior derivative,in particular, d θ = d Q I ( P ∗ ξ ∧ ω ) = Q I d( P ∗ ξ ∧ ω ) = Q I ( P ∗ (d ξ ) ∧ ω ) . To conclude, we obtain ψ (d ξ ) = (cid:90) [0 , d Q I ( P ∗ ξ ∧ ω ) = − d ψ ( ξ ) + T P ( · , ,ω ( · , ξ − T P ( · , ,ω ( · , ξ, Proof of Proposition 3.13 . We need to construct (cid:102) W and P as above satisfying T P ( · , ,ω ( · , = T p ,ω and T P ( · , ,ω ( · , = T p ,ω .Let h : Y × [0 , → X be a re-parametrized G -homotopy between p ( · ,
0) and p ( · , h isa G -equivariant smooth map satisfying h ( y, t ) = p ( y,
0) for t ∈ [0 , h ( y, t ) = p ( y,
0) for t ∈ [2 , . G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds here G acts on [0 ,
3] trivially. Moreover, consider the following fiber product W × Y W = { ( y , w ) , ( y , w ) ∈ W × W | y = y } .Let us introduce a smooth map χ : [0 , → [0 ,
1] satisfying that χ ( t ) = 0 for t ∈ (cid:2) , (cid:1) ∪ (cid:0) , (cid:3) and χ ( t ) = 1 for t ∈ (cid:0) , (cid:1) . Then (cid:101) h : ( W × Y W ) × [0 , → X (( y, t ) , w , w ) (cid:55)→ p ( y, (1 − χ ( t )) w ) for t ∈ [0 , ,h ( y, t ) for t ∈ [1 , ,p ( y, (1 − χ ( t )) w ) for t ∈ [2 , . This (cid:101) h is submersion as long as χ ( t ) (cid:54) = 1 due to the submergence of p and p . Let BX := { v ∈ T X | (cid:107) v (cid:107) < } be the unit disk tangent bundle and consider the pull-back bundle (cid:102) W := (cid:101) h ∗ BX and let us regard it as a bundle over Y × [0 ,
3] and set P : (cid:102) W → X (( y, t ) , w , w , v ) (cid:55)→ exp (cid:101) h (( y,t ) ,w ,w ) ( χ ( t ) v ) . Due to the ( χ ( t ) v )-component, P is submersion also when χ ( t ) (cid:54) = 0 not only when χ ( t ) (cid:54) = 1.Moreover, define ω ∈ Ω ∗ ( W ) as ω := π ∗ ω ∧ π ∗ ω ∧ (cid:101) h ∗ ω BX , where π j : (cid:102) W (cid:16) W j for j = 1 , ω BX ∈ Ω ∗ ( BX ) is a G -invariant differential with fiber-wisely compact support satisfying (cid:82) BX x ω BX =1. These (cid:102) W , P and ω satisfy the assumption of Lemma 3.14.It is easy to see that T P ( · , ,ω ( · , = T p ,ω and T P ( · , ,ω ( · , = T p ,ω as follows. For the simplicity, let π : (cid:102) W Y ×{ } (cid:16) W denote the projection. Note that P ( y,
0) = p π and we can write ω ( · ,
0) = π ∗ ω ∧ (cid:101) ω ,using some (cid:101) ω ∈ Ω ∗ ( (cid:102) W Y ×{ } ) satisfying π I (cid:101) ω = 1. Then we obtain that T P ( · , ,ω ( · , ( ξ ) = ( q ) I π I ( π ∗ p ∗ ξ ∧ π ∗ ω ∧ (cid:101) ω )= ( q ) I π I ( π ∗ ( p ∗ ξ ∧ ω ) ∧ (cid:101) ω )= ( q ) I (( p ∗ ξ ∧ ω ) ∧ π I (cid:101) ω )= ( q ) I ( p ∗ ξ ∧ ω ) = T p ,ω ( ξ ) , and similarly, T P ( · , ,ω ( · , = T p ,ω .Now let us define a map T ∈ L ( E X , E Y ) which satisfies the assumption of Lemma 2.10. First, remarkthat our map f : Y → X is a proper map by Lemma 1.5. Definition 3.15
Let BX := { v ∈ T X | (cid:107) v (cid:107) < } be the unit disk tangent bundle and let W := f ∗ BX be the pull-back on Y , that is, W = (cid:110) ( y, v ) ∈ Y × BX (cid:12)(cid:12)(cid:12) v ∈ BX | f ( y ) (cid:111) . Let (cid:101) f : W → BX be a mapgiven by (cid:101) f ( x, v ) := ( f ( x ) , v ). Since the action of G on X is isometric and f is G -equivariant, G actson BX and also on W . Consider a G -equivariant submersion given by the formula; p : W → X ( y, v ) (cid:55)→ exp f ( y ) ( v ) . (3.6)Let us fix a G -invariant R -valued closed n -form ω ∈ Ω n ( BX ) with fiber-wisely compact supportwhose integral along the fiber is always equal to 1, and let ω := (cid:101) f ∗ ω ∈ Ω n ( W ) For these W , p and ω ,let us set T := T p,ω . . Fukumoto Lemma 3.16
The adjoint with respect to quadratic forms Q X and Q Y is given by T (cid:48) = p I e ω q ∗ .Proof. Note that deg( ω ) = dim( X ) is even, hence, ω commutes with other differential forms. For ν ∈ Ω kc ( Y ) and ξ ∈ Ω n − kc ( X ), (cid:90) X p I e ω q ∗ ( ν ) ∧ ξ = (cid:90) X p I ( q ∗ ν ∧ ω ) ∧ ξ = (cid:90) X p I ( q ∗ ν ∧ ω ∧ p ∗ ξ ) = (cid:90) BX q ∗ ν ∧ ω ∧ p ∗ ξ = (cid:90) Y q I ( q ∗ ν ∧ p ∗ ξ ∧ ω ) = (cid:90) Y ν ∧ q I ( p ∗ ξ ∧ ω ) = (cid:90) Y ν ∧ T ( ξ ) . Since Q X ( ν, ξ )( γ ) := i k ( n − k ) ∆( γ ) − (cid:82) X ¯ ν ∧ γ [ ξ ], the proof complete replacing ν and ξ by ¯ ν and γ [ ξ ]respectively and using the G -invariance of T . Proposition 3.17
There exists φ ∈ L ( E X ) such that − T (cid:48) T = d X φ + φ d X .Proof. Consider the fiber product W × Y W and let q and q : W × Y W → W denote the projectionsgiven by q j ( y, v , v ) := ( y, v j ). Take ζ ∈ Ω ∗ c ( W ), here W is regarded as the first component of W × Y W . Using the commutativity of the diagram (3.5),Ω ∗ ( W × Y W ) ( q ) I (cid:15) (cid:15) Ω ∗ ( W ) Ω ∗ c ( W ) q I (cid:15) (cid:15) q ∗ (cid:103) (cid:103) Ω ∗ ( Y ) q ∗ (cid:103) (cid:103) we obtain that e ω q ∗ q I ( ζ ) = e ω ( q ) I q ∗ ( ζ ) = ( q ) I ( q ∗ ζ ) ∧ ω = ( q ) I ( q ∗ ζ ∧ q ∗ ω )= ( q ) I e q ∗ ω q ∗ ( ζ ) , and hence, T (cid:48) T = p I e ω q ∗ q I e ω p ∗ = p I ( q ) I e q ∗ ω q ∗ e ω p ∗ . On the other hand, since q ( y,
