Higher rho invariant and delocalized eta invariant at infinity
aa r X i v : . [ m a t h . K T ] A p r HIGHER RHO INVARIANT AND DELOCALIZED ETAINVARIANT AT INFINITY
XIAOMAN CHEN, HONGZHI LIU, HANG WANG, AND GUOLIANG YU
Abstract.
In this paper, we introduce several new secondary invariants forDirac operators on a complete Riemannian manifold with a uniform positivescalar curvature metric outside a compact set and use these secondary invari-ants to establish a higher index theorem for the Dirac operators. We applyour theory to study the secondary invariants for a manifold with corner withpositive scalar curvature metric on each boundary face.
Contents
1. Introduction 22. Preliminary 62.1. Geometric C ∗ -algebras 62.2. Engel-Samukar¸s smooth subalgebra 82.3. The Connes-Moscovici smooth subalgebra 103. Secondary invariants at infinity 113.1. Higher index 113.2. Higher index: a different approach 123.3. Higher rho invariant at infinity 173.4. Delocalized eta invariant at infinity 203.5. A formula for the higher index 213.6. Formula for the delocalized eta-invariant at infinity 234. Manifold with corner of codimension 2 274.1. M¨uller’s Atiyah-Patodi-Singer type theorem 284.2. Higher rho invariant at infinity on manifold with corner. 33References 41 Date : April 2, 2020.The first author is partially supported by NSFC 11420101001.The second author is partially supported by NSFC 11901374.The third author is partially supported by the Shanghai Rising-Star Program grant19QA1403200, and by NSFC 11801178.The fourth author is partially supported by NSF 1700021, NSF 1564398, and Simons FellowsProgram. Introduction
In this article, we introduce a new theory of secondary invariants for Dirac oper-ators on noncompact spin manifolds endowed with metrics with uniform positivescalar curvature at infinity, i.e. higher rho invariants and the delocalized etainvariants. Let X be a complete spin manifold with metric admitting uniformpositive scalar curvature outside a compact set Z , D be the Dirac operator onthe universal covering space e X of X , and let G be π ( X ), the fundamental groupof X . We obtain a higher index formula for the Dirac operator D and express thedelocalized trace of the higher index in terms of delocalized secondary invariantsat infinity, as follows: Theorem 1.1.
Let D be the Dirac operator on the universal covering space e X of a spin manifold X with uniform positive scalar curvature outside a compactsubset Z defined as above. Let G be the fundamental group of X . Let g ∈ G be anontrivial element whose conjugacy class has polynomial growth. Let ind G ( D ) bethe higher index of the Dirac operator D . The delocalized trace at g of the higherindex of D , tr g (ind G ( D )) equals the half of the negative delocalized eta invariantat infinity, η g, ∞ ( D ) , i.e. (1.1) tr g (ind G ( D )) = − η g, ∞ ( D ) . Furthermore, we have (1.2) 12 η g, ∞ ( D ) = lim t → Z ∞ t tr g ( e − sD − c D + c D − c [ D + , ψ ]) ds. where D c is the invertible Dirac operator on e X \ e Z , and ψ is a G -invariant cutofffunction from e X to [0 , , which equals on a cocompact neighbourhood of e Z , the G -Galois covering space of Z , and equals outside a cocompact set larger than Z . The integral on the right hand side of (1.2) is independent of the choice ofthe cutoff function ψ . The higher index ind G ( D ) of D was first introduced by Bunke (cf. [6]). Themain significance and subtlety of the higher index formula in Theorem 1.1 is thattr g (ind G ( D )) vanishes when X itself is compact (cf. [27]). Furthermore, when X is a cylindrical manifold obtained from a compact manifold with boundary ∂M ,and with positive scalar curvature on the boundary which is collared near ∂M ,then tr g (ind G ( D )) equals the delocalized eta invariant of the Dirac operator on ∂M .Equation (1.1) is obtained by considering the higher rho invariant at infinity.These secondary invariants we introduce demonstrate the nonlocality of the higher IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 3 index of D . We also apply the higher rho invariant at infinity to analyze the ge-ometry of manifolds with corner of codimension 2 with positive scalar curvaturemetrics on all boundary faces.The classical eta invariant is a nonlocal spectral invariant of Dirac type operators(cf. [1, 2, 3, 4]). Let M be a closed odd dimensional spin manifold with a positivescalar curvature metric, D M be the Dirac operator on M . When the followingintegral converges, the eta invariant of D M can be expressed as2 √ π Z ∞ tr( e − t D M D M ) dt where tr is the operator trace. To take into account of the information of thefundamental group, Lott introduced the delocalized eta invariant (cf. [14]). Moreprecisely, let G be the fundamental group of M , f M be the universal covering spaceof M , and e D M be the lifting of D M to f M . Let g be a nontrivial element of G ,whose conjugacy class h g i has polynomial growth. The delocalized eta invariantof e D M at g , η g ( e D M ), introduced by Lott in [14], is defined by the followingintegration 2 √ π Z ∞ tr g ( e − t e D M e D M ) dt. Here tr g is the following trace maptr g ( A ) = X h ∈h g i Z F A ( x, hx ) dx, on G -equivariant Schwartz kernels A ∈ C ∞ ( f M × f M ), where F is a fundamentaldomain of f M under the G -action. One can also define tr g for continuous group,see [13] for example. There is also a higher generalization of the pairing betweentr g and K -theory of geometric C ∗ -algebras, introduced by [17], [7], [19], and [23],which is to consider cyclic cocycles.Since the metric m on M admits positive scalar curvature, it follows from theLichnerowicz formula that the higher index of D M , ind G ( e D M ), in the K -theoryof the group C ∗ -algebra C ∗ r ( G ), is trivial with a specific trivialization. In thiscase, Higson and Roe proposed to study a secondary invariant in K -theory of acertain C ∗ -algebra, the higher rho invariant of e D M , ρ ( e D M ) (cf. [21, 11]). Thehigher rho invariant of the Dirac operator has been applied to estimate the lowerbound of how many positive scalar curvature metrics a manifold can bear (cf.[18, 31, 30, 32, 27]). The higher rho invariant is closely related to the delocalizedeta invariant. Let g be a nontrivial element of G , whose conjugacy class h g i haspolynomial growth. In [29], Xie and Yu defined a canonical determinant map τ g associated to g , and showed that τ g ( ρ ( e D M )) = η g ( e D M ). XIAOMAN CHEN, HONGZHI LIU, HANG WANG, AND GUOLIANG YU
The eta invariants and the higher rho invariants appear in the study on thegeometry of manifolds with boundary naturally. Let M be an even dimensionalspin manifold with boundary N , with the metric m having product structurenear N , and admitting positive scalar curvature when restricted to N . Thenthe eta invariant of the Dirac operator D N on the boundary, η ( D N ), is the thecorrection term in the formula of the Fredholm index of the Dirac operator D M ∞ on M ∞ := M ⊔ ∂M × [0 , ∞ ). Let index( D M ∞ ) be the Fredholm index of D M ∞ .Then(1.3) index( D M ∞ ) = Z M ˆ A ( M ) − η ( D N )2 , where ˆ A ( M ) is the ˆ A genus (cf. [1, 2, 3, 4]). The eta invariant explains thenonlocality of the Fredholm index. Equation 1.3 is often referred to as theAtiyah-Patodi-Singer index theorem. On the other hand, the delocalized etainvariant and the higher rho invariant capture the nonlocality of the higher in-dex of D M ∞ . In fact, in [29, 7], Xie-Yu and Chen-Wang-Xie-Yu obtained severalhigher generalizations of the Atiyah-Patodi-Singer index theorem. In particular,for any nontrivial element g ∈ G = π ( M ) whose conjugacy class is of polynomialgrowth, they proved(1.4) tr g (ind G ( e D M ∞ )) = − τ g ( ρ ( e D N )) = − η g ( e D N ) . Since the metric on M ∞ has positive scalar curvature outside the compact set M , the Dirac operator D M ∞ is invertible at infinity. In this particular case, thedelocalized eta invariant and the higher rho invariant of D N can be viewed assecondary invariants at infinity associated to D M ∞ . The study of the secondaryinvariant at infinity had led to several major breakthroughs to the estimate of thelower bound of the rank of the abelian group formed by the concordance classesof positive scalar curvature metric (cf. [30, 31, 32, 33]). In the meanwhile, aparallel study to the secondary invariants at infinity allows one to estimate thelower bound of the topological structure group ([26]).The first main result, Theorem 1.1, is a generalization of the above Atiyah-Patodi-Singer index theorem and its higher counterpart to the case of a noncompactcomplete manifold with uniform positive scalar curvature metric at infinity.Let X be an n -dimensional complete spin manifold and Z be a compact subset of X , let m be a metric on X , which has uniformly positive scalar curvature outside Z , and G be the fundamental group of X . Let e X be the universal covering spaceof X and e Z be the induced G -Galois covering space of Z . In this case, following[22] and [30], one can define the higher index of the Dirac operator D on e X in IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 5 K n ( C ∗ r ( G )). See Subsections 3.1 and 3.2 for details. The higher index of D isdenoted as ind G ( D ) or ind e X, e Z,T ( D ), where T is any sufficiently large number.We emphasize that the definition of the higher index of D is independent of T .However, the subscript T manifests the particular choice of a representative classof ind G ( D ). In Subsection 3.3, the higher rho invariant at infinity of D , denotedas ρ e X, e Z,T ( D ), is defined as the image of ind e X, e Z,T ( D ) under the connecting mapof a K -theoretic six-term exact sequence associated to a canonical long exactsequence of geometric C ∗ -algebras. Moreover, for nontrivial element g ∈ G ,whose conjugacy class is of polynomial growth, the delocalized eta invariant atinfinity of D is simply defined to be(1.5) 2 τ g ( ρ e X, e Z,T ( D )) . As in the case of manifold with cylindrical end, the number in line (1.5) is essentialfor us to obtain Equation (1.1) in Theorem 1.1.In the meanwhile, Equation (1.2) in Theorem 1.1 is established by applying themethod in [13], which is invented when Hochs, Wang and Wang was to develop anew approach to obtain a refinement of the Atiyah-Patodi-Singer index theorem.See Section 3.6 for details.The higher rho invariant at infinity and the delocalized eta invariant at infinityfor D exhibit the nonlocality of the higher index of D . They can be appliedto the study of the geometry of noncompact complete manifolds with uniformpositive scalar curvature outside a compact set. As an example, we apply ourtheory to study manifolds with corners. Our theory on manifolds with cornersare inspired by [9], [15] and [16]. In Section 4, we first obtain a new proof and an L generalization of M¨uller’s Atiyah-Patodi-Singer type index theorem (cf. [16,Theorem 0.1]). Then we generalize M¨uller’s Atiyah-Patodi-Singer type indextheorem to higher case by considering the higher rho invariant at infinity. Moreprecisely, let M be a compact manifold with corner with ∂M = ∂ M ∪ ∂ M ,where ∂ M and ∂ M are manifolds with boundary and ∂∂ M = ∂∂ M = Y . Let Z i be ∂ i M ⊔ Y × R + , i = 1 ,
2. The metric of M is assumed to be collared near ∂M and has positive scalar curvature on ∂ M and ∂ M . Furthermore, the metric on ∂ M and ∂ M is collared near Y . Let G be the fundamental group of M . Thenthe naturally induced metric on CM := M ⊔ ∂ M × R + ⊔ ∂ M × R + ⊔ Y × R admits uniform positive scalar curvature outside a compact set. Note that π ( CM ) ∼ = π ( M ) = G . To obtain the higher version of M¨uller’s Atiyah-Patodi-Singer type index theorem, we consider the higher rho invariant at infinity of theDirac operator D on g CM . For a spin manifold N with boundary ∂N endowedwith a positive scalar curvature metric, which is collared near the boundary, we XIAOMAN CHEN, HONGZHI LIU, HANG WANG, AND GUOLIANG YU define the higher rho invariant of the Dirac operator D e N on e N as ρ e N, g ∂N ( D e N ).With the higher rho invariant for Dirac operator on manifold with boundary, weobtain the following combinatorial formula for the higher rho invariant at infinity: Theorem 1.2.
