A Variant of Roe Algebras for Spaces with Cylindrical Ends with Applications in Relative Higher Index Theory
aa r X i v : . [ m a t h . K T ] M a r A Variant of Roe Algebras for Spaces withCylindrical Ends with Applications in RelativeHigher Index Theory
Mehran Seyedhosseini
Abstract
In this paper we define a variant of Roe algebras for spaces with cylin-drical ends and use this to study questions regarding existence and clas-sification of metrics of positive scalar curvature on such manifolds whichare collared on the cylindrical end. We discuss how our constructionsare related to relative higher index theory as developed by Chang, Wein-berger, and Yu and use this relationship to define higher rho-invariants forpositive scalar curvature metrics on manifolds with boundary. This pavesthe way for classification of these metrics. Finally, we use the machinerydeveloped here to give a concise proof of a result of Schick and the author,which relates the relative higher index with indices defined in the presenceof positive scalar curvature on the boundary.
The question whether a given manifold admits a metric of positive scalar cur-vature has spurred much activity in recent years. One of the main approachesto partially answer this question is index theory. On a closed spin manifold M the Schr¨odinger-Lichnerowicz formula implies that the nonvanishing of theFredholm index of the Dirac operator is an obstruction to the existence of pos-itive scalar curvature metric. However, this does not tell the whole story, sincethere exist spin manifolds with vanishing Fredholm index of the Dirac operator,which however do not admit metrics with positive scalar curvature. One way toobtain more refined invariants from the Dirac operator is to not only considerthe dimensions of its kernel and cokernel, but also to consider the action of thefundamental group on them. This gives rise to a higher index for the Diracoperator which is an element of the K -theory of the group C ∗ -algebra of thefundamental group. In general, one can associate a class in the K -homology ofthe manifold to the spin Dirac operator and the higher index is obtained as theimage of this class under the index map µ π ( M ) : K ∗ ( M ) → K ∗ ( C ∗ ( π ( M ))) . The nonvanishing of the higher index gives an obstruction to the existence ofpositive scalar curvature metrics. In order to prove this one can use the factthat the index map fits in the Higson-Roe exact sequence . . . → S π ( M ) ∗ ( M ) → K ∗ ( M ) → K ∗ ( C ∗ ( π ( M ))) → . . . S π ( M ) ∗ ( M ). Given two positive scalar curvaturemetrics on M , one can also define an index difference in K ∗ +1 ( C ∗ ( π ( M ))).These secondary invariants can then also be used for classification of positivescalar curvature metrics up to concordance and bordism. More concretely, in[9] and [10] the authors use these invariants to prove concrete results on the sizeof the space of positive scalar metrics on closed manifolds.In [1] Chang, Weinberger and Yu recently considered the question on com-pact spin manifolds with boundary. Let M be a compact spin manifold withboundary N . They constructed a relative index map µ π ( M ) ,π ( N ) : K ∗ ( M, N ) → K ∗ ( C ∗ ( π ( M ) , π ( N ))) , where K ∗ ( M, N ) and C ∗ ( π ( M ) , π ( N )) denote the relative K -homology groupand the so called relative group C ∗ -algebra. One can define a relative class forthe Dirac operator on M in the relative K -homology group. The relative indexis then the image of the latter relative class under the relative index map. Givena positive scalar curvature metric on M which is collared at the boundary, itwas shown in [1] that the relative index vanishes. A general Riemannian metricwhich is collared at the boundary and has positive scalar curvature there, alsodefines an index in K ∗ ( C ∗ ( π ( M ))), which vanishes if the metric has positivescalar curvature everywhere. It was shown in [2] and [8] that the latter indexmaps to the relative index under a certain group homomorphism. Apart fromrelating previously defined indices to the relative index, this fact also gives aconceptual proof that the relative index is an obstruction to the existence ofpositive scalar curvature metrics which are collared at the boundary.The relative index map fits in an exact sequence . . . → S π ( M ) ,π ( N ) ∗ ( M, N ) → K ∗ ( M, N ) → K ∗ ( C ∗ ( π ( M ) , π ( N ))) → . . . , where S π ( M ) ,π ( N ) ∗ ( M, N ) is the relative analytic structure group and has differ-ent realisations. The main aim of the following paper is to answer the followingnatural question: given a positive scalar curvature metric, which is collared atthe boundary, can one define a secondary invariant in S π ( M ) ,π ( N ) ∗ ( M, N ) whichlifts the relative fundamental class and is useful for classification purposes? Us-ing the machinery we develop in this paper, we will be able to answer the latterquestion in the positive. Furthermore, the same machinery allows us to define ahigher index difference associated to positive scalar curvature