0) = q ( y, G -equivariant L -bounded operator ψ W : Ω ∗ c ( W ) → Ω ∗ c ( W ) satisfying( q ) I e q ∗ ω q ∗ − ( q ) I e q ∗ ω q ∗ = d ψ W + ψ W d . Moreover, it is obvious that ( q ) I e q ∗ ω q ∗ = id Ω c ( W ) , so we obtain p I e ω p ∗ − T (cid:48) T = p I (cid:16) id Ω c ( W ) − ( q ) I e q ∗ ω q ∗ (cid:17) e ω p ∗ = p I (d ψ W + ψ W d) e ω p ∗ = d ◦ p I ψ W e ω p ∗ + p I ψ W e ω p ∗ ◦ d . (3.7)Remark that p I ◦ d = d ◦ p I because the act on differential forms with compact support, and e ω ◦ d = d ◦ e ω because ω is a closed form.Next let us consider submersion p X : BX → X given by ( x, v ) (cid:55)→ exp x ( v ). Note that p = p X (cid:101) f . f ∗ BX p (cid:36) (cid:36) (cid:101) f (cid:47) (cid:47) BX p X (cid:15) (cid:15) X G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds Now we want to check that p I e ω p ∗ = ( p X ) I e ω p ∗ X . For any ν ∈ Ω ∗ c ( X ) and ζ ∈ Ω ∗ c ( BX ), (cid:90) X ν ∧ p I (cid:16) (cid:101) f ∗ ζ (cid:17) = (cid:90) W p ∗ ν ∧ (cid:101) f ∗ ζ = (cid:90) W (cid:101) f ∗ ( p ∗ X ν ∧ ζ )= deg (cid:16) (cid:101) f (cid:17) (cid:90) BX p ∗ X ν ∧ ζ = (cid:90) BX p ∗ X ν ∧ ζ = (cid:90) X ν ∧ ( p X ) I ( ζ ) , since f is an orientation preserving proper homotopy equivalent. In particular, we obtain p I (cid:16) (cid:101) f ∗ ζ (cid:17) = ( p X ) I ( ζ ) . Put ζ := p ∗ X ξ ∧ ω for ξ ∈ Ω ∗ c ( X ) to obtain p I e ω p ∗ ( ξ ) = p I (cid:16) (cid:101) f ∗ p ∗ X ξ ∧ (cid:101) f ∗ ω (cid:17) = p I (cid:16) (cid:101) f ∗ ( p ∗ X ξ ∧ ω ) (cid:17) = ( p X ) I ( p ∗ X ξ ∧ ω ) = ( p X ) I e ω p ∗ X ( ξ ) . (3.8)Let π : BX → X be the natural projection. Since p X ( x,
0) = π ( x, G -equivariant L -bounded operator ψ X : Ω ∗ c ( X ) → Ω ∗ c ( X ) satisfying π I e ω π ∗ − ( p X ) I e ω p ∗ X = d ψ X + ψ X d . (3.9)On the other hand, it is obvious that π I e ω π ∗ = id Ω c ( X ) . Therefore, combining (3.7), (3.8) and (3.9),we conclude id Ω c ( X ) − T (cid:48) T = d φ + φ d , where φ = p I ψ W e ω p ∗ + ψ X . Since φ is properly supported G -invariant L -bounded operator, it definesan element in L ( E X ). Proof of Theorem A . First, let us check that T satisfies the assumption (1) of Lemma 2.10. Since ω is a closed form and has fiber-wisely compact support, it follows that T δ X = δ Y T . Let g : X → Y bethe G -equivariant homotopy inverse of f and consider a map S ∈ L ( E Y , E X ) constructed in the samemethod as T from g instead of f in Definition 3.15. By 3.12, the composition ST is equal to the map T p ∈ L ( E X ) for p satisfying that p ( · ,
0) is G -equivariant homotopic to id X . Then by Proposition 3.13,there exists φ X ∈ L ( E X ) satisfying that ST − ( δ X φ X + φ X δ X ) = T id X = id E X . Thus, ST induces theidentity map on Ker( δ X ) / Im( δ X ). Similarly T S induces the identity map on Ker( δ Y ) / Im( δ Y ), andhence, T induces an isomorphism Ker( δ X ) / Im( δ X ) → Ker( δ Y ) / Im( δ Y ).The assumption (2) of Lemma 2.10 is obtained from 3.17.Finally, let ε ( ξ ) := ( − k ξ for ξ ∈ Ω kc ( X ). Clearly, ε determines an operator ε ∈ L ( E X ), ε = 1 andsatisfies ε (cid:48) = ε , ε (dom( δ X )) ⊂ dom( δ X ) and εδ X = − δ X ε . Moreover since neither T nor T (cid:48) changesthe order of the differential forms, ε commutes with 1 − T (cid:48) T . Thus ε satisfies the assumption (3) ofLemma 2.10. To conclude, we obtain ind G ( ∂ X ) = Ψ( E X , Q X , δ X ) = Ψ( E Y , Q Y , δ Y ) = ind G ( ∂ Y ). To prove Corollary B, we will combine [F, Theorem A] with Theorem A. Suppose, in addition, that G is unimodular and H ( X ; R ) = H ( Y ; R ) = { } . Let f : Y → X be a G -equivariant orientationpreserving homotopy invariant map and consider a G -manifold Z := X (cid:116) ( − Y ), the disjoint unionof X and orientation reversed Y . Let ∂ Z be the signature operator, then we have that ind G ( ∂ Z ) =ind G ( ∂ X ) − ind G ( ∂ Y ) = 0 ∈ K ( C ∗ ( G )). Although the G -manifold should be connected in [F, TheoremA], however in this case, we can apply it to Z after replacing some arguments in [F] as follows. . Fukumoto U (1)-valued cocycle α ∈ Z ( G ; U (1)) from thegiven line bundle, we just use a line bundle L over X ignoring f ∗ over Y . When constructing familyof line bundles { L t } on which the central extension group G α t acts, just construct a family of linebundles { L t } over X in the same way and pull back on Y to obtain a family { f ∗ L t } . To be specific, f ∗ L t is a trivial bundle Y × C , equipped with the connection given by ∇ t = d + itf ∗ η and the actionof G α t is given by( γ, u )( y, z ) = ( γy, exp[ − itf ∗ ψ γ ( x )] uz ) for ( γ, u ) ∈ G α t , y ∈ Y, z ∈ C = ( L t ) x . Then consider a family of G α t -line bundles { L t (cid:116) f ∗ L t } over Z . We also need the similar replacementin [F, Definition 7.19] to obtain the global section on L t (cid:116) f ∗ L t . Then the rest parts proceed similarly. Now we will discuss on the Dirac operators twisted by a family of Hilbert module bundles { E k } whosecurvature tend to zero. and prove Theorem C. Such an family is called a family of almost flat bundles.In this section, it is convenient to formulate the index map using KK -theory. G -index map in KK -theory Lemma 4.1 [Kas88, Theorem 3.11]
Let G be a second countable locally compact Hausdorff group. Forany G -algebras A and B there exists a natural homomorphism j G : KK G ( A, B ) → KK ( C ∗ ( G ; A ) , C ∗ ( G ; B )) Furthermore if x ∈ KK G ( A, B ) and y ∈ KK G ( B, D ) , then j G ( x (cid:98) ⊗ B y ) = j G ( x ) (cid:98) ⊗ C ∗ ( G ; B ) j G ( y ) . Lemma 4.2