For any sufficiently large T , we have that ρ g CM, f M,T ( D ) = ρ g ∂ M, ^ ∂∂ M ( D g ∂ M ) + ρ g ∂ M, ^ ∂∂ M ( D g ∂ M ) . See Theorem 4.6 for details.The paper is organized as follows. In Section 2, we recall basic concepts, in-cluding geometric C ∗ -algebras and their smooth subalgebras, which will be usedlater in the paper. In Section 3, we define two secondary invariants at infinityfor the Dirac operator on a complete manifold with uniform positive scalar cur-vature metric outside a compact set, the higher rho invariant at infinity and thedelocalized eta invariant at infinity. We establish a formula for the delocalizedeta invariant at infinity in Subsection 3.6, together with which we generalize theAtiyah-Patodi-Singer index theorem. In Section 4 we apply the theory developedin Section 3 to study invariants associated to the Dirac operator on a manifoldwith corner endowed with positive scalar curvature metrics on all boundary faces.2. Preliminary
In this section, we introduce some notions and concepts used in this paper, in-cluding geometric C ∗ -algebras and their smooth subalgebras. All the groupsconsidered in this paper are finitely generated discrete groups. Denote by | · | theword length metric of a group G . For an element g in G , we say its conjugacyclass, h g i has polynomial growth if there exist constants C and d , such that ♯ { h ∈ h g i , | g | ≤ n } ≤ Cn d . Geometric C ∗ -algebras. In this subsection, we recall definitions of severalgeometric C ∗ -algebras, including equivariant Roe, localization and obstructionalgebras (see [12, 20, 34, 22, 28] for more details).Let X be a complete Riemannian manifold where G acts properly, cocompactlyand freely. An X -module is a separable Hilbert space equipped with a ∗ repre-sentation of C ( X ). It is nondegenerate if the ∗ representation is nondegenerate.It is called standard if no nonzero function in C ( X ) acts as a compact operator.Let H X be a standard nondegenerate X -module, where H X admits a unitaryrepresentation of the group G , and the representation of C ( X ) is covariant tothe group representation.We first recall some ingredients for constructing the geometric C ∗ -algebras. IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 7
Definition 2.1. let H X be an X -module and T ∈ B ( H X ) be a bounded linearoperator. • The propagation of T is defined to besup { d ( x, y ) | ( x, y ) ∈ Supp( T ) } , where Supp( T ) is the complement in X × X of the set of points ( x, y ) ∈ X × X for which there exist f, g ∈ C ( X ) satisfying f ( x ) = 0 , g ( y ) = 0such that gT f = 0. T is said to have finite propagation if its propagationis finite. The propagation of a finite propagation operator T is denotedas propagation( T ). • T is said to be locally compact if f T and T f are compact for all f ∈ C ( X ). • T is said to be pseudo-local if [ T, f ] is compact for all f ∈ C ( X ). • Let Z be a G -invariant subspace of X . We say T is supported near Z ifthere is λ >
0, such that for all f ∈ C ( X ) whose support is at least λ away from Z , the operators T f and f T are zero.The following are definitions of the geometric C ∗ -algebras that will be used inthis paper. Definition 2.2.
Let H X be a standard nondegenerate X -module and B ( H X ) bethe operator algebra of all bounded linear operators on H X . • G -equivariant Roe algebra C ∗ ( X ) G is the C ∗ -algebra generated by locallycompact G -invariant operators with finite propagation. • Let Z be a G -invariant subspace of X . The localized G -equivariant Roealgebra at Z , C ∗ ( X, Z ) G , is defined to be the C ∗ -algebra generated by G -invariant, locally compact operators with finite propagation , supportednear Z . • G -equivariant localization algebra C ∗ L ( X ) G is the C ∗ -algebra generated byall bounded, uniformly norm-continuous functions f : [0 , ∞ ) → C ∗ ( X ) G such that lim t →∞ propagation of f ( t ) = 0 . • The kernel of the evaluation map ev : C ∗ L ( X ) G → C ∗ ( X ) G f → f (0)is called the G -equivariant obstruction algebra, and denoted by C ∗ L, ( X ) G . XIAOMAN CHEN, HONGZHI LIU, HANG WANG, AND GUOLIANG YU • Let Z be a G -invariant subspace of X , then C ∗ L ( X, Z ) G (resp. C ∗ L, ( X, Z ) G )is defined to be the closed subalgebra of C ∗ L ( X ) G (resp. C ∗ L, ( X ) G ) gen-erated by all elements f such that there exists c t > t →∞ c t = 0, and Supp( f ( t )) ⊂ { ( x, z ) ∈ X × X | d (( x, z ) , Z × Z ) ≤ c t } . In general, if Z is a cocompact G -invariant subspace of X , then we have thefollowing isomorphism K ∗ ( C ∗ ( Z ) G ) ∼ = K ∗ ( C ∗ ( X, Z ) G ) ∼ = K ∗ ( C ∗ r ( G )) , where the first isomorphism is induced by the obvious embedding map.In the same time, by Lemma 3.10 of [34], we know(2.1) K ∗ ( C ∗ L ( X, Z ) G ) ∼ = K ∗ ( C ∗ L ( Z ) G ) . Moreover, by the following two K -theoretic six-term exact sequences(2.2) K ( C ∗ L, ( Z ) G ) / / K ( C ∗ L ( Z ) G ) / / K ( C ∗ ( Z ) G ) (cid:15) (cid:15) K ( C ∗ ( Z ) G ) O O K ( C ∗ L ( Z ) G ) o o K ( C ∗ L, ( Z ) G ) o o K ( C ∗ L, ( X, Z ) G ) / / K ( C ∗ L ( X, Z ) G ) / / K ( C ∗ ( X, Z ) G ) (cid:15) (cid:15) K ( C ∗ ( X, Z ) G ) O O K ( C ∗ L ( X, Z ) G ) o o K ( C ∗ L, ( X, Z ) G ) o o and a standard five lemma argument, one can see that(2.3) K ∗ ( C ∗ L, ( X, Z ) G ) ∼ = K ∗ ( C ∗ L, ( Z ) G ) . Zeidler gave a constructive proof of the above isomorphisms in (2.1), (2.2) and(2.3). See Lemma 3.7 of [36] for details.2.2.
Engel-Samukar¸s smooth subalgebra.
In this subsection, we introducethe smooth dense subalgebra of C ∗ r ( G ) defined by Kaˇgan Samurka¸s in [24], theconstruction of which is inspired by Engel [10]. This subalgebra depends onthe choice of a nontrivial element g of G , whose conjugacy class has polynomialgrowth. For S ⊂ G and f ∈ l ( G ), let k f k S be the l norm of f restricted to S .For r >
0, set B r ( S ) as { h ∈ G, ∀ s ∈ S, d G ( e, hs − ) < r } , where d G is the word length metric.Choose C > d ∈ N such that for all k ∈ N , ♯ { h ∈ h g i , d G ( e, h ) = k } ≤ Ck d . IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 9
For a ∈ B ( l ( G )) and r >
0, define µ a ( r ) = inf { c > , ∀ f ∈ l ( G ) , k af k G \ B r (supp( f )) ≤ c k f k G } , and k a k g = inf { A > , ∀ r > , µ a ( r ) ≤ Ar − d/ } . Let k · k B ( l ( G )) be the operator norm on B ( l ( G )). For f ∈ C G , we also denoteby f ∈ B ( l ( G )) the convolution operator by f from the left.Define C polg ( G ) to be the completion of C G in the norm k f k = k f k B ( l ( G )) + k f k g . In [24], Samurka¸s showed that C polg ( G ) is closed under holomorphic functionalcalculus, hence C polg ( G ) is a smooth dense subalgebra of C ∗ r ( G ). We denote C polg ( G ) by A g ( G ).Let us recall the delocalized trace map tr g from the Kaˇgan Samurka¸s algebra to C . Denote by tr g : A g ( G ) → C the orbital integral trace given by(2.4) tr g X γ ∈ G a γ γ ! = X h ∈h g i a h where h g i stands for the conjugacy class of g in G. Moreover, the map tr g inducesthe following orbital integral trace map on K -theory:tr g : K ( A g ( G )) ∼ = K ( C ∗ r ( G )) → C . At last, we recall the notion of g -trace class operators. Let F be a fundamentaldomain of X with respect to the G -action. Compare the following definition toDefinition 2.5 of [13]. Definition 2.3.
Let T be a G -invariant operator having Schwartz kernel in C ∞ ( X × X ) and g be an element of G . Then T is g -trace class if X h ∈h g i Z F | T ( x, hx ) | dx converges. The g -trace of T is defined astr g ( T ) , X h ∈h g i Z F T ( x, hx ) dx. Note that when G is trivial and g = e , this is weaker than T being trace class,which is equivalent to the condition R M | T | ( x, x ) dx < ∞ . For a general g distinctfrom identity, tr g is not a positive trace, hence positivity is not required in thedefinition of g -trace class operators. The Connes-Moscovici smooth subalgebra.
In this subsection, let M be a complete Riemannian manifold with a proper, free and cocompact actionby a discrete group G . Let us recall Xie and Yu’s construction of a particularsmooth dense subalgebra of C ∗ L, ( M ) G ([29]).The following construction of a smooth dense subalgebra of C ∗ r ( G ) ⊗ K is due toConnes and Moscovici (cf. [8]). Let R be the algebra of G -invariant smoothingoperators on M/G . Under the isomorphism L ( M/G ) ∼ = l ( N ), R is identifiedwith the algebra of infinite matrices ( a ij ) i,j ∈ N , such that ∀ k, l ∈ N , sup i,j i k j l | a ij | < ∞ . For a finitely generated discrete group G , let ∆ G be an unbounded operator on l ( G ) defined by ∆ G g = | g | g, for g ∈ G. Particularly, let ∆ : l ( N ) → l ( N ) be the unbounded operator defined by∆( δ j ) = jδ j . Denote by ∂ G = [∆ G , · ] the unbounded derivation of B ( l ( G )), and e ∂ G = ∂ G ⊗ I the unbounded derivation of B ( l ( G ) ⊗ l ( N )). Set B ( M ) G = { A ∈ C ∗ r ( G ) ⊗ K , for all k ∈ N , e ∂ kG ( A ) ◦ ( I ⊗ ∆) is bounded } for us a dense smooth subalgebra of C ∗ r ( G ) ⊗ K . Lemma 2.7 of [29] shows that if h g i has polynomial growth, then the map tr g can be extended to a trace map on B ( M ) G such that for any A = P g A g g ∈ B ( M ) G , tr g ( A ) equals X h ∈h g i trace( A h ) . Note that this tr g map induces the same map on K ( C ∗ r ( G )) with the correspond-ing one defined in the line (2.4).In the meanwhile, the following algebra B L, ( M ) G , { f ∈ C ∗ L, ( M ) G , ∀ t ∈ [0 , ∞ ) , f ( t ) ∈ B ( M ) G } is a smooth dense subalgebra of C ∗ L, ( M ) G . Let g ∈ G be a nontrivial elementwhose conjugacy class has polynomial growth. Define the map τ g : B L, ( M ) G → C to be τ g ( A ) = 12 πi Z ∞ tr g ( dA ( t ) dt A − ( t )) dt. As shown in [29], τ g induces the following determinant map(2.5) τ g : K ( B L, ( M ) G ) ∼ = K ( C ∗ L, ( M ) G ) → C , which is also denoted as τ g . IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 11
One may think it is sufficient to recall the definition of either the Engel-Samukar¸ssmooth subalgebra or the Connes-Moscovici smooth subalgebra, however, thereis a reason for us not to do so. The advantage of the Engel-Samukar¸s smooth sub-algebra is that it allows us to do analysis in a noncocompact setting and to applythe method of Hochs, Wang and Wang (cf. [13]) to develop the index formula forthe Dirac operator on noncocompact complete manifold with uniform positivescalar curvature metric at infinity, while the Connes-Moscovici smooth subalge-bra is necessary for us to apply the theory of Xie and Yu on delocalized trace (cf.[29]) to study the higher rho invariant at infinity and define the delocalized etainvariant at infinity, where a cocompact setting is sufficient.3.
Secondary invariants at infinity
In this section, we introduce two new secondary invariants for the Dirac operatoron a complete spin manifold endowed with a metric with uniform positive scalarcurvature outside a cocompact set, i.e. the higher rho invariant at infinity andthe delocalized eta invariant at infinity. We begin with two approaches to thedefinition of the higher index of the Dirac operator. We also establish a formulafor the delocalized eta invariant, along with which we generalize the Atiyah-Patodi-Singer index theorem.3.1.
Higher index.
In this subsection, we define the higher index of the Diracoperator on a complete manifold having a metric with uniform positive scalarcurvature outside a cocompact set. Higher index of the Dirac operator on acomplete manifold with positive scalar curvature at infinity was first introducedby Bunke in [6]. There is a nice description of this higher index in [5]. However,our construction in this subsection follows [20] [22].Let X be a complete Riemannian manifold (even dimensional) with a spin struc-ture on which a discrete group G acts freely, properly and preserving the metric.Let M ⊂ X be a G -invariant subset. Assume that the complement X \ M admitsa metric µ having uniform positive scalar curvature h , with strictly positive lowerbound h > D be the spin Dirac operator associated to the metric µ on X . Denote by D M , D c the restriction of D to M and C := X \ M respectively. Because X haseven dimension, the spinor bundle S is Z -graded, i.e. S = S + ⊕ S − , and theDirac operator is odd with respect to the grading L ( X, S + ) ⊕ L ( X, S − ): D = (cid:20) D − D + (cid:21) ( D + ) ∗ = D − . M X \ M Figure 1.
Complete manifold X , with G cocompact subset M .Let b : ( −∞ , + ∞ ) → [ − ,
1] be an odd smooth function such that b ( x ) = 1 when x > √ h . Lemma 2.1 and Lemma 2.3 of [22] state that b ( D ) is pseudo-local,and b ( D ) − C ∗ ( X, M ) G . Since b is an odd function, b ( D ) can also bedecomposed as (cid:20) b + b − (cid:21) . Let V be (cid:20) b + (cid:21) (cid:20) − b − (cid:21) (cid:20) b + (cid:21) (cid:20) − (cid:21) . Definition 3.1.
The G -equivariant coarse index of D , ind G,X,M ( D ) is defined tobe the K -theory class in K ( C ∗ ( X, M ) G ), represented by the formal difference ofidempotents V (cid:20) (cid:21) V − − (cid:20) (cid:21) . Furthermore, if
M/G is compact, the K -theory classind G,X,M ( D ) ∈ K ( C ∗ ( X, M ) G ) ∼ = K ( C ∗ r ( G )) , is called the higher index of D , and denoted as ind G ( D ).Note that direct computation shows that V (cid:20) (cid:21) V − = (cid:20) (1 − b + b − ) b + b − + b + b − (2 − b + b − ) b + (1 − b − b + ) b − (1 − b + b − ) (1 − b − b + ) (cid:21) . Higher index: a different approach.
In this subsection, we introducea new approach to define the higher index of the Dirac operator on a completemanifold with metric admitting uniform positive scalar curvature at infinity. Thisapproach is necessary for us to define the higher rho invariant at infinity. This
IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 13 approach is inspired by [30], [22], and has been used to define the higher indexin [37].Set X [ a,b ] , a, b ∈ [0 , ∞ ] , to be the G -invariant subset of X defined by { x ∈ X | a ≤ dist( x, M ) ≤ b } . For simplicity, X [0 ,b ] , b < ∞ , is denoted as X ≤ b , and X [ a, ∞ ] , a >
0, is denoted as X ≥ a . Note that X [0 , ∞ ] = X . The set X ( a,b ) are defined similarly.Let χ : ( −∞ , ∞ ) → [ − ,
1] be a normalizing function satisfying the followingconditions,(1) χ is a smooth odd function, such that lim s →±∞ χ ( s ) = ± s → χ ( s ) s = 1.(3) ˆ χ is a compactly supported distribution on R , such that the diameter ofits support is bounded by real number δ , i.e. diam(Supp( ˆ χ )) ≤ δ .(4) k χ − b k ≤ ǫ , where ǫ is a positive number less than .Furthermore, let T ≥ F T to be the self-adjoint bounded operator χ ( T D ) = Z ∞−∞ ˆ χ ( s ) e πisT D ds. Since χ is an odd function, F T is odd with respect to the grading L ( X, S + ) ⊕ L ( X, S − ): F T = (cid:20) U T U ∗ T (cid:21) . Let N be a sufficiently large integer such that for any x ∈ [ − , ∞ X n = N k (32 πi ) n n ! k ≤ ǫ , k N X n =1 (32 πix ) n n ! k ≤ ǫ . Let r >
N r ≤ δ . For any n ∈ N , we choose a G -invariant locally finite open cover { U n,j } j , and a G -invariant partition of unity { φ n,j } j subordinate to { U n,j } j , such that(1) if U n,j ⊂ X ≤ T δ , then the diameter of U n,j is less than r n .(2) for any fixed n , there are precisely two open sets U n,j such that U n,j T X ≥ T δ is nonempty. For convenience, we denote them by W n, = X (100 T δ − r n , T δ ) and W = X > T δ . Set F T ( n ) = X j p φ n,j F T p φ n,j , ∀ n ∈ N , and F T ( t ) = ( n + 1 − t ) F T ( n + 1) + ( t − n ) F T ( n ) . For any t , F T ( t ) is odd with respect to the grading L ( X, S + ) ⊕ L ( X, S − ): F T ( t ) = (cid:20) U T ( t ) U ∗ T ( t ) 0 (cid:21) . As shown by [35, Lemma 2.6], k U T ( t ) k ≤ k U T k = 4. Consider W T ( t ) = (cid:20) U T ( t )0 1 (cid:21) (cid:20) − U ∗ T ( t ) 1 (cid:21) (cid:20) U T ( t )0 1 (cid:21) (cid:20) − (cid:21) . Note that F T has finite propagation T δ .Decompose L ( X, S ) into L ( X ≤ T δ , S ) ⊕ L ( X ≥ T δ , S ). Lemma 3.2.
Write U T ( t ) U ∗ T ( t ) = (cid:20) B ( t ) B ( t ) B ( t ) B ( t ) (cid:21) and U ∗ T ( t ) U T ( t ) = (cid:20) C ( t ) C ( t ) C ( t ) C ( t ) (cid:21) with respect to the decomposition L ( X, S ) = L ( X ≤ T δ , S ) ⊕ L ( X ≥ T δ , S ) . Then one has the following inequalities k B ( t ) k = k B ( t ) k ≤ ǫ, k B ( t ) − k ≤ ǫ, k C ( t ) k = k C ( t ) k ≤ ǫ, k C ( t ) − k ≤ ǫ. for any t ∈ [0 , ∞ ) . Proof : By assumption, D c is bounded below by h . Therefore it has a Friedrich’sextension E on the Hilbert space L ( X ≥ T δ , S ), which is a selfadjoint opera-tor bounded below by h . For the normalizing function χ , define an operator χ ( T √ E ).Since the propagation of F T is controlled by T δ , for all f ∈ L ( X ≥ T δ , S ), wehave F T ( t )( f ) ∈ L ( X ≥ T δ , S ) , F T ( t ) ( f ) ∈ L ( X ≥ T δ , S ) . By a standard finite propagation argument (cf. the proof of [22, Lemma 2.5]),we have k F T ( t ) ( f ) − k = k χ ( T √ E )( f ) − k ≤ ǫ k f k , which implies the Lemma. (cid:4) IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 15
Now let us consider the formal difference W T ( t ) (cid:20) (cid:21) W − T ( t ) − (cid:20) (cid:21) . Direct computation shows that W T ( t ) (cid:20) (cid:21) W T ( t ) − = (cid:20) (1 − U T ( t ) U ∗ T ( t )) U T ( t ) U ∗ T ( t ) + U T ( t ) U ∗ T ( t ) (2 − U T ( t ) U ∗ T ( t )) U T ( t )(1 − U ∗ T ( t ) U T ( t )) U T ( t ) ∗ (1 − U T ( t ) U ∗ T ( t )) (1 − U ∗ T ( t ) U T ( t )) (cid:21) = (cid:20) − (1 − U T ( t ) U ∗ T ( t )) (2 − U T ( t ) U ∗ T ( t )) U T ( t )(1 − U ∗ T ( t ) U T ( t )) U ∗ T ( t )(1 − U T ( t ) U ∗ T ( t )) (1 − U ∗ T ( t ) U T ( t )) (cid:21) . This formal difference represents a K -theory class in K ( C ∗ ( X ) G ), which is notisomorphic to K ( C ∗ ( M ) G ). To define the equivariant coarse index of D , weconstruct an idempotent sufficiently closed to W T ( t ) (cid:20) (cid:21) W T ( t ) − by the operator norm.Set Z ( t ) = (cid:20) − B ( t ) 00 0 (cid:21) , and Z ( t ) = (cid:20) − C ( t ) 00 0 (cid:21) . By Lemma 3.2, we have k Z ( t ) − (1 − U T ( t ) U ∗ T ( t )) k ≤ ǫ, and k Z ( t ) − (1 − U ∗ T ( t ) U T ( t )) k ≤ ǫ. Define P T ( t ) to be the operator(3.2) (cid:20) − Z ( t ) (2 − U T ( t ) U ∗ T ( t )) U T ( t ) Z ( t ) U ∗ T ( t ) Z ( t ) Z ( t ) (cid:21) . Then we have k P T ( t ) − W T ( t ) (cid:20) (cid:21) W T ( t ) − k ≤ ǫ, which implies k P T ( t ) − P T ( t ) k ≤ ǫ .The following lemma explains why we construct P T ( t ) in Equation (3.2). Proposition 3.3.
Let P T ( t ) be as in Equation (3.2) , then P T ( t ) preserves thedecomposition L ( X, S ) = L ( X ≤ T δ , S ) ⊕ L ( X ≥ T δ , S ) . Moreover, P T ( t ) has the form P T ( t ) = P ′ T ( t ) 00 (cid:20) (cid:21) with respect to the decomposition L ( X, S ) = L ( X ≤ T δ , S ) ⊕ L ( X ≥ T δ , S ) . Proof : The lemma is proved by direct computation.For any f ∈ L ( X ≥ T δ , S ), by definitions of Z and Z , we have Z ( t ) f = 0 , (2 − U T ( t ) U ∗ T ( t )) U T ( t ) Z ( t ) f = 0 ,U ∗ T ( t ) Z ( t ) f = 0 ,Z ( t ) f = 0 . On the other hand, for any f ∈ L ( X ≤ T δ , S ), we have Z ( t ) f ∈ L ( X ≤ T δ , S ) ,Z ( t ) f ∈ L ( X ≤ T δ , S ) ,Z ( t ) f ∈ L ( X ≤ T δ , S ) ,Z ( t ) f ∈ L ( X ≤ T δ , S ) . For any t , by definition of U T ( t ), we know that the propagation of U T ( t ) and U ∗ T ( t ) is less than T δ . Thus one can see(2 − U T ( t ) U ∗ T ( t )) U T ( t ) Z ( t ) f ∈ L ( X ≤ T δ , S ) ,U ∗ T ( t ) Z ( t ) f ∈ L ( X ≤ T δ , S ) . This completes the proof. (cid:4)
Definition 3.4.
By Proposition 3.3, for any t ∈ [0 , ∞ ), P ′ T ( t ) − (cid:20) (cid:21) represents an element in K ( C ∗ ( X ≤ T δ ) G ), which is denoted by ind X,M,T ( D ).Without loss of generality, for any t ∈ [0 , ∞ ), we assume P ′ T ( t ) to be a genuineprojection, which will be denoted by P X,M,T ( t ) (see Figure 2).When M/G is compact, ind
X,M,T ( D ) defines an element in K ( C ∗ r ( G )) and coin-cides with the higher index defined in Definition 3.1. IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 17 MX ≤ T δ X ≥ T δ
Figure 2.
For all t ∈ [0 , ∞ ), the operator P X,M,T ( t ) is supportedon X ≤ T δ .3.3.
Higher rho invariant at infinity.
In this subsection, we define the higherrho invariant at infinity. This invariant gives a K -theoretic demonstration of thenonlocality of the higher index of the Dirac operator on noncompact spin manifoldendowed with a metric admitting uniform positive scalar curvature at infinity.The following theorem is necessary in order to introduce the definition of thehigher rho invariant at infinity. Theorem 3.5.