Using a cut-off function c ∈ C c ( X ) , one can define an idempotent p ∈ C c ( G ; C ( X )) bythe formula; ˇ c ( γ )( x ) = (cid:112) c ( x ) c ( γ − x )∆( γ ) − . In particular it defines an element of K-homology denoted by [ c ] ∈ K ( C ∗ ( G ; C ( X ))) . Moreover theelement of K -homology [ c ] ∈ K ( C ∗ ( G ; C ( X ))) does not depend on the choice of cut-off functions. Definition 4.3 ( G -Index) [Kas16, Theorem 5.6.] Define µ G : KK G ( C ( X ) , C ) → K ( C ∗ ( G ))as the composition of • j G : KK G ( C ( X ) , C ) → KK ( C ∗ ( G ; C ( X )) , C ∗ ( G )) and • [ c ] (cid:98) ⊗ : KK ( C ∗ ( G ; C ( X )) , C ∗ ( G )) → KK ( C , C ∗ ( G )) (cid:39) K ( C ∗ ( G )), i.e., µ G (-) := [ c ] (cid:98) ⊗ C ∗ ( G ; C ( X )) j G (-) ∈ K ( C ∗ ( G )) . Remark 4.4
As in [Kas16, Remark 4.4.] or [F, Subsection 5.2], it is sufficient to consider only in thecase of Dirac type operators for calculating the index.Let B be a unital C ∗ -algebra. Following the definition 4.3, we define the index maps with coefficients; Definition 4.5
For unital C ∗ -algebras B , define the index mapind G : KK G ( C ( X ) , B ) → K ( C ∗ ( G ; B ))as the composition of0 G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds • j G : KK G ( C ( X ) , B ) → KK ( C ∗ ( G ; C ( X )) , C ∗ ( G ; B )) and • [ c ] (cid:98) ⊗ : KK ( C ∗ ( G ; C ( X )) , C ∗ ( G ; B )) → K ( C ∗ ( G ; B )), i.e.,ind G (-) := [ c ] (cid:98) ⊗ C ∗ ( G ; C ( X )) j G (-) ∈ K ( C ∗ ( G ; B )) . The crossed product C ∗ ( G ; B ) is either maximal or reduced one. In this paper, we assume that G actson B trivially. Then C ∗ Max ( G ; B ) and C ∗ red ( G ; B ) will be naturally identified with C ∗ Max ( G ) ⊗ Max B and C ∗ red ( G ) ⊗ min B respectively. Moreover if B is nuclear, ⊗ Max B and ⊗ min B are identified. Definition 4.6
Let E be a finitely generated projective ( Z / Z )-graded Hilbert B -module G -bundle.Define C ( X ; E ) as a space consisting of sections s : X → E vanishing at infinity. It is considered as a Z / Z -graded Hilbert C ( X ; B )-module with the right action given by point-wise multiplications andthe scalar product given by (cid:104) s , s (cid:105) ( x ) := (cid:104) s ( x ) , s ( x ) (cid:105) E x ∈ C ( X ; B ) . Remark 4.7
The C ∗ -algebra C ( X ; B ) consisting of B -valued function vanishing at infinity is nat-urally identified with C ( X ) (cid:98) ⊗ B by [We, 6.4.17. Theorem]. Similarly, if E = X × E is a trivialHilbert B -module bundle over X , then C ( X ; E ) is naturally identified with C ( X ) (cid:98) ⊗ E as Hilbert (cid:0) C ( X ; B ) ∼ = C ( X ) (cid:98) ⊗ B (cid:1) -modules. Definition 4.8 E define an element in KK -theory[ E ] = ( C ( X ; E ) , ∈ KK G (cid:0) C ( X ) , C ( X ) (cid:98) ⊗ B (cid:1) . The action of C ( X ) on C ( X ; E ) is the point-wise multiplication. Definition 4.9
Let E be a finitely generated Hilbert B -module bundle over X equipped with aHermitian connection ∇ E . Let R E ∈ C ∞ (cid:16) X ; End( E ) ⊗ (cid:86) ( T ∗ ( X )) (cid:17) denote its curvature. Thendefine its norm as follows: First, define the point-wise norm as the operator norm given by (cid:13)(cid:13) R E (cid:13)(cid:13) x := sup (cid:110)(cid:13)(cid:13) R E ( u ∧ v ) (cid:13)(cid:13) L (E) (cid:12)(cid:12)(cid:12) u, v ∈ T x X, (cid:107) u ∧ v (cid:107) = 1 (cid:111) for x ∈ X. Then define the global norm as the supremum in x ∈ X of the point-wise norm; (cid:13)(cid:13) R E (cid:13)(cid:13) := sup x ∈ X (cid:13)(cid:13) R E (cid:13)(cid:13) x To describe the Theorem which we will prove,
Theorem C
Let X be a complete oriented Riemannian manifold and let G be a locally compactHausdorff group acting on X isometrically, properly and co-compactly. Moreover we assume that X is simply connected. Let D be a G -invariant properly supported elliptic operator of order on G -Hermitian vector bundle over X .Then there exists ε > satisfying the following: for any finitely generated projective Hilbert B -module G -bundle E over X equipped with a G -invariant Hermitian connection such that (cid:13)(cid:13) R E (cid:13)(cid:13) < ε ,we have ind G (cid:16) [ E ] (cid:98) ⊗ C ( X ) [ D ] (cid:17) = 0 ∈ K ( C ∗ Max ( G ) ⊗ Max B ) if ind G ([ D ]) = 0 ∈ K ( C ∗ Max ( G )) . If we only consider commutative C ∗ -algebras for B , then the sameconclusion is also valid for C ∗ red ( G ) . . Fukumoto C ∗ -algebras Definition 4.10
Let B k be a sequence of C ∗ -algebras. • Define (cid:81) k ∈ N B k as the C ∗ -algebra consisting of norm-bounded sequences (cid:89) k ∈ N B k := (cid:26) { b , b , . . . } (cid:12)(cid:12)(cid:12)(cid:12) b k ∈ B k , sup k {(cid:107) b k (cid:107) B k } < ∞ (cid:27) . The norm of B k is given by (cid:107){ b , b , . . . }(cid:107) (cid:81) B k := sup k {(cid:107) b k (cid:107) B k } . • Let (cid:76) k ∈ N B k a closed two-sided ideal in (cid:81) k ∈ N B k consisting of sequences vanishing at infinity (cid:77) k ∈ N B k := (cid:26) { b , b , . . . } (cid:12)(cid:12)(cid:12)(cid:12) b k ∈ B k , lim k →∞ (cid:107) b k (cid:107) = 0 (cid:27) . In other words, (cid:76) k ∈ N B k is a closure of the sub-space in (cid:81) k ∈ N B k consisting of sequences { b , b , . . . , , , . . . } whose entries are zero except for finitely many of them. • Define Q k ∈ N B k as the quotient algebra given by Q k ∈ N B k := (cid:16)(cid:89) B k (cid:17) (cid:14) (cid:16)(cid:77) B k (cid:17) . The norm of Q B k is given by (cid:107){ b , b , . . . }(cid:107) Q B k := lim sup k →∞ (cid:107) b k (cid:107) B k . • If E k are Hilbert B k -modules, one can similarly define (cid:81) E k as a Hilbert (cid:81) B k -module consistingof bounded sequences (cid:89) k ∈ N E k := (cid:26) { s , s , . . . } (cid:12)(cid:12)(cid:12)(cid:12) s k ∈ E k , sup k {(cid:107) s k (cid:107) E k } < ∞ (cid:27) . The action of (cid:81) B k and (cid:81) B k -valued scalar product are defined as follows; { s k } · { b k } := { s k · b k } ∈ (cid:89) E k for { s k } ∈ (cid:89) E k , { b k } ∈ (cid:89) B k , (cid:10) { s k } , { s k } (cid:11) (cid:81) E k := (cid:110)(cid:10) s k , s k (cid:11) E k (cid:111) ∈ (cid:89) B k for { s k } , { s k } ∈ (cid:89) E k . One can define similraly (cid:77) k ∈ N E k := (cid:26) { s , s , . . . } (cid:12)(cid:12)(cid:12)(cid:12) s k ∈ E k , lim k →∞ (cid:107) s k (cid:107) E k = 0 (cid:27) as a Hilbert (cid:81) B k -module, and define Q k ∈ N E k := (cid:16)(cid:89) E k (cid:17) (cid:98) ⊗ π ( Q B k ) = (cid:16)(cid:89) E k (cid:17) (cid:14) (cid:16)(cid:77) E k (cid:17) as a Hilbert Q B k -module, where π : (cid:81) B k → Q B k denotes the projection. Example 4.11
If all of B k are C , then, (cid:81) C = (cid:96) ∞ ( N ) and (cid:76) C = C ( N ).Following [Ha12, Section 3.], we will construct “infinite product bundle (cid:81) E k ” over X which hasa structure of finite generated projective (cid:81) B k -module.2 G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds Definition 4.12
Let us fix some notations about the holonomy. • Two paths p and p from x to y in X are thin homotopic to each other if there exists an endpoints preserving homotopy h : [0 , × [0 , → X with h ( · , j ) = p j that factors through a finitetree T , h : [0 , × [0 , → T → X such that both restrictions of the first map [0 , × { j } → T are piecewise-linear for j = 0 , • The path groupoid P ( X ) is a groupoid consisting of all the points in X as objects. Themorphism from x to y are the equivalence class of piece-wise smooth paths connecting given twopoints P ( X )[ x, y ] := { p : [0 , → X | p (0) = x, p (1) = y } / ∼ . The equivalent relationship is given by re-parametrization and thin homotopy. • If a Hilbert B -module G -bundle E over X is given, the transport groupoid T ( X ; E ) is a groupoidwith the same objects as P ( X ). The morphism from x to y are the unitary isomorphismsbetween the fibers T ( X ; E )[ x, y ] := Iso B ( E x , E y ). Definition 4.13
A parallel transport of E is a continuous functor Φ E : P ( X ) → T ( X ; E ). Φ E iscalled ε -close to the identity if for each x ∈ X and contractible loop p ∈ P ( X )[ x, x ], it follows that (cid:13)(cid:13) Φ Ep − id E x (cid:13)(cid:13) < ε · area( D )for any two dimensional disk D ⊂ X spanning p . D may be degenerated partially or completely. Remark 4.14
Let E be a Hermitian vector bundle, in other words, a finitely generated Hilbert C -module bundle, equipped with a compatible connection ∇ . Let Φ E be the parallel transport withrespect to ∇ in the usual sense. If its curvature R E ∈ C ∞ (cid:16) X ; End( E ) ⊗ (cid:86) ( T ∗ ( X )) (cid:17) has uniformlybounded operator norm (cid:13)(cid:13) R E (cid:13)(cid:13) < C , then for any loop p ∈ P ( X )[ x, x ] and any two dimensional disk D ⊂ X spanning p , it follows that (cid:13)(cid:13) Φ Ep − id E x (cid:13)(cid:13) < (cid:82) D (cid:13)(cid:13) R E (cid:13)(cid:13) < C · area( D ) so it is C -closed to identity. Proposition 4.15
Let { E k } be a sequence of Hilbert B k -module G -bundles over X with B k unital C ∗ -algebras. Assume that each parallel transport Φ k for E k is ε -close to the identity uniformly, thatis, ε is independent of k .Then there exists a finitely generated Hilbert ( (cid:81) k B k ) -module G -bundle V over X with Lipschitzcontinuous transition functions in diagonal form and so that the k -th component of this bundle isisomorphic to the original E k .Moreover, if the parallel transport Φ k for each of E k comes from the G -invariant connection ∇ k on E k , V is equipped with a continuous G -invariant connection induced by E k .Proof. We will essentially follow the proof of [Ha12, Proposition 3.12.]. For each x ∈ X take a openball U x ⊂ X of radius (cid:28) x . Assume that each U x is geodesically convex. Due to thecorollary 1.4 of the slice theorem, there exists a sub-family of finitely many open subsets { U x , . . . U x N } such that X = (cid:83) γ ∈ G (cid:83) Ni =1 γ ( U x i ).Fix k . In order to simplify the notation, let U i := U x i and Φ y ; x : E ky → E kx denote the paralleltransport of E k along the minimal geodesic from y to x for x and y in the same neighborhood γ ( U i ).Trivialize E k via Φ y ; x i : E ky → E kx i on each U i . Similarly trivialize E k on each γ ( U i ) for γ ∈ G viaΦ γy ; γx i : E kγy → E kγx i . Note that since parallel transport commute with the action of G , it follows thatΦ γy,γx i = γ ◦ Φ y ; x i ◦ γ − .These provide a local trivializations for E k whose transition functions have uniformly boundedLipschitz constants. More precisely we have to fix unitary isomorphisms φ γx i : E kγx i → E k between . Fukumoto γx i and the typical fiber E k . Our local trivialization is φ γx