Let ind
X,M,T ( D ) be as in Definition 3.4, then ∂ (ind X,M,T ( D )) represents an element in K ( C ∗ L, ( X [90 T δ,
T δ ] ) G ) , where ∂ is the connecting mapin the following exact sequence K ( C ∗ L, ( X ≤ T δ ) G ) / / K ( C ∗ L ( X ≤ T δ ) G ) / / K ( C ∗ ( X ≤ T δ ) G ) (cid:15) (cid:15) K ( C ∗ ( X ≤ T δ ) G ) O O K ( C ∗ L ( X ≤ T δ ) G ) o o K ( C ∗ L, ( X ≤ T δ ) G ) o o . Proof : Fix t = 0. To compute ∂ (ind X,M,T ( D )), one need to lift P X,M,T (0) to C ∗ L ( X ≤ T δ ) G first.Recall that N is an integer sufficiently large such that for any x ∈ [ − , ∞ X n = N k (32 πi ) n n ! k ≤ ǫ , k N X n =1 (32 πix ) n n ! k ≤ ǫ . and r is a positive number such that(3.3) N r ≤ δ . Thus for any x ∈ [ − , k N X n =2 n − X j =0 (2 πi ) n n ! x j ! (cid:0) x − x (cid:1) − e πix k ≤ ǫ. For each t ∈ [0 , ∞ ), consider a G -invariant partition of unity { ζ t , θ t } on X ≤ T δ satisfying(1) ζ t + θ t = 1.(2) for n ≤ t < n + 1, ζ t ≡ X ≤ T δ − r/ n − and ζ t ≡ X ≥ T δ − r/ n .Let ξ ∈ C ∞ ( −∞ , ∞ ) be a decreasing function such that ξ | [ −∞ , ≡ ξ | [2 , ∞ ] ≡ t P ′ X,M,T ( t ) forms a lifting of P X,M,T (0) in C ∗ L ( X ≤ T δ ) G , where P ′ X,M,T ( t ) is defined as (1 − t ) P X,M,T (0) + t ( √ ζ P X,M,T (0) √ ζ + √ θ P X,M,T (0) √ θ ) , t ∈ [0 , , (cid:20) (cid:21) + p ζ t − ( P X,M,T ( t − − (cid:20) (cid:21) ) p ζ t − + ξ ( t ) p θ t − ( P X,M,T ( t − − (cid:20) (cid:21) ) p θ t − , t ∈ [1 , ∞ ) . Then we have ∂ (ind X,M,T ( D )) = [ e πiP ′ X,M,T ( · ) ] . Define a path of invertible operators u T ( t ) as1 + N X n =2 n − X j =0 (2 πi ) n n ! ( P ′ X,M,T ( t )) j ! (cid:16)(cid:0) P ′ X,M,T ( t ) (cid:1) − (cid:0) P ′ X,M,T ( t ) (cid:1)(cid:17) . It is straight-forward to verify that k u T ( t ) − e πiP ′ X,M,T ( t ) k ≤ ǫ. Thus u T ( t ) is invertible for any t and the path u ′ T ( t ) = (cid:26) (1 − t ) + tu T (0) t ∈ [0 , u T ( t − t ∈ [1 , ∞ )represents the same element in K ( C ∗ L, ( X ) G ) as the path e πiP ′ X,M,T ( · ) . In otherwords, we have ∂ (ind X,M,T ( D )) = [ u ′ T ( t )] . Note that the propagation of u ′ T ( · ) restricting on X ≤ T δ is less than δ by thechoice of r in line (3.3). For any h ∈ L ( X ≤ T δ , S ) and any t ∈ [0 , ∞ ), wehave u ′ T ( t ) h lies in L ( X ≤ T δ , S ). As mentioned in Definition 3.4, P X,M,T ( t ) is IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 19 a genuine projection. Thus P ′ X,M,T ( t ) is a genuine projection on L ( X ≤ T δ , S ).Thus e πiP ′ X,M,T ( t ) equals identity. One can see that for any h ∈ L ( X ≤ T δ , S ) k u ′ T ( t ) h − h k = k h + N X n =2 n − X j =0 (2 πi ) n n ! ( P ′ X,M,T ( t )) j ! (cid:16)(cid:0) P ′ X,M,T ( t ) (cid:1) − (cid:0) P ′ X,M,T ( t ) (cid:1)(cid:17) h − h k = k h + N X n =2 n − X j =0 (2 πi ) n n ! ( P X,M,T ( t )) j ! (cid:0) ( P X,M,T ( t )) − ( P X,M,T ( t )) (cid:1) h − h k = k h + N X n =2 n − X j =0 (2 πi ) n n ! ( P X,M,T ( t )) j ! (cid:0) ( P X,M,T ( t )) − ( P X,M,T ( t )) (cid:1) h − e πiP X,M,T h k≤ ǫ k h k . By the same reason, for any h ∈ L ( X [80 T δ, T δ ] , S ), we have k u ′ T ( t ) h − h k = k u ′ T ( t ) h − e πiP ′ X,M,T ( t ) h k ≤ ǫ k h k . Finally, for all h ∈ L ( X [95 T δ,
T δ ] , S ), u ′ T ( t ) h belongs to L ( X [90 T δ,
T δ ] , S ). Ac-cording to the decomposition L ( X ≤ T δ , S ) = L ( X ≤ T δ , S ) ⊕ L ( X [90 T δ,
T δ ] , S ) , the path u ′ T ( t ) can be perturbed into the following path of invertible operators (cid:20) I u X,M,T ( t ) (cid:21) , such that k u ′ T ( t ) − (cid:20) I u X,M,T ( t ) (cid:21) k ≤ ǫ. Hence we have(3.4) ∂ (ind X,M,T ( D )) = [ u X,M,T ( · )] , where the path u X,M,T ( t ) is supported on X [90 T δ,
T δ ] (see Figure 3). The proofis then complete. (cid:4) Definition 3.6.
For any T sufficiently large, the element in K ( C ∗ L, ( X [90 T δ,
T δ ] ) G )represented by u X,M,T ( · ) (defined in Equation (3.4) ) is the higher rho invariantat infinity, denoted by ρ X,M,T ( D ). Remark 3.7. If X has a metric with uniform positive scalar curvature at infinity,then ρ X,M,T ( D ) is trivial. Higher rho invariant at infinity is an obstruction of amanifold having a metric with uniform positive acalar curvature. MX ≤ T δ X ≥ T δ X [90 T δ,
T δ ] Figure 3. for all t ∈ [0 , ∞ ), operator u X,M,T ( t ) is supported on X [90 T δ,
T δ ] .3.4. Delocalized eta invariant at infinity.
In this subsection, we define thedelocalized eta invariant at infinity, which measures the nonlocality of higherindex of the Dirac operators on noncompact spin manifolds endowed with a metricadmitting positive scalar curvature at infinity numerically.Consider the connecting map ∂ : K ( C ∗ r ( G )) → K ( C ∗ L, ( X [90 T δ,
T δ ] ) G )in the K -theory six-term exact sequence associated to the short exact sequence0 → C ∗ L, ( X [90 T δ,
T δ ] ) G → C ∗ L ( X [90 T δ,
T δ ] ) G → C ∗ ( X [90 T δ,
T δ ] ) G → . Here K ( C ∗ r ( G )) is identified with K ( C ∗ ( X [90 T δ,
T δ ] ) G ). The following lemmais similar to Lemma 3.9 of [29], which shows that the orbital integral trace on K ( C ∗ r ( G )), defined in line (2.4), and the determinant map, defined in line (2.5),are compatible with the K -theory boundary map. Note that although X [90 T δ,
T δ ] is not a complete manifold, it can be embedded into a complete manifold withproper, free and cocompact G -action. For example, let N ⊂ X be a manifoldwith boundary contains X ≤ T δ , then a perturbation around ∂N makes N ∪ N op into a complete smooth manifold, to which one can embedding X [90 T δ,
T δ ] . Thusfor any nontrivial element g ∈ G whose conjucgay class has polynomial growth,the determinant map τ g : K ( C ∗ L, ( X [90 T δ,
T δ ] ) G ) → C is still well defined. IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 21
Lemma 3.8.
Let g ∈ G be a nontrivial element whose conjugacy class has poly-nomial growth, then the following diagram commutes: K ( C ∗ r ( G )) ∂ / / tr g (cid:15) (cid:15) K ( C ∗ L, ( X [90 T δ,
T δ ] ) G ) − τ g (cid:15) (cid:15) C / / C . Proof : For any element [ p ] ∈ K ( C ∗ r ( G )) ∼ = K ( C ∗ ( X [90 T δ,
T δ ] ) G ), ∂ [ p ] = [ u ( · )]is defined by u ( t ) = e πia ( t ) , t ∈ [0 , ∞ ) , where a ( t ) is a lift of p in C ∗ L ( X [90 T δ,
T δ ] ) G with a (0) = p . By definition, we have τ g ( u ) = 12 πi Z ∞ tr g ( u ( t ) − u ′ ( t )) dt = Z ∞ tr g ( a ′ ( t )) dt. Since g is not the identity element, tr g ( a ( t )) = 0 when t is sufficiently large suchthat the propagation of a ( t ) is small enough. It follows that τ g ( ∂ [ p ]) = τ g ([ u ]) = − tr g ([ p ]) . The lemma is then proved. (cid:4)
Lemma 3.9 of [29] implies that for any T sufficiently large,tr g (ind G ( D )) = − τ g ( ρ X,M,T ( D )) . Inspired by Theorem 4.3 of [29], we define the delocalized higher eta invariant atinfinity as follows.
Definition 3.9.
Let g ∈ G be a nontrivial element. The delocalized eta invariantat g ∈ G at infinity, denoted by η g, ∞ ( D ), is defined to be η g, ∞ ( D ) := − g (ind G ( D )) = 2 τ g ( ρ X,M,T ( D )) . A formula for the higher index.
In this subsection, we establish a for-mula for the higher index of the Dirac operator we considered above for thepurpose of computing the delocalized eta invariant defined in Definition 3.9. InSubsections 3.5 and 3.6, we assume in addition that X has bounded geometry,and consider only finitely generated discrete group. This is due to a special choiceof parametrix of the Dirac operator.Compare Subsection 3.5 and 3.6 with [13, Section 5].Consider the Dirac operator on X (even dimensional) which is odd with respectto the grading L ( X, S + ) ⊕ L ( X, S − ): D = (cid:20) D − D + (cid:21) ( D + ) ∗ = D − . Denote by Q ( t ) := − e − tD − D + D − D + D − , t ≥
0. Operators Q ( t ) are well defined since X is a complete manifold, D − D + admits a unique self-adjoint extension and thefunctional calculus can be applied . Choose a parametrix for D + c to be its inverse Q c ( t ) := ( D − c D + c ) − D − c , t ≥ . Operators Q c ( t ) are well defined since D − c D + c isbounded below by a positive number and admits a unique Friedrich’s extension(a self-adjoint extension).Without loss of generality, we assume that M is a manifold with boundary ∂M ,and assume that ( − , × ∂M = { x ∈ X, d( x, ∂M ) ≤ } is a tubular neighbourhood of ∂M . Let ψ be a smooth function from X to [0 , ψ ≡ X [0 , ] and ψ ≡ X ≥ . Set ψ = 1 − ψ . Let ϕ , ϕ be smooth functions from X to [0 ,
1] such that ϕ ≡ X [0 , ] and ϕ ≡ X ≥ , while ϕ ≡ X [0 , ] and ϕ ≡ X ≥ . Note that ϕ j ψ j = ψ j and[ D, ϕ j ] ψ j = 0, j = 1 , M ∂M∂M × ( − , Figure 4.
Tubular neighbourhood of ∂M .We construct parametrices for D + as follows, R = ϕ Qψ + ϕ Q c ψ ,R ′ = ψ Qϕ + ψ Q c ϕ . Denote S = 1 − RD + ,S = 1 − D + R,S ′ = 1 − R ′ D + . Denote by C ∗ ( X, M ) G the equivariant Roe algebra supported near M. IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 23
The proof for the following Lemma goes verbatim as the one for Lemma 4.5 in[13].
Lemma 3.10.
The operators S , S , S ′ have smooth kernels and S , S , S ′ ∈ C ∗ ( X, M ) G . Furthermore, by the same argument in [13], one can show that S , S , ( S ′ ) and S , S , S ′ are all g -trace classes with smooth Schwartz kernel when the conjugacyclass of g has polynomial growth, and when the manifold X is assumed to havebounded geometry. Note that we have assumed that the positive scalar curvatureoutside compact subset M is uniformly bounded below. Proposition 3.11.
The higher index for D is given by ind G ( D ) = (cid:20) S S ( S + 1) RS D + − S (cid:21) − (cid:20) (cid:21) Proof : This can be seen by choosing the real function b : ( −∞ , ∞ ) → [ − ,
1] inSubsection 3.1 to be b ( x ) = (cid:26) p − f ( x ) x ≥ − p − f ( x ) x < , where f : ( −∞ , ∞ ) → [ − ,
1] is a real function defined as f ( x ) = x ≥ √ h − x √ h x ∈ [0 , √ h ]1 + x √ h x ∈ [ − √ h , x < − √ h . (cid:4) Formula for the delocalized eta-invariant at infinity.
The main goalof this subsection is to prove a formula for the delocalized eta invariant at infinityin light of the techniques developed in [13], along with which we establish anAtiyah-Patodi-Singer type index theorem for Dirac operator on manifold witha metric with uniform positive scalar curvature at infinity. We mention thatwith the technique developed in [25], one can generalize the main result of thissubsection to the case of proper actions.
Theorem 3.12.
Let g be an element of G whose conjugacy class has polynomialgrowth. Then (3.5) tr g (ind G ( D )) = Z M g I ( g ) − lim t → Z ∞ t tr g ( e − sD − c D + c D − c [ D + , ψ ]) ds, where I ( g ) is the integrand as in the Atiyah-Segal-Singer fixed point theorem. Inparticular, if g = e , we have (3.6) 12 η g, ∞ ( D ) = − tr g (ind G ( D )) = lim t → Z ∞ t tr g ( e − sD − c D + c D − c [ D + , ψ ]) ds. The second integral on the right hand side of (3.5) and the integral on the righthand side of (3.6) are independent from the choice of the cutoff function ψ . We divide the proof of Theorem 3.12 into the following series of lemmas.
Lemma 3.13.
We have S = φ (1 − QD + ) ψ + φ Q [ D + , ψ ] + φ Q c [ D + , ψ ] , (3.7) S ′ = ψ (1 − QD + ) φ + ψ Q [ D + , φ ] + ψ Q c [ D + , φ ] , (3.8) S = φ (1 − D + Q ) ψ − [ D + , φ ] Qψ − [ D + , φ ] Q c ψ . (3.9) Proof : Direct computation. (cid:4)
From now on, to emphasize the dependence of operators Q (resp. Q c , S , S ′ , S )on t , we write Q ( t ) (resp. Q c ( t ) , S ( t ) , S ′ ( t ) , S ( t )) for Q (resp. Q c , S , S ′ , S ). Lemma 3.14.
We have tr g ( S ) − tr g ( S ) = tr g ( S ) − tr g ( S ) . Proof : To begin with, note thattr g ( S ) − tr g ( S ) = tr g ( S − S ) = tr g ( S RD + ) . By definition, S RD + = RS D + . Moreover, applying trace property of tr g , weobtain tr g ( RS D + ) = tr g ( S D + R ) = tr g ( S (1 − S )) = tr g ( S ) − tr g ( S ) . Hence the lemma follows. (cid:4)
Lemma 3.15. lim t → + [tr g ( S ′ ) − tr g ( S )] = Z M g I ( g ) Proof : By Lemma 3.13, one can seetr g ( S ′ ) = tr g ( ψ (1 − QD + ) φ + ψ Q [ D + , φ ] + ψ Q c [ D + , φ ]) . However, since [ D + , φ ] ψ = [ D + , φ ] ψ = 0,tr g ( S ′ ) = tr g ( ψ (1 − QD + ) φ ) = tr g ((1 − QD + ) ψ ) . Similarly, tr g ( S ) = tr g ((1 − D + Q ) ψ ) . IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 25
Then a standard heat kernel argument shows thatlim t → + [tr g ( S ′ ) − tr g ( S )] = Z M g I ( g ) . This proves the Lemma. (cid:4)
Lemma 3.16. lim t → + [tr g ( S ) − tr g ( S ′ )] = − lim t → + Z ∞ t tr g ( e − sD − c D + c D − c [ D + , ψ ]) ds Proof : By Lemma 3.13, the following equalitytr g ( S ) − tr g ( S ′ ) = tr g ( φ Q [ D + , ψ ]) + tr g ( φ Q c [ D + , ψ ])holds. Construct a new parametrix for D c Q ′ c := 1 − e − tD − c D + c D − c D + c D − c . Choose φ ∈ C ∞ ( X ) such that φ = 1 on the support of [ D + , ψ j ] , j = 1 , φ = 0 on the support of 1 − φ j . This implies thattr g ( φ Q [ D + , ψ ] − φ Q ′ c [ D + , ψ ])(3.10) = tr g ( φ ( Q − Q ′ c )[ D + , ψ ]) + tr g ((1 − φ )( φ Q − φ Q ′ c )[ D + , ψ ])(3.11) = tr g ( φ ( Q − Q ′ c )[ D + , ψ ]) . (3.12)Hence we have tr g ( φ ( Q − Q ′ c )[ D + , ψ ])(3.13) = tr g ( φ ( Q ( t ) − Q ′ c ( t ))[ D + , ψ ])(3.14) = tr g ( φ Z t e − sD − D + D − − e − sD − c D + c D − c ds [ D + , ψ ])(3.15) = tr g ( Z t φ ( e − sD − D + D − − e − sD − c D + c D − c )[ D + , ψ ] ds ) . (3.16)However, by a finite propagation argument, one can show φ ( e − sD − D + D − − e − sD − c D + c D − c )[ D + , ψ ] = 0when s is sufficiently small. Thustr g ( φ Q [ D + , ψ ] − φ Q ′ c [ D + , ψ ]) = tr g ( φ ( Q − Q ′ c )[ D + , ψ ]) = 0 . On the other hand, we havelim t → + [tr g ( φ Q c [ D + , ψ ] − φ Q ′ c [ D + , ψ ])](3.17) = lim t → + [tr g ( φ ( Q c − − e − tD − c D + c D − c D + c D − c )[ D + , ψ ])](3.18) = lim t → + [tr g ( φ ( Q c − − e − tD − c D + c D − c D + c D − c D + c Q c )[ D + , ψ ])](3.19) = lim t → + [tr g ( φ e − tD − c D + c Q c [ D + , ψ ])](3.20) = − lim t → + [tr g ( φ Z ∞ t e − sD − c D + c D − c dt [ D + , ψ ])](3.21) = − lim t → + Z ∞ t tr g ( e − sD − c D + c D − c [ D + , ψ ]) dt. (3.22)Note that [ D + , ψ ] + [ D + , ψ ] = 0, thus we havelim t → + [tr g ( S ) − tr g ( S ′ )]= lim t → + [tr g ( φ Q [ D + , ψ ]) + tr g ( φ Q c [ D + , ψ ])]= lim t → + [[tr g ( φ Q [ D + , ψ ]) − tr g ( φ Q ′ c [ D + , ψ ])]+ lim t → + [tr g ( φ Q c [ D + , ψ ]) − tr g ( φ Q ′ c [ D + , ψ ])]= − lim t → + Z ∞ t tr g ( e − tD − c D + c D − c [ D + , ψ ]) dt. The lemma is then proved. (cid:4)
Lemma 3.17.
By the isomorphism between C ∗ r ( G ) ⊗ K and C ∗ ( X, M ) G , andbasic index theory, we have tr g (ind G D ) = tr g ( S ) − tr g ( S ) . Proof of Theorem 3.12 .
Combine Lemma 3.14, 3.15, 3.16 and 3.17, we obtaintr g (ind G ( D )) = tr g ( S ) − tr g ( S )(3.23) = tr g ( S ) − tr g ( S )(3.24) = tr g ( S ) − tr g ( S ′ ) + tr g ( S ′ ) − tr g ( S )(3.25) = Z M g I ( g ) − lim t → Z ∞ t tr g ( e − sD − c D + c D − c [ D + , ψ ]) dt. (3.26)The proof is completed. (cid:3) IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 27 Manifold with corner of codimension 2
In this section, we apply the tools and techniques developed in the previoussections to study the Atiyah-Patodi-Singer type index theorem and the higherrho invariant at infinity for manifolds with corner of codimension 2 and where allboundary faces have metrics of positive scalar curvature.Let us first fix some notions. Without loss of generality, let M be an evendimensional manifold with corner with ∂M = ∂ M ∪ ∂ M , where ∂ M and ∂ M are manifolds with common boundary, i.e. ∂∂ M = ∂∂ M = Y . The metricof M is collared near ∂M and have positive scalar curvature on ∂ M and ∂ M .Furthermore, the metric on ∂ M and ∂ M is collared near Y . Let G be a discretegroup acting on M properly, freely and cocompactly by isometries. ∂ M ∂ MYM
Figure 5.
Manifold with corner of codimension 2.Let Z i , i = 1 , ∂ i M ∪ ( Y × R + ) and Z i, ≤ T , i = 1 , ∂ i M ∪ ( Y × [0 , T ]). Denote C i , i = 1 , ∂ i M × R + , i = 1 , C i, ≤ T , i = 1 , ∂ i M × [0 , T ] , i = 1 ,
2. Let C be Y × R and C , ≤ T be Y × [0 , T ] × [0 , T ]. Let M i , i = 1 , M ∪ C i , i = 1 , M i, ≤ T , i = 1 , M ∪ C i, ≤ T , i = 1 , M C (= ∂ M × R + ) C (= ∂ M × R + ) C (= Y × R ) Figure 6.
Complete manifold X .Set a complete manifold(4.1) X = M ∪ ( Z × R + ) = M ∪ ( Z × R + ) and(4.2) X ≤ T = M , ≤ T ∪ Z , ≤ T × [0 , T ] = M , ≤ T ∪ Z , ≤ T × [0 , T ] . We assume in addition G is a finitely generated discrete group acting on M ,properly, cocompactly, and freely by isometries.Let σ be the grading operator on the spinor bundle of X .4.1. M¨uller’s Atiyah-Patodi-Singer type theorem.