i Φ y ; γx i : E ky → E k . If y, z ∈ γ ( U i ) ∩ η ( U j ) (cid:54) = ∅ , we can consider the transition function y (cid:55)→ ψ γ ( U i ) ,η ( U j ) ( y ) := (cid:0) φ ηx j ◦ Φ y ; ηx j (cid:1) ( φ γx i ◦ Φ y ; γx i ) − ∈ End B k ( E k ) . Now we will estimate its Lipschitz constant as follows; ψ γ ( U i ) ,η ( U j ) ( y ) − ψ γ ( U i ) ,η ( U j ) ( z )= (cid:0) φ ηx j Φ y ; ηx j (cid:1) ( φ γx i Φ y ; γx i ) − − (cid:0) φ ηx j Φ z ; ηx j (cid:1) ( φ γx i Φ z ; γx i ) − = φ ηx j (cid:110)(cid:16) Φ y ; ηx j (cid:17) (cid:0) Φ − y ; γx i (cid:1) − (cid:16) Φ z ; ηx j Φ y ; z (cid:17) (cid:0) Φ − y ; z Φ − z ; γx i (cid:1)(cid:111) φ − γx i = φ ηx j (cid:110)(cid:16) Φ y ; ηx j − Φ z ; ηx j Φ y ; z (cid:17) (cid:0) Φ − y ; γx i (cid:1) + (cid:16) Φ z ; ηx j Φ y ; z (cid:17) (cid:0) Φ − y ; γx i − Φ − y ; z Φ − z ; γx i (cid:1)(cid:111) φ − γx i . Since φ ’s and Φ’s are isometry, it follows that (cid:13)(cid:13)(cid:13) ψ γ ( U i ) ,η ( U j ) ( y ) − ψ γ ( U i ) ,η ( U j ) ( z ) (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) Φ y ; ηx j − Φ z ; ηx j Φ y ; z (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13) Φ − y ; γx i − Φ − y ; z Φ − z ; γx i (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) Φ y ; ηx j Φ z ; y Φ ηx j ; z − id E kηxj (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) Φ z ; γx i Φ y ; z Φ γx i ; y − id E kγxi (cid:13)(cid:13)(cid:13) ≤ ε · (area( D ) + area( D )) . (4.1)Here D ⊂ η ( U j ) is a two dimensional disk spanning the piece-wise geodesic loops connecting ηx j , y , z , and ηx j and D ⊂ γ ( U i ) is a two dimensional disk spanning the piece-wise geodesic loop connecting γx i , y , z , and γx i .We claim that there exists a constant C depending only on X such thatarea( D ) , area( D ) ≤ C · dist( y, z ) (4.2)if we choose suitable disks D and D .We verify this using the geodesic coordinate exp − ηx j : η ( U j ) → T ηx j X centered at ηx j (cid:55)→
0. Moreprecisely, let p denote the minimal geodesic from y = p (0) to z = p (dist( y, z )) with unit speed.Consider D := (cid:8) ( r cos θ, r sin θ ) ∈ R (cid:12)(cid:12) ≤ r , ≤ θ ≤ dist( y, z ) (cid:9) ⊂ R and F : D → η ( U j ) ⊂ X given by F ( r cos θ, r sin θ ) := exp ηx j (cid:16) r exp − ηx j ( p ( θ )) (cid:17) . Set D := F ( D ). F is injective if exp − ηx j ( y ) and ± exp − ηx j ( z ) are on different radial directions, in whichcase F is a homeomorphism onto its image, and hence F ( D ) is a two dimensional disk spanning thetarget loop. The Lipschitz constant of F is bounded by a constant depending on the curvature on η ( U j ), so there exists a constant C η,j depending on the Riemannian curvature on η ( U j ) satisfyingarea( D ) ≤ C η,j · area( D ) ≤ C η,j · dist( y, z ) . However, the constant C η,j can be taken independent of η ( U j ) due to the bounded geometry of X implied by the slice theorem (Corollary 1.4). In the case of exp − ηx j ( y ) and ± exp − ηx j ( z ) are on the sameradial direction, D is completely degenerated and area( D ) = 0. We can construct D in the samemanner so the claim (4.2) has been verified.Therefore combining (4.1) and (4.2), we conclude that the Lipschitz constants of the transitionfunctions of these local trivialization are less than 2 Cε , which are independent of E k , U i and γ ∈ G ,in particular, the product of themΨ γ ( U i ) ,η ( U j ) := (cid:110) ψ kγ ( U i ) ,η ( U j ) (cid:111) k ∈ N : γ ( U i ) ∩ η ( U j ) → L ( (cid:81) B k ) (cid:32)(cid:89) k E k (cid:33) G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds are Lipschitz continuous. So it is allowed to use them to define the Hilbert (cid:81) k B k -module bundle V as required. Precisely V can be constructed as follows; V := (cid:71) γ,i (cid:32) γ ( U i ) × (cid:89) k E k (cid:33) (cid:46) ∼ . Here, ( x, v ) ∈ γ ( U i ) × (cid:81) k E k and ( y, w ) ∈ η ( U j ) × (cid:81) k E k are equivalent if and only if x = y ∈ γ ( U i ) ∩ η ( U j ) (cid:54) = ∅ and Ψ γ ( U i ) ,η ( U j ) ( v ) = w . By the construction of V , if p n : (cid:81) k B k → B n denotes theprojection onto the n -th component, V (cid:98) ⊗ p n B n is isomorphic to the original n -th component E n .In order to verify the continuity of the induced connection, let { e i } be any orthonormal local frameon U i for an arbitrarily fixed E k obtained by the parallel transport along the minimal geodesic fromthe center x i ∈ U i , namely, e i ( y ) = Φ x i ; y e i ( x i ). It is sufficient to verify that (cid:13)(cid:13) ∇ k e i (cid:13)(cid:13) < C . Let v ∈ T y X be a unit tangent vector and p ( t ) := exp y ( tv ) be the geodesic of unit speed with direction v . ∇ kv e i ( y ) = lim t → t (cid:0) Φ p ( t ); p (0) e i ( p ( t )) − e i ( p (0)) (cid:1) = lim t → t (cid:0) Φ p ( t ); p (0) Φ x i ; p ( t ) − Φ x i ; p (0) (cid:1) e i ( x i ) , (cid:13)(cid:13)(cid:13) ∇ kv e i ( y ) (cid:13)(cid:13)(cid:13) ≤ lim t → | t | (cid:13)(cid:13) Φ p ( t ); p (0) Φ x i ; p ( t ) − Φ x i ; p (0) (cid:13)(cid:13) ≤ lim t → | t | ε · area( D ( t )) , where D ( t ) is a 2-dimensional disk in U i spanning the piece-wise geodesic connecting x i , p (0) = y , p ( t ) and x i . As above, we can find a constant C > D ( t ) satisfyingarea( D ( t )) ≤ C · dist( p (0) , p ( t )) = C | t | for | t | (cid:28)