In this subsection, weapply the technique from Subsection 3.6 to develop a Atiyah-Patodi-Singer typetheorem on manifold with corner of codimension 2. Our computation can beeasily generalized to the case of manifolds with corners of codimensions k ≥ L generalization of Muller’s extraordinarypioneer work in [15] and [16] in the particular case of manifolds with metric withuniform positive scalar curvature at infinity.Recall that Z × R + = C ∪ C and Z × R + = C ∪ C (see Figures 6 and 7). M Z × R + Z × R + C (= Y × R ) Figure 7. Z × R + and Z × R + .Let ψ : R + → [0 ,
1] be a smooth function such that ψ ≡ , ) and ψ ≡ , ∞ ). Set ψ = 1 − ψ . Let ϕ , ϕ : R + → [0 ,
1] be smooth functions suchthat ϕ ≡ , ) and ϕ ≡ , ∞ ), while ϕ ≡ , ) and ϕ ≡ , ∞ ). Note that ϕ j ψ j = ψ j and ϕ ′ j ψ j = 0, j = 1 ,
2. Denote ψ i,j by ψ j ⊗ χ Z i ,and ϕ i,j by ϕ j ⊗ χ Z i for i, j = 1 ,
2, where χ Z i is the characteristic function of Z i .Note that the range of ϕ i,j is contained in [0 , D X be the Dirac operator on X , D Z i × R + and D be the Dirac operator on Z i × R + and C respectively. Let Q i be defined using ( D − Z i × R + D + Z i × R + ) − D − Z i × R + ,the inverse of D + Z i × R + , and Q be defined using the inverse of D +0 . The existence of IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 29 Q , Q , Q are ensured by the uniform positive scalar curvature on all boundarypieces of M . For any t ∈ [0 , ∞ ), set Q ( t ) as1 − e − tD − X D + X D − X D + X D − X ,Q ′ i ( t ) as 1 − e − tD − Zi × R + D + Zi × R + D − Z i × R + D + Z i × R + D − Z i × R + , and Q ′ ( t ) as 1 − e − tD − D +0 D − D +0 D − . Similarly as in Section 3.5, we construct two paramatices as following R ( t ) = ϕ , ϕ , Q ( t ) ψ , ψ , + ϕ , Q ( t ) ψ , + ϕ , Q ( t ) ψ , − ϕ , ϕ , Q ( t ) ψ , ψ , ,R ′ ( t ) = ψ , ψ , Q ( t ) ϕ , ϕ , + ψ , Q ( t ) ϕ , + ψ , Q ( t ) ϕ , − ψ , ψ , Q ( t ) ϕ , ϕ , . Write S ( t ) = 1 − R ( t ) D + X ,S ′ ( t ) = 1 − R ′ ( t ) D + X ,S ( t ) = 1 − D + X R ( t ) . From ψ , + ψ , = 1 and ψ , + ψ , = 1 , we obtain ψ , ψ , + ψ , + ψ , − ψ , ψ , = 1 . Replacing 1 in S , S ′ , S by ψ , ψ , + ψ , + ψ , − ψ , ψ , , and direct computationshows that S ( t ) = ϕ , ϕ , e − tD − X D + X ψ , ψ , + ϕ , ϕ , Q ( t )[ D + X , ψ , ψ , ]+ ϕ , Q ( t )[ D + X , ψ , ] + ϕ , Q ( t )[ D + X , ψ , ] − ϕ , ϕ , Q ( t )[ D + X , ψ , ψ , ] ,S ′ ( t ) = ψ , ψ , e − tD − X D + X ϕ , ϕ , + ψ , ψ , Q ( t )[ D + X , ϕ , ϕ , ]+ ψ , Q ( t )[ D + X , ϕ , ] + ψ , Q ( t )[ D + X , ϕ , ] − ψ , ψ , Q ( t )[ D + X , ϕ , ϕ , ] ,S ( t ) = ϕ , ϕ , e − tD − X D + X ψ , ψ , + [ D + X , ϕ , ϕ , ] Q ( t ) ψ , ψ , +[ D + X , ϕ , ] Q ( t ) ψ , + [ D + X , ϕ , ] Q ( t ) ψ , − [ D + X , ϕ , φ , )] Q ( t ) ψ , ψ , . Note that S , S ′ , S and their squares are e -trace class operators with smoothSchwartz Kernels. Then taking the trace of the K -theoretic index, we obtain(4.3) tr e (ind G ( D M )) = tr e ( S ( t )) − tr e ( S ( t )) . Exactly the same argument of Lemma 3.14 shows that(4.4) tr e ( S ( t )) − tr e ( S ( t )) = tr e ( S ( t )) − tr e ( S ( t )) . For the same reason as in the proof of Lemma 3.15, we havelim t → + (tr e ( S ′ ( t )) − tr e ( S ( t ))) = Z M ˆ A ( M ) . Similarly as in the proof of Lemma 3.16, one can also obtainlim t → + tr e ( S ( t ) − S ′ ( t ))= lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , ) + lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , ) − lim t → tr e ( ϕ , ϕ , ( Q ( t ) − Q ′ ( t ))[ D + X , ψ , ψ , ])= lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , ) + lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , ) − lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , ψ , ) − lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ , ψ ′ , ) . Let χ ψ i, =1 , i = 1 , { x ∈ X | ψ i, ( x ) = 1 } ,and χ Z i × R + , i = 1 , Z i × R + . Then we havelim t → + tr e ( S ( t ) − S ′ ( t ))= lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , − ( Q ( t ) − Q ′ ( t )) σψ ′ , χ ψ , =1 )+ lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , − ( Q ( t ) − Q ′ ( t )) σψ ′ , χ ψ , =1 ) − lim t → tr e (( Q ( t ) − Q ′ ( t )) σ ( ψ ′ , ψ , (1 − χ ψ , =1 ) + ψ , ψ ′ , (1 − χ ψ , =1 )))= lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , − ( Q ( t ) − Q ′ ( t )) σψ ′ , χ Z × R + )+ lim t → tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , − ( Q ( t ) − Q ′ ( t )) σψ ′ , χ Z × R + ) − lim t → tr e ( σ ( ψ ′ , ψ , (1 − χ ψ , =1 ) + ψ , ψ ′ , (1 − χ ψ , =1 ))( Q ( t ) − Q ′ ( t )) − lim t → tr e ( σ ( ψ ′ , ( χ Z × R + − χ ψ , =1 ) + ψ ′ , ( χ Z × R + − χ ψ , =1 ))( Q ( t ) − Q ′ ( t )) . Lemma 4.1.
We have(1) [tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , − ( Q ( t ) − Q ′ ( t )) σψ ′ , χ Z × R + )]= − √ π Z ∞ t √ s tr e ( e − sD Z D Z − e − sD Y × R + D Y × R + ) ds (2) [tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , − ( Q ( t ) − Q ′ ( t )) σψ ′ , χ Z × R + )]= − √ π Z ∞ t √ s tr e ( e − sD Z D Z − e − sD Y × R + D Y × R + ) ds IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 31
Proof : It is sufficient to prove the first item. By the proof of Lemma 3.16, wehave [tr e (( Q ( t ) − Q ′ ( t )) σψ ′ , − ( Q ( t ) − Q ′ ( t )) σψ ′ , χ Z × R + )]= − Z ∞ t tr ( e − sD − Z × R + D + Z × R + D − Z × R + − e − sD − D +0 D − χ Z × R + ) σψ ′ , ds. Note that D Z × R + = (cid:20) D − Z × R + D + Z × R + (cid:21) = (cid:20) ∂∂x + D Z − ∂∂x + D Z (cid:21) , and D = (cid:20) D − D +0 (cid:21) = (cid:20) ∂∂x + D Y × R + − ∂∂x + D Y × R + (cid:21) . Hence Z ∞ t tr e ( e − sD − Z × R + D + Z × R + D − Z × R + − e − sD − D +0 D − χ Z × R + ) σψ ′ , ds = Z ∞ t tr e ( e s ∂ ∂ x ( e − sD Z D Z − e − sD Y × R + D Y × R + ) ψ ′ , ds − Z ∞ t tr e ( e s ∂ ∂ x ∂∂x ( e − sD Z − e − sD Y × R + ) ψ ′ , ds = Z ∞ t tr e ( e s ∂ ∂ x ψ ′ )tr e ( e − sD Z D Z − e − sD Y × R + D Y × R + ) ds − Z ∞ t tr e ( e s ∂ ∂ x ∂∂x ψ ′ )tr e ( e − sD Z − e − sD Y × R + ) ds. However, the kernel of e s ∂ ∂ x on R equals κ e s ∂ ∂ x ( u, u ′ ) := 1 √ πs e −| u − u ′| s , while the kernel of e s ∂ ∂ x ∂∂x equals κ e s ∂ ∂ x ∂∂x ( u, u ′ ) := − √ πs | u − u ′ | s e −| u − u ′| s . Hence tr e ( e s ∂ ∂ x ψ ′ ) = Z ∞−∞ κ e s ∂ ∂ x ( x, x ) ψ ′ dx = 1 √ πs and tr e ( e s ∂ ∂ x ∂∂x ψ ′ ) = Z ∞−∞ κ e s ∂ ∂ x ∂∂x ( x, x ) ψ ′ dx = 0 . This completes the proof. (cid:4)
Lemma 4.2.
We have (1) tr e ( σ ( ψ ′ , ψ , (1 − χ ψ , =1 ) + ψ , ψ ′ , (1 − χ ψ , =1 ))( Q ( t ) − Q ′ ( t ))) = 0 (2) tr e ( σ ( ψ ′ , ( χ Z × R + − χ ψ , =1 ) + ψ ′ , ( χ Z × R + − χ ψ , =1 ))( Q ( t ) − Q ′ ( t ))) = 0 Proof :Actually we have(4.5) tr e ( σψ ′ , ψ , (1 − χ ψ , =1 )( Q ( t ) − Q ′ ( t ))) = 0(4.6) tr e ( σψ , ψ ′ , (1 − χ ψ , =1 )( Q ( t ) − Q ′ ( t ))) = 0(4.7) tr e ( σψ ′ , ( χ Z × R + − χ ψ , =1 )( Q ( t ) − Q ′ ( t ))) = 0(4.8) tr e ( σψ ′ , ( χ Z × R + − χ ψ , =1 )( Q ( t ) − Q ′ ( t ))) = 0We will prove Equation (4.5) only, the other three are totally parallel.Now, we havetr e (( Q ( t ) − Q ′ ( t )) σψ ′ , ψ , (1 − χ ψ , =1 )) = Z ∞ t tr e ( e − sD − D +0 D − σψ ′ , ψ , (1 − χ ψ , =1 )) ds. By definition, we have D − = (cid:20) ∂∂x + i ∂∂y D − Y − D + Y − ∂∂x + i ∂∂y (cid:21) D +0 = (cid:20) − ∂∂x + i ∂∂y − D − Y D + Y ∂∂x + i ∂∂y (cid:21) . Direct computation showstr e ( e − sD − D +0 D − σψ , ψ ′ , (1 − χ ψ , =1 ))= tr e ( e − s ∂ ∂ x ∂∂x ψ ′ )tr e ( e − s ∂ ∂ y σψ (1 − χ ψ =1 ))tr e ( (cid:20) e − sD − Y D + Y e − sD + Y D − Y (cid:21) )+tr e ( e − s ∂ ∂ x ψ ′ )tr e ( e − s ∂ ∂ y ∂∂y ψ (1 − χ ψ =1 ))tr e ( (cid:20) e − sD − Y D + Y e − sD + Y D − Y (cid:21) )+tr e ( e − s ∂ ∂ x ψ ′ )tr e ( e − s ∂ ∂ y ψ (1 − χ ψ =1 ))tr e ( (cid:20) e − sD − Y D + Y D − Y − e − sD + Y D − Y D + Y (cid:21) )In the proof of Lemma 4.1, we have shown thattr e ( e − s ∂ ∂ x ∂∂x ψ ′ ) = tr e ( e − s ∂ ∂ y ∂∂y ψ (1 − χ ψ =1 )) = 0 . At the same time, we havetr e ( (cid:20) e − sD − Y D + Y D − Y − e − sD + Y D − Y D + Y (cid:21) ) = 0 . The proof is completed. (cid:4)
IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 33
In a word, by the above argument and Lemmas 4.1 and 4.2, we have
Theorem 4.3.
The following equality holds: tr e (ind G D M ) = Z M I ( e ) − lim t → √ π Z ∞ t tr e ( e − sD Z D Z − e − sD Y × R + D Y × R + ) 1 √ s ds − lim t → √ π Z ∞ t tr e ( e − sD Z D Z − e − sD Y × R + D Y × R + ) 1 √ s ds. Higher rho invariant at infinity on manifold with corner.
In thissubsection, we develop a theory for higher rho invariants at infinity on manifoldwith corner of codimension 2. We first define higher rho invariants associatingto metrics with positive scalar curvature on manifolds with boundary. Then weprove a combinatorial formula for higher rho invariants at infinity on manifoldswith corner of codimension 2. The formula can be viewed as a higher generaliza-tion of Muller’s formulas in [15] and [16].4.2.1.
Higher rho invariant on manifold with boundary.
In this subsubsection,we introduce the higher rho invariant of the Dirac operator on a spin manifoldwith boundary endowed with a positive scalar curvature metric collared near theboundary.Let (
M, ∂M ) be a manifold with boundary. Let m be a positive scalar curvaturemetric on M and collared near the boundary ∂M , and G be a discrete groupacting properly, freely and cocompactly on ( M, ∂M ) by isometries, such that(
M/G, ( ∂M ) /G ) is a compact manifold with boundary. Furthermore, Let M ∞ be the complete manifold M ∪ ( ∂M × R + ), with subspaces M [ a,b ] , M ≤ T and M ≥ T similarly defined in the beginning of Subsection 3.2. Let D be the Dirac operatoron M .Suppose that the scalar curvature k of m are uniform bounded below by a constant k >
0. Let χ be a normalizing function such that ∀ x, | x | > k , k χ ( x ) − k ≤ ǫ, and ˆ χ has compact support. Suppose that the diameter of the support of ˆ χ is K .We will address the odd dimensional case, i.e. M is of odd dimensional, in detailsonly. The even case is totally parallel.For any t ∈ [1 , ∞ ), denote P M ∞ ( t ) as χ ( DM ∞ t )+12 and P ∂M × R + ( t ) as χ ( D∂M × R + t )+12 .Operators P ∂M × R + ( t ) are well defined since D ∂M × R + is bounded below by a pos-itive number, which implies that D ∂M × R + has a unique Friedrich’s extension. By definition, the following path of operators(4.9) ρ M ∞ ( t ) = (cid:26) (1 − t ) I + t exp(2 πiP M ∞ (1)) , t ∈ [0 , πiP M ∞ ( t )) , t ∈ [1 , ∞ )defines an invertible element in C ∗ L, ( M ∞ ) G . On the other hand, the path(4.10) ρ ∂M × R + ( t ) = (cid:26) (1 − t ) I + t exp(2 πiP ∂M × R + (1)) , t ∈ [0 , πiP ∂M × R + ( t )) , t ∈ [1 , ∞ )defines an invertible element in C ∗ L, ( M ∞ ) G . Under the decomposition L ( M ∞ , S )as L ( M, S ) ⊕ L ( ∂M × R + , S ), we can extend ρ ∂M ( · ) to the invertible element (cid:20) I ρ ∂M × R + ( · ) (cid:21) ∈ C ∗ L, ( M ∞ ) G , which will be denoted as ρ ∂M × R + ( · ) for the sake of simplifying notations.In the following, we show that the path of invertible operators ρ M ∞ ( t ) ρ − ∂M × R + ( t )actually defines an element in K ( C ∗ L, ( M ) G ). Theorem 4.4.
Let ρ M ∞ ( · ) and ρ ∂M × R + ( · ) be as defined in (4.9) and (4.10) . Then ρ M ∞ ρ − ∂M × R + defines a class in K ( C ∗ L, ( M ) G ) , which will be denoted as ρ M,∂M ( D ) . Proof : For any ǫ < , let N ∈ N + be a sufficiently large number suchthat(1) k P ∞ n = N (2 πiP M ∞ ( t )) n n ! k ≤ ǫ , for all t ∈ [0 , ∞ ),(2) k P ∞ n = N ( − πiP ∂M × R + ( t )) n n ! k ≤ ǫ , for all t ∈ [0 , ∞ ),(3) | P Nn =1 (2 πi ) n n ! | < ǫ and | P Nn =1 ( − πi ) n n ! | < ǫ .We have N X n =0 (2 πiP M ∞ ( t )) n n != I + ( N X n =1 (2 πi ) n n ! ) P M ∞ ( t ) + N X n =2 (2 πi ) n n ! ( P M ∞ ( t ) n − P M ∞ ( t )) . Define u M ∞ ( t ) , t ∈ [0 , ∞ ) to be u M ∞ ( t ) = (cid:26) (1 − t ) I + t ( I + P Nn =2 (2 πi ) n n ! ( P M ∞ (1) n − P M ∞ (1)) , t ∈ [0 , I + P Nn =2 (2 πi ) n n ! ( P M ∞ ( t ) n − P M ∞ ( t )) , t ∈ [1 , ∞ ) . IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 35
Comparing (4.9) and using the above inequalities shows that u M ∞ ( · ) is an in-vertible element in C ∗ L, ( M ∞ ) G . Similarly, one can define an invertible element v − ∂M × R + ( · ) ∈ C ∗ L, ( ∂M × R + ) G by v − ∂M × R + ( t ) = (cid:26) (1 − t ) I + t ( I + P Nn =2 ( − πi ) n n ! ( P ∂M × R + (1) n − P M ∞ (1)) , t ∈ [0 , I + P Nn =2 ( − πi ) n n ! ( P ∂M × R + ( t ) n − P ∂M × R + ( t )) , t ∈ [1 , ∞ ) . It is straightforward to check that k ρ M ∞ ρ − ∂M × R + − u M ∞ v − ∂M × R + k ≤ ǫ. Note that the propagation of u M ∞ ( t ) , t ≥ , and v − ∂M × R + ( t ) , t ≥ N Ks ( t ) , where(4.11) s ( t ) = (cid:26) , ≤ t ≤ t, t ≥ . For t ≥
0, decompose L ( M ∞ , S ) as(4.12) L ( M ≤ N Ks ( t ) , S ) ⊕ L ( M [4 N Ks ( t ) , N Ks ( t ) ] , S ) ⊕ L ( M [8 N Ks ( t ) , N Ks ( t ) ] , S ) ⊕ L ( M ≥ N Ks ( t ) , S ) . Under the above decomposition, u M ∞ v − ∂M × R + ( t ) can be written as the followingmatrix a ( t ) a ( t ) 0 0 a ( t ) a ( t ) a ( t ) 00 a ( t ) a ( t ) a ( t )0 0 a ( t ) a ( t ) . For any h ∈ L ( M ≥ N Ks ( t ) , S ), u M ∞ v − ∂M × R + ( t )) h lies in L ( M ≥ N Ks ( t ) , S ), thus u M ∞ ( t ) v − ∂M × R + ( t ) h = u ∂M × R + ( t ) v − ∂M × R + ( t ) h where u ∂M ( t ) is defined to be u ∂M × R + ( t ) = (cid:26) (1 − t ) I + t ( I + P Nn =2 (2 πi ) n n ! ( P ∂M × R + (1) n − P ∂M × R + (1)) , t ∈ [0 , I + P Nn =2 (2 πi ) n n ! ( P ∂M × R + ( t ) n − P ∂M × R + ( t )) , t ∈ [1 , ∞ ) . Consequently, we have k u M ∞ ( t ) v − ∂M × R + ( t ) h − h k = k u ∂M × R + ( t ) v − ∂M × R + ( t ) h − ρ ∂M × R + ρ − ∂M × R + h k ≤ ǫ, which implies that k a ( t ) − I k ≤ ǫ, k a ( t ) k ≤ ǫ. For the same reason, one can see that k a ( t ) k ≤ ǫ. For any t ≥
1, set the following matrix A ( t ) = a ( t ) a ( t ) 0 0 a ( t ) a ( t ) a ( t ) 00 a ( t ) a ( t ) 00 0 0 I with respect to the decomposition (4.12), L ( M ≤ N Ks ( t ) , S ) ⊕ L ( M [4 N Ks ( t ) , N Ks ( t ) ] , S ) ⊕ L ( M [8 N Ks ( t ) , N Ks ( t ) ] , S ) ⊕ L ( M ≥ N Ks ( t ) , S ) .M M ≤ N Ks ( t ) M [4 N Ks ( t ) , N Ks ( t ) ] M [8 N Ks ( t ) , N Ks ( t ) ] M ≥ N Ks ( t ) Figure 8.
Decomposition of M ∞ .Thus up to a trivial cycle of K ( C ∗ L, ( M ∞ ) G ), A ( t ) is supported on is supportedon M ≤ N Kt × M ≤ N Kt , and sup t ∈ [0 , ∞ ) k A ( t ) − u M ∞ ( t ) v − ∂M ∞ × R + ( t ) k ≤ ǫ. Hence ρ M ∞ ρ − ∂M × R + represents an element in K ( C ∗ L, ( M ∞ ; M ) G ). However, as shown inthe end of Subsection 2.1, we have K ( C ∗ L, ( M ∞ ; M ) G ) ∼ = K ( C ∗ L, ( M ) G ). Thiscompletes the proof. (cid:4) When M is an even dimensional manifold with boundary, by essentially the sameargument as above, one can show that the following path of formal difference ofprojections ρ M ∞ ( t ) − ρ ∂M × R + ( t )actually defines an element in K ( C ∗ L, ( M ) G ), which will be denoted as ρ M,∂M ( D ). Definition 4.5.
Let (
M, ∂M ) be a spin manifold with boundary with proper,cocompact and free G action by isometries. Let g be a positive scalar curvaturemetric on M which is collared near ∂M . Let D be the Dirac operator on M .The K -theory class ρ M,∂M ( D ) defined in Theorem 4.4 is called the higher rhoinvariant for D associated to g . IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 37
Combinatorial formula.
In this subsubsection, we consider the higher rhoinvariant at infinity for the Dirac operator on a spin manifold M with corner ofcodimension 2. In this subsubsection, we use the notions defined at the beginningof Section 4.Recall X is the complete manifold given by Equation (4.1). Although the defini-tion of X ≤ T is slightly different from the one given by Equation (4.2) in Section3, all the constructions and proofs in Section 3 can be verbatim applied to X and X we defined here. More precisely, the metric on X has uniformly positive scalarcurvature outside X ≤ . Let D be the Dirac operator on X . The higher index of D , ind X,X ≤ ,T ( D ) is defined in K ( C ∗ ( X ≤ S ) G ), where S depends linearly on T . Asshown in the Proposition 3.3 and Definition 3.4, for any t ∈ [0 , ∞ ), ind X,X ≤ ,T ( D )is represented by projection P X,X ≤ ,T ( t ). Furthermore, for sufficiently large T , thehigher rho invariant ρ X,X ≤ ,T ( D ) is defined in K ( C ∗ L, ( X [ √ S,S ] ) G ), where X [ √ S,S ] is X ≤ S \ X < √ S . As shown in the proof of Theorem 3.5, ρ X,X ≤ ,T ( D ) is representedby the path of invertible operators u X,X ≤ ,T ( t ) = e πiP ′ X,X ≤ ,T ( t ) , t ∈ [0 , ∞ ) . Note that for all S greater than 1, X [ √ S,S ] is strongly Lipschitz homotopy equiv-alent to ∂M , thus we have K ( C ∗ L, ( X [ √ S,S ] ) G ) ∼ = K ( C ∗ L, ( ∂M ) G ). Obviouslyby the definition of the isomorphism of K -groups induced by strongly Lipschitzhomotopy equivalence, one can see that ρ X,X ≤ ,T ( D ) = ρ X,X ≤ ,T ( D ) ∈ K ( C ∗ L, ( ∂M ) G ) , ∀ T , T > . Put ρ X,X ≤ , ∞ ( D ) := ρ X,X ≤ ,T ( D ) ∈ K ( C ∗ L, ( ∂M ) G ) for any T > C and C , ≤ .Recall that ρ ∂ i M,∂∂ i M ( D ∂ i M ) defines an element in K ( C ∗ L, ( ∂ i M ) G ) , i = 1 ,
2, thusan element in K ( C ∗ L, ( ∂M ) G ) under the obvious K -theoretic map induced byembedding. The following is the main theorem of this subsection. Theorem 4.6.
We have that ρ X,X ≤ , ∞ ( D ) = ρ ∂ M,∂∂ M ( D ∂ M ) + ρ ∂ M,∂∂ M ( D ∂ M ) ∈ K ( C ∗ L, ( ∂M ) G ) . To prepare for the proof of Theorem 4.6, we introduce some notations. Let i = 1 ,
2. Consider the C ∗ -algebras C ∗ L, ( ∂ i M × R ) G . Denote by C ∗ L, ( ∂ i M × R ) G + the C ∗ -algebra ∪ n ∈ N C ∗ L, ( ∂ i M × R , ∂ i M × [ − n, ∞ )) G and by C ∗ L, ( ∂ i M × R ) G − the C ∗ -algebra ∪ n ∈ N C ∗ L, ( ∂ i M × R , ∂ i M × ( −∞ , n ]) G . It is clear to see that both of C ∗ L, ( ∂ i M × R ) G ± are closed two-sided ideals of C ∗ L, ( ∂ i M × R ) G , such that C ∗ L, ( ∂ i M × R ) G − + C ∗ L, ( ∂ i M × R ) G + = C ∗ L, ( ∂ i M × R ) G . Since ∪ n ∈ N C ∗ L, ( ∂ i M × R , ∂ i M × [ − n, n ]) G = C ∗ L, ( ∂ i M × R ) G + ∩ C ∗ L, ( ∂ i M × R ) G − and K ∗ ( ∪ n ∈ N C ∗ L, ( ∂ i M × R , ∂ i M × [ − n, n ]) G ) = K ∗ ( C ∗ L, ( ∂ i M ) G ) , we have the following Mayer-Vietories sequence of K -theory(4.13) K ( C ∗ L, ( ∂ i M ) G ) / / K ( C ∗ L, ( ∂ i M × R ) G + ) ⊕ K ( C ∗ L, ( ∂ i M × R ) G − ) / / K ( C ∗ L, ( ∂ i M × R ) G ) ∂ MV,i (cid:15) (cid:15) K ( C ∗ L, ( ∂ i M × R ) G ) ∂ MV,i O O K ( C ∗ L, ( ∂ i M × R ) G + ) ⊕ K ( C ∗ L, ( ∂ i M × R ) G − ) o o K ( C ∗ L, ( ∂ i M ) G ) o o . Similarly, consider C ∗ L ( R ) + := ∪ n ∈ N C ∗ L ( R , [ − n, ∞ )) and C ∗ L ( R ) − := ∪ n ∈ N C ∗ L ( R , ( −∞ , n ]),we obtain the following Mayer-Vietories sequence of K -theory(4.14) K ( C ∗ L (pt)) / / K ( C ∗ L ( R ) + )) ⊕ K ( C ∗ L ( R ) − ) / / K ( C ∗ L, ( R )) ∂ MV (cid:15) (cid:15) K ( C ∗ L ( R )) ∂ MV O O K ( C ∗ L ( R ) + ) ⊕ K ( C ∗ L ( R ) − ) o o K ( C ∗ L (pt)) o o Furthermore, there is a natural homomorphism α : C ∗ L, ( ∂ i M ) G ⊗ C ∗ L ( R ) → C ∗ L, ( ∂ i M × R ) G such that for any element a ⊗ b ∈ K n ( C ∗ L, ( ∂ i M ) G ) ⊗ K ( C ∗ L ( R )), α ( a ⊗ ∂ MV ( b )) = ∂ MV,i ( α ( a ⊗ b )) ∈ K n ( C ∗ L, ( ∂ i M ) G ) . Now let us prove Theorem 4.6.