1. Hence, we obtain (cid:13)(cid:13) ∇ kv e i ( y ) (cid:13)(cid:13) ≤ Cε.
Definition 4.16
Let us define a Hilbert ( Q B k )-module bundle W := V (cid:98) ⊗ π ( Q B k ) , where π : (cid:81) B k (cid:16) Q B k denotes the projection.The family of parallel transport of E k induces the parallel transport Φ W of W which commutes withthe action of G . Proposition 4.17
If the parallel transport of E k is C k -close to the identity with C k (cid:38) , thenthe G -bundle W constructed above is a flat bundle. More precisely the parallel transport Φ W ( p ) ∈ Hom( W x , W y ) depends only on the ends-fixing homotopy class of p ∈ P ( X )[ x, y ] .Proof. It is sufficient to prove that for any contractive loop p ∈ P ( X )[ x, x ], it satisfies Φ W ( p ) = id W x .Fix a two dimensional disk D ⊂ X spanning the loop p . For arbitrary ε > n such thatevery k ≥ n satisfies that Φ E k is ε D ) -close to the identity. (cid:13)(cid:13) Φ W ( p ) − id W x (cid:13)(cid:13) = lim sup k →∞ (cid:13)(cid:13)(cid:13) Φ E k ( p ) − id (cid:13)(cid:13)(cid:13) ≤ sup k ≥ n (cid:13)(cid:13)(cid:13) Φ E k ( p ) − id (cid:13)(cid:13)(cid:13) ≤ ε D ) · area( D ) ≤ ε This implies Φ W ( p ) = id W x . . Fukumoto Proposition 4.18 (1) Let p n : (cid:81) B k → B n denote the projection onto the n -th component and consider (1 ⊗ p n ) ∗ : K (cid:16) C ∗ ( G ) (cid:98) ⊗ (cid:16)(cid:89) B k (cid:17)(cid:17) → K (cid:0) C ∗ ( G ) (cid:98) ⊗ B n (cid:1) . Then (1 ⊗ p n ) ∗ ind G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) = ind G (cid:0) [ E n ] (cid:98) ⊗ [ D ] (cid:1) . (2) Let π : (cid:81) B k → Q B k denote the quotient map and consider (1 ⊗ π ) ∗ : K (cid:16) C ∗ ( G ) (cid:98) ⊗ (cid:16)(cid:89) B k (cid:17)(cid:17) → K (cid:0) C ∗ ( G ) (cid:98) ⊗ ( Q B k ) (cid:1) . Then (1 ⊗ π ) ∗ ind G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) = ind G ([ W ] (cid:98) ⊗ [ D ]) . Proof.
As for the first part, [ E n ] = ( p n ) ∗ (cid:2)(cid:81) E k (cid:3) ∈ KK G ( C ( X ) , C ( X ) (cid:98) ⊗ B n ) by the construction of (cid:81) E k . Then it follows thatind G (cid:0) [ E n ] (cid:98) ⊗ [ D ] (cid:1) = ind G (cid:16) ( p n ) ∗ (cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) = [ c ] (cid:98) ⊗ j G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:98) ⊗ p n (cid:17) = [ c ] (cid:98) ⊗ j G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) (cid:98) ⊗ j G ( p n )= ind G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) (cid:98) ⊗ j G ( p n )= (1 ⊗ p n ) ∗ ind G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) , where p n = ( B n , p n , ∈ KK (cid:0)(cid:81) k ∈ N B k , B n (cid:1) . Then note that j G ( p n ) = ( C ∗ ( G ) (cid:98) ⊗ B n , (cid:98) ⊗ p n , ∈ KK (cid:0) C ∗ ( G ) (cid:98) ⊗ (cid:0)(cid:81) k ∈ N B k (cid:1) , C ∗ ( G ) (cid:98) ⊗ B n (cid:1) . Since π ∗ (cid:2)(cid:81) E k (cid:3) = [ W ] ∈ KK G (cid:0) C ( X ) , C ( X ) (cid:98) ⊗ ( Q B k ) (cid:1) bythe construction of W , the second part can be proved in the similar way. Proposition 4.19
Let [ D ] be a K -homology element in KK G ( C ( X ) , C ) determined by a Dirac oper-ator on a G -Hermitian vector bundle V over X . Suppose that W is a finitely generated flat B -module G -bundle. Assume that X is simply connected.Then ind G (cid:0) [ W ] (cid:98) ⊗ [ D ] (cid:1) = 0 ∈ K (cid:0) C ∗ ( G ) (cid:98) ⊗ B (cid:1) if ind G ([ D ]) = 0 . In order to prove this, we introduce an element [ W ] rpn ∈ KK G ( C , B ) using the holonomy representa-tion. Definition 4.20 • Let Φ x ; y denote the parallel transport of W along an arbitrary path from x ∈ X to y ∈ X . Since X is simply connected and W is flat, it depends only on the ends of the path. • Let us fix a base point x ∈ X and W x be the fiber on x . Define [ W ] rpn as[ W ] rpn := ( W x , ∈ KK G ( C , B )The action of G on W x is given by the holonomy ρ : G → End Q ( W x ) ρ [ γ ]( w ) = (Φ x ; γx ) − γ ( w ) for γ ∈ G, w ∈ W x G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds Lemma 4.21 [ W ] (cid:98) ⊗ C ( X ) [ D ] = [ D ] (cid:98) ⊗ C [ W ] rpn ∈ KK G ( C ( X ) , B ) . Proof.
Recall that [ D ] ∈ KK G ( C ( X ) , C ) is given by (cid:0) L ( X ; V ) , F D (cid:1) , where F D denotes the operator D √ D , and that [ W ] (cid:98) ⊗ C ( X ) [ D ] = (cid:16) C ( X ; W ) (cid:98) ⊗ C ( X ) L ( X ; V ) , F D W (cid:17) , where D W is the Dirac operator twisted by W acting on L ( X ; W ⊗ V ) (cid:39) C ( X ; W ) (cid:98) ⊗ C ( X ) L ( X ; V ),that is, D W = (cid:88) j (id W ⊗ c ( e j )) (cid:16) ∇ We j ⊗ id V + id W ⊗ ∇ V e j (cid:17) , where { e j } denotes a orthogonal basis for T X and c ( · ) denotes the Clifford multiplication by Cliff( T X )on V . The action of C ( X ) on C ( X ; W ) and L ( X ; V ) are the point-wise multiplications. On theother hand, [ D ] (cid:98) ⊗ C [ W ] rpn = (cid:0) L ( X ; V ) (cid:98) ⊗ C W x , F D (cid:98) ⊗ (cid:1) . The action of C ( X ) is the point-wise multiplications. Note that the action of G on W x is given bythe holonomy representation ρ . It is sufficient to give a G -equivariant isomorphism ϕ : L ( X ; V ) (cid:98) ⊗ C W x → C ( X ; W ) (cid:98) ⊗ C ( X ) L ( X ; V ) , which is compatible with D W and D (cid:98) ⊗