Proof : Let T be a sufficiently large positive number.Note that since the metric on C admits a uniformly positive scalar curvature, ρ − C ,C , ≤ ,T ( D C ) = I ∈ K ( C ∗ L, ( X [ √ S,S ] ) G ) , where S is a positive number depending on T linearly. Hence we actually have ρ X,X ≤ ,T ( D ) = ρ X,X ≤ ,T ( D ) ρ − C ,C , ≤ ,T ( D C ) ∈ K ( C ∗ L, ( X [ √ S,S ] ) G ) . IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 39
Note that ρ X,X ≤ ,T ( D ) ρ − C ,C , ≤ ,T ( D C ) is represented by(4.15) e πiP ′ X,X ≤ ,T e − πiP ′ C ,C , ≤ ,T . However, by the same argument as in Theorem 4.4, one can see that the rep-resentative in Equation (4.15) actually has its support on ( Z , ≤ S / × [ √ S, S ]) ∪ ( Z , ≤ S / × [ √ S, S ]). That is, there exists two paths of operators ρ , ρ ∈ C ∗ L, ( X [ √ S,S ] ) G ),supported on Z , ≤ S / × [ √ S, S ] and Z , ≤ S / × [ √ S, S ] respectively (see Figure9), such that ρ + ρ = [ e πiP ′ X,X ≤ ,T e − πiP ′ C ,C , ≤ ,T ] ∈ K ( C ∗ L, ( X [ √ S,S ] ) G ) . √ SSS / S / √ S S
Support of e πiP ′ X,X ≤ ,T √ SSS / S / √ S S
Support of e − πiP ′ C ,C , ≤ ,T √ SSS / S / √ S S
Support of ρ Support of ρ e πiP ′ X,X ≤ ,T e − πiP ′ C ,C , ≤ ,T Figure 9. ρ + ρ = e πiP ′ X,X ≤ ,T e − πiP ′ C ,C , ≤ ,T Moreover, by direct finite propagation argument, one can show that[ ρ ] = [ e πiP ′ X,M ,T e − πiP ′ C ∪ C ,C ,T ][ ρ ] = [ e πiP ′ X,M ,T e − πiP ′ C ∪ C ,C ,T ]Furthermore, we actually have(4.16) [ e πiP ′ X,M ,T e − πiP ′ C ∪ C ,C ,T ] = [ e πiP ′ Z × R ,Z × R − ,T e − πiP ′ Y × R + × R ,Y × R + × R − ,T ] , (4.17) [ e πiP ′ X,M ,T e − πiP ′ C ∪ C ,C ,T ] = [ e πiP ′ Z × R ,Z × R − ,T e − πiP ′ Y × R + × R ,Y × R + × R − ,T ] . However, by definition of the connecting map, the path of operators e πiP ′ Zi × R ,Zi × R − ,T e − πiP ′ Y × R + × R ,Y × R + × R − ,T represents ∂ MV,i ( ρ ∂ i M,∂∂ i M ( D ∂M i )) ∈ K ( C ∗ L, ( Z i , ∂ i M )) G ) ∼ = K ( C ∗ L, ( ∂ i M ) G )where ∂ MV,i , i = 1 , ρ ] = ∂ MV,i ( ρ ∂ i M,∂∂ i M ( D ∂M i )) . At last, it is straightforward to see that ρ ∂ i M,∂∂ i M ( D ∂M i ) ⊗ [ D R ] = ρ ∂ i M × R ,∂∂ i M × R ( D ∂M i × R ) . Thus we have ∂ MV,i ( ρ ∂ i M × R + ,∂∂ i M × R + ( D ∂M i × R ))(4.18) = ∂ MV,i ( ρ ∂ i M,∂∂ i M ( D ∂M i ) ⊗ [ D R ])(4.19) = ρ ∂ i M,∂∂ i M ( D ∂M i ) ⊗ ∂ MV,i [ D R ](4.20) = ρ ∂ i M,∂∂ i M ( D ∂M i ) . (4.21)Finally, we have ρ X,X ≤ ,T ( D )= ρ X,X ≤ ,T ( D ) ρ − C ,C , ≤ ,T ( D C )= [ ρ ] + [ ρ ]= ∂ MV, ( ρ ∂ M × R ,∂∂ M × R ( D ∂M × R ))+ ∂ MV, ( ρ ∂ M × R ,∂∂ M × R ( D ∂M × R ))= ρ ∂ M,∂∂ M ( D ∂M ) + ρ ∂ M,∂∂ M ( D ∂M ) . This completes the proof. (cid:4)
Corollary 4.7.
Let g ∈ G be a nontrivial element whose conjugacy class is ofpolynomial growth, then we have (4.22) η g, ∞ ( D X ) = 2 τ g ( ρ ∂ M,∂∂ M ( D ∂ M )) + 2 τ g ( ρ ∂ M,∂∂ M ( D ∂ M )) . At last, we briefly explain why Theorem 4.6 is a higher generalization of Muller’stheorem. For the odd dimensional manifold M with boundary and with positive IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 41 scalar curvature metric, in light of Theorem 4.3 of [29], one may guess that thereis a formula for τ g ( ρ M,∂M ( D M )) looks like(4.23) 1 √ π Z ∞ tr g ( D M ∞ e − t D M ∞ − D ∂M × R + e − t D ∂M × R + ) dt. Then for the manifold M with corner, equation (4.22) can be reformulated as η g, ∞ ( D X ) = 2 √ π Z ∞ tr g ( D ( ∂ M ) ∞ e − t D ∂ M ) ∞ − D ∂∂ M × R + e − t D ∂∂ M × R + ) dt + 2 √ π Z ∞ tr g ( D ( ∂ M ) ∞ e − t D ∂ M ) ∞ − D ∂∂ M × R + e − t D ∂∂ M × R + ) dt. We will prove Equation (4.23) in a forthcoming paper.
References [1] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geom-etry.
Bull. London Math. Soc. , 5:229–234, 1973.[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geom-etry. I.
Math. Proc. Cambridge Philos. Soc. , 77:43–69, 1975.[3] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geom-etry. II.
Math. Proc. Cambridge Philos. Soc. , 78(3):405–432, 1975.[4] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geom-etry. III.
Math. Proc. Cambridge Philos. Soc. , 79(1):71–99, 1976.[5] Jonathan Block and Shmuel Weinberger. Arithmetic manifolds of positive scalar curvature.
J. Differential Geom. , 52(2):375–406, 1999.[6] Ulrich Bunke. A K-theoretic relative index theorem and Callias-type Dirac operators.
Math.Ann. , 303:241–279, 1995.[7] Xiaoman Chen, Jinmin Wang, Zhizhang Xie, and Guoliang Yu. Delocalized eta invariants,cyclic cohomology and higher rho invariants. arXiv:1901.02378.[8] Alain Connes and Henri Moscovici. Cyclic cohomology, the Novikov conjecture and hyper-bolic groups.
Topology , 29(3):345–388, 1990.[9] Ronald G. Douglas and Krzysztof P. Wojciechowski. Adiabatic limits of the η -invariants.The odd-dimensional Atiyah-Patodi-Singer problem. Comm. Math. Phys. , 142(1):139–168,1991.[10] Alexander Engel. Banach strong Novikov conjecture for polynomially contractible groups.
Adv. Math. , 330:148–172, 2018.[11] Nigel Higson and John Roe. K -homology, assembly and rigidity theorems for relative etainvariants. Pure Appl. Math. Q. , 6(2, Special Issue: In honor of Michael Atiyah and IsadoreSinger):555–601, 2010.[12] Nigel Higson, John Roe, and Guoliang Yu. A coarse Mayer-Vietoris principle.
MathematicalProceedings of the Cambridge Philosophical Society , 114(1):85–97, 1993.[13] Peter Hochs, Bai-ling Wang, and Hang Wang. An equivariant Atiyah-Patodi-Singer indextheorem for proper actions. arXiv:1904.11146v1.[14] John Lott. Delocalized L -invariants. J. Funct. Anal. , 169(1):1–31, 1999.[15] Werner M¨uller. Eta invariants and manifolds with boundary.
J. Differential Geom. ,40(2):311–377, 1994.[16] Werner M¨uller. On the L -index of Dirac operators on manifolds with corners of codimen-sion two. I. J. Differential Geom. , 44(1):97–177, 1996. [17] Markus J. Pflaum, Hessel Posthuma, and Xiang Tang. The transverse index theorem forproper cocompact actions of Lie groupoids.
J. Differential Geom. , 99(3):443–472, 2015.[18] Paolo Piazza and Thomas Schick. Rho-classes, index theory and Stolz’ positive scalarcurvature sequence.
J. Topol. , 7(4):965–1004, 2014.[19] Paolo Piazza, Thomas Schick, and Vito Felice Zenobi. Higher rho numbers and the mappingof analytic surgery to homology. arXiv:1905.11861.[20] John Roe. Coarse cohomology and index theory on complete Riemannian manifolds.
Mem.Amer. Math. Soc. , 104(497):x+90, 1993.[21] John Roe.
Index theory, coarse geometry, and topology of manifolds , volume 90 of
CBMSRegional Conference Series in Mathematics . Published for the Conference Board of theMathematical Sciences, Washington, DC; by the American Mathematical Society, Provi-dence, RI, 1996.[22] John Roe. Positive curvature, partial vanishing theorems and coarse indices.
Proc. Edinb.Math. Soc. (2) , 59(1):223–233, 2016.[23] Yanli Song and Xiang Tang. Higher Orbit Integrals, Cyclic Cocyles, and K-theory ofReduced Group C ∗ -algebra. arXiv:1910.00175.[24] Kaˇgan Samurka¸s S¨uleyman. Bounds for the rank of the finite part of operator K -theory.Published online, J. Noncommut. Geom.[25] Bai-Ling Wang and Hang Wang. Localized index and L -Lefschetz fixed-point formula fororbifolds. J. Differential Geom. , 102(2):285–349, 2016.[26] Shmuel Weinberger, Zhizhang Xie, and Guoliang Yu. Additivity of higher rho invariantsand nonrigidity of topological manifolds.
112 pages, to appear in Communications in Pureand Applied Mathematics .[27] Shmuel Weinberger and Guoliang Yu. Finite part of operator K -theory for groups finitelyembeddable into Hilbert space and the degree of nonrigidity of manifolds. Geom. Topol. ,19(5):2767–2799, 2015.[28] Rufus Willett and Guoliang Yu.
Higher index theory . Cambridge University Press, 2020.[29] Zhizhang Xie and Guoliang Yu. Delocalized eta invariants, algebraicity, and K -theoryof group C ∗ -algebras. Published online, https://doi.org/10.1093/imrn/rnz170, Int. Math.Res. Not.[30] Zhizhang Xie and Guoliang Yu. Positive scalar curvature, higher rho invariants and local-ization algebras. Adv. Math. , 262:823–866, 2014.[31] Zhizhang Xie and Guoliang Yu. A relative higher index theorem, diffeomorphisms andpositive scalar curvature.
Adv. Math. , 250:35–73, 2014.[32] Zhizhang Xie and Guoliang Yu. Higher rho invariants and the moduli space of positivescalar curvature metrics.
Adv. Math. , 307:1046–1069, 2017.[33] Zhizhang Xie, Guoliang Yu, and Rudolf Zeidler. On the range of the relative higher indexand the higher rho-invariant for positive scalar curvature. arXiv:1712.03722.[34] Guoliang Yu. Localization algebras and the coarse Baum-Connes conjecture. K -Theory ,11(4):307–318, 1997.[35] Guoliang Yu. The Novikov conjecture for groups with finite asymptotic dimension. Ann.of Math. (2) , 147(2):325–355, 1998.[36] Rudolf Zeidler. Positive scalar curvature and product formulas for secondary index invari-ants.
J. Topol. , 9(3):687–724, 2016.[37] Xiaofei Zhang, Yanlin Liu, and Hongzhi Liu. Metrics with positive scalar curvature atinfinity and localization algebra. Preprint.
IGHER RHO INVARIANT AND DELOCALIZED ETA INVARIANT AT INFINITY 43 (Xiaoman Chen)
School of Mathematical Sciences, Fudan University
E-mail address : [email protected] (Hongzhi Liu) School of Mathematics, Shanghai University of Finance and Eco-nomics
E-mail address : [email protected] (Hang Wang) School of Mathematical Sciences, East China Normal University
E-mail address : [email protected] (Guoliang Yu) Department of Mathematics, Texas A&M University
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