1. Set a section for W given by w : x (cid:55)→ Φ x ; x w ∈ W x (4.3)and define ϕ on a dense sub space C c ( X ; V ) (cid:98) ⊗ W x as ϕ ( s ⊗ w ) := w · χ ⊗ s for s ∈ C c ( X ; V ) and w ∈ W x , where χ ∈ C ( X ) is an arbitrary compactly supported function on X with values in [0 ,
1] satisfyingthat χ ( x ) = 1 for all x ∈ supp( s ). ϕ is independent of the choice of χ and hence well-defined. Indeed, Let χ (cid:48) ∈ C c ( X ) be another suchfunction, and let ρ ∈ C c ( X ) be a compactly supported function on X with values in [0 ,
1] satisfyingthat ρ ( x ) = 1 for all x ∈ supp( χ ) ∪ supp( χ (cid:48) ). Then in C ( X ; W ) (cid:98) ⊗ C ( X ) C c ( X ; V ), w · χ ⊗ s − w · χ (cid:48) ⊗ s = w · (cid:0) χ − χ (cid:48) (cid:1) ⊗ s = w · ρ · (cid:0) χ − χ (cid:48) (cid:1) ⊗ s = w · ρ ⊗ (cid:0) χ − χ (cid:48) (cid:1) s = 0 . Now we obtain that D W ◦ ϕ ( s ⊗ w ) = D W ( w ⊗ s ) = w ⊗ D ( s ) = ϕ ◦ ( D (cid:98) ⊗ s ⊗ w )for s ∈ C c ( V ) and w ∈ W x . This is because ∇ W w = 0 by its construction.Compatibility with the action of G is verified as follows; ϕ ( γ ( s ⊗ w ))( x ) = Φ x ; x ( ρ [ γ ]( w )) ⊗ γ ( s ( γ − x )) = Φ x ; x (Φ x ; γx ) − γ ( w ) ⊗ γ ( s ( γ − x ))= Φ γx ; x ( γ ( b )) ⊗ γ ( s ( γ − x )) ,γ ( ϕ ( s ⊗ w ))( x ) = γ ((Φ x ; γ − x )( w ) ⊗ s ( γ − x )) = Φ γx ; x ( γ ( w )) ⊗ γ ( s ( γ − x )) . . Fukumoto ϕ induces an isomorphism. For s ⊗ w , s ⊗ w ∈ C c ( X ; V ) (cid:98) ⊗ C W x , it followsthat (cid:68) ϕ ( s ⊗ w ) , ϕ ( s ⊗ w ) (cid:69) C ( X ; W ) (cid:98) ⊗ C X ; X ) L ( X ; V ) = (cid:68) s , (cid:104) w · χ, w · χ (cid:105) C ( X ; W ) s (cid:69) L ( X ; V ) = (cid:90) X (cid:68) s ( x ) , (cid:10) (Φ x ; x w ) χ ( x ) , (Φ x ; x w ) χ ( x ) (cid:11) W x s ( x ) (cid:69) V x dvol( x )= (cid:90) X (cid:104) w , w (cid:105) W χ ( x ) (cid:104) s ( x ) , s ( x ) (cid:105) V x dvol( x )= (cid:104) w , w (cid:105) W (cid:104) s , s (cid:105) L ( X ; V ) = (cid:104) s ⊗ w , s ⊗ w (cid:105) L ( X ; V ) (cid:98) ⊗ W x , where χ ∈ C ( X ) is a compactly supported function on X satisfying that χ ( x ) = 1 for all x ∈ supp( s ) ∪ supp( s ). This implies that ϕ is continuous and injective.Moreover, choose arbitrary F ∈ C c ( X ; W ) and s ∈ C c ( X ; V ). Since Φ − x ; x provides a trivializationof W (cid:39) X × W x , we have an isomorphism C c ( X ; W ) (cid:39) C c ( X ) (cid:98) ⊗ C W x . Remark that, however, this isnot a G -equivariant isomorphism, just as pre-Hilbert (cid:0) C ( X ; B ) ∼ = C ( X ) (cid:98) ⊗ B (cid:1) -modules. Then thereexist countable subsets { f , f , . . . } ⊂ C c ( X ) and { w , w , . . . } ⊂ W x satisfying that (cid:80) j ∈ N f j w j = F in C ( X ; W ). Now it follows that ϕ (cid:88) j ∈ N f j s ⊗ w j = (cid:88) j ∈ N (cid:16) w j · χ ⊗ f j s (cid:17) = (cid:88) j ∈ N (cid:16) w j · χf j ⊗ s (cid:17) = (cid:88) j ∈ N w j f j · χ ⊗ s = F ⊗ s, where χ ∈ C ( X ) is a compactly supported function on X satisfying that χ ( x ) = 1 for all x ∈ supp( F ) ∪ supp( s ). This implies that the image of ϕ is dense in C ( X ; W ) (cid:98) ⊗ L ( X ; V ). Therefore ϕ induces an isomorphism. Proof of the Proposition 4.19 . Due to the previous lemma, it follows thatind G (cid:0) [ W ] (cid:98) ⊗ [ D ] (cid:1) = ind G ([ D ] (cid:98) ⊗ [ W ] rpn )= [ c ] (cid:98) ⊗ j G (cid:0) [ D ] (cid:98) ⊗ [ W ] rpn (cid:1) = [ c ] (cid:98) ⊗ (cid:0) j G [ D ] (cid:1) (cid:98) ⊗ (cid:0) j G [ W ] rpn (cid:1) = (ind G [ D ]) (cid:98) ⊗ (cid:0) j G [ W ] rpn (cid:1) . Thus the assumption ind G [ D ] = 0 implies ind G (cid:0) [ W ] (cid:98) ⊗ [ D ] (cid:1) = 0 Proof of Theorem C . As Remark 4.4, we may assume that D is a Dirac type operator. Assume thatind G [ D ] = 0 and we assume the converse. that is, for each k ∈ N there exits a Hilbert B k -module G -bundle E k over X whose curvature norm is less than k satisfying thatind G ([ E k ] (cid:98) ⊗ [ D ]) (cid:54) = 0 ∈ K ( C ∗ ( G ) (cid:98) ⊗ B k ) . To begin with, we have an exact sequence;0 → (cid:77) B k ι −−−→ (cid:89) B k π −−−→ Q B k → , G -Homotopy Invariance of the Analytic Signature of Proper Co-compact G -manifolds where ι and π are natural inclusion and projection. We also have the following exact sequence [We,Theorem T.6.26];0 → C ∗ Max ( G ) (cid:98) ⊗ Max (cid:16)(cid:77) B k (cid:17) (cid:98) ⊗ ι −−−−→ C ∗ Max ( G ) (cid:98) ⊗ Max (cid:16)(cid:89) B k (cid:17) (cid:98) ⊗ π −−−−−→ C ∗ Max ( G ) (cid:98) ⊗ Max ( Q B k ) → . We have the exact sequence of K -groups K (cid:16) C ∗ Max ( G ) (cid:98) ⊗ Max (cid:16)(cid:77) B k (cid:17)(cid:17) → K (cid:16) C ∗ Max ( G ) (cid:98) ⊗ Max (cid:16)(cid:89) B k (cid:17)(cid:17) → K (cid:0) C ∗ Max ( G ) (cid:98) ⊗ Max ( Q B k ) (cid:1) . If all of B k are commutative, then Q B k is also commutative and hence nuclear. In that case, wealso have the same exact sequences in which C ∗ Max ( G ) and (cid:98) ⊗ Max are replaced by C ∗ red ( G ) and (cid:98) ⊗ min respectively.Let us start with ind G (cid:0)(cid:2)(cid:81) E k (cid:3) (cid:98) ⊗ [ D ] (cid:1) ∈ K (cid:0) C ∗ ( G ) (cid:98) ⊗ ( (cid:81) B k ) (cid:1) .Due to the flatness of W (Proposition 4.17) and Proposition 4.19 and 4.18, we have(1 (cid:98) ⊗ π ) ∗ ind G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) = ind G (cid:0) [ W ] (cid:98) ⊗ [ D ] (cid:1) = 0 . It follows from the exactness that there exists ζ ∈ K (cid:0) C ∗ ( G ) (cid:98) ⊗ ( (cid:76) B k ) (cid:1) such that(1 (cid:98) ⊗ ι ) ∗ ( ζ ) = ind G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17) . Lemma 4.22 A (cid:98) ⊗ (cid:0)(cid:76) k ∈ N B k (cid:1) is naturally isomorphic to (cid:76) k ∈ N (cid:0) A (cid:98) ⊗ B k (cid:1) = lim −→ n (cid:76) nk =1 (cid:0) A (cid:98) ⊗ B k (cid:1) .Proof. Let C denote the direct product lim −→ n (cid:76) nk =1 (cid:0) A (cid:98) ⊗ B k (cid:1) . Note that for the finite direct product,we have the natural isomorphism (cid:76) nk =1 (cid:0) A (cid:98) ⊗ B k (cid:1) ∼ = A (cid:98) ⊗ ( (cid:76) nk =1 B k ). For each n ∈ N , we have thefollowing commutative diagram: A (cid:98) ⊗ ( (cid:76) nk =1 B k ) id A (cid:98) ⊗ ι n +1 n (cid:47) (cid:47) id A (cid:98) ⊗ ι n (cid:40) (cid:40) A (cid:98) ⊗ (cid:16)(cid:76) n +1 k =1 B k (cid:17) id A (cid:98) ⊗ ι n +1 (cid:15) (cid:15) A (cid:98) ⊗ (cid:0)(cid:76) k ∈ N B k (cid:1) . Now by using the universal property of the direct limit, we obtain a map φ : A (cid:98) ⊗ ( (cid:76) nk =1 B k ) (cid:47) (cid:47) (cid:40) (cid:40) lim −→ n A (cid:98) ⊗ ( (cid:76) nk =1 B k ) φ (cid:15) (cid:15) A (cid:98) ⊗ (cid:0)(cid:76) k ∈ N B k (cid:1) . Since id A (cid:98) ⊗ ι n are isometric and injective, φ is isometric and injective on each sub-space A (cid:98) ⊗ ( (cid:76) nk =1 B k ) ⊂ lim −→ n A (cid:98) ⊗ ( (cid:76) nk =1 B k ). Since the union of such sub-spaces is dense in lim −→ n A (cid:98) ⊗ ( (cid:76) nk =1 B k ), it follows that φ itself is isometric and injective.As for the surjectivity of φ , take any a (cid:98) ⊗{ b k } ∈ A (cid:98) ⊗ (cid:0)(cid:76) k ∈ N B k (cid:1) . For any ε >
0, there exists n ∈ N such that (cid:107) b k (cid:107) < ε (cid:107) a (cid:107) for k ≥ n . Then replace b k by 0 for all k ≥ n to obtain an element β := { b , b , . . . n , n +1 , . . . } ∈ (cid:76) k ∈ N B k . Now we have that a (cid:98) ⊗ β = (id A (cid:98) ⊗ ι n )( a (cid:98) ⊗{ b , b , . . . b n − } ) = φ ( a (cid:98) ⊗{ b , b , . . . b n − } ) ∈ Im( φ )and (cid:13)(cid:13) a (cid:98) ⊗{ b k } − a (cid:98) ⊗ β (cid:13)(cid:13) ≤ (cid:107) a (cid:107) (cid:107){ b k } − β (cid:107) ≤ ε . These imply that Im( φ ) is dense in A (cid:98) ⊗ (cid:0)(cid:76) k ∈ N B k (cid:1) andhence, φ is surjective since it has closed range. . Fukumoto C ∗ ( G ) (cid:98) ⊗ ( (cid:76) B k ) is naturally isomorphic to (cid:76) (cid:0) C ∗ ( G ) (cid:98) ⊗ B k (cid:1) . Besides, we have the nat-ural isomorphism K (cid:0)(cid:76) (cid:0) C ∗ ( G ) (cid:98) ⊗ B k (cid:1)(cid:1) (cid:39) (cid:76) K (cid:0) C ∗ ( G ) (cid:98) ⊗ B k (cid:1) [HR, 4.1.15 Proposition, 4.2.3 Remark],with the last (cid:76) meaning the algebraic direct sum. Thus we can consider the following diagram; K (cid:0) C ∗ ( G ) (cid:98) ⊗ ( (cid:76) B k ) (cid:1) ι ∗ −−−−→ K ( C ∗ ( G ) ⊗ ( (cid:81) B k )) (1 ⊗ π ) ∗ −−−−→ K ( C ∗ ( G ) ⊗ Q B k ) { (1 ⊗ p k ) ∗ } (cid:121) ∼ = { (1 ⊗ p k ) ∗ } (cid:121)(cid:76) K (cid:0) C ∗ ( G ) (cid:98) ⊗ B k (cid:1) −−−−−→ inclusion (cid:81) K (cid:0) C ∗ ( G ) (cid:98) ⊗ B k (cid:1) Since p k = ιp k , this diagram commutes. Note that both (cid:76) and (cid:81) in the bottom row are in thealgebraic sense. Again due to Proposition 4.18, (cid:110) ind G (cid:16) [ E k ] (cid:98) ⊗ [ D ] (cid:17)(cid:111) k ∈ N = { (1 ⊗ p k ) ∗ } (cid:16) ind G (cid:16)(cid:104)(cid:89) E k (cid:105) (cid:98) ⊗ [ D ] (cid:17)(cid:17) = { (1 ⊗ p k ) ∗ } ((1 ⊗ ι ) ∗ ( ζ ))= { (1 ⊗ p k ) ∗ } ( ζ ) ∈ (cid:77) K (cid:0) C ∗ ( G ) (cid:98) ⊗ B k (cid:1) . This implies that all of ind G (cid:0) [ E n ] (cid:98) ⊗ [ D ] (cid:1) ∈ K (cid:0) C ∗ ( G ) (cid:98) ⊗ B n (cid:1) are equal to zero except for finitely many n ∈ N , which contradicts to our assumption. To prove Corollary D, we will combine Theorem C with Theorem A. Consider the same conditionsas Theorem A on X , Y and G and assume additionally that X and Y are simply connected. Let f : Y → X be a G -equivariant orientation preserving homotopy invariant map. Assume that for each k ∈ N there exits a Hilbert B k -module G -bundle E k over X whose curvature norm is less than k satisfying that ind G ([ E k ] (cid:98) ⊗ [ ∂ X ]) (cid:54) = ind G ([ f ∗ E k ] (cid:98) ⊗ [ ∂ Y ]) ∈ K ( C ∗ ( G ) (cid:98) ⊗ B k ) . as in the proof of Theorem C. Consider a G -manifold Z := X (cid:116) ( − Y ), the disjoint union of X andorientation reversed Y and the signature operator ∂ Z on it. Although Z is not connected, however,we may apply Theorem C to ∂ Z , after replacing some argument in the proof as follows. Consider afamily of Hilbert B k -module bundles { E k (cid:116) f ∗ E k } over Z and obtain a flat bundle W (cid:116) f ∗ W as insubsection 4.2. In order to obtain a global section w as in (4.3) in the proof of Lemma 4.21, we haveused the connectedness of the base space. In this case, construct a section w : X → W on X in thesame way and pull back it on Y by f to obtain a global section on Z . The other parts are the sameas above. Acknowledgements
The author is supported by Natural Science Foundation of China (NSFC) Grant Number 11771143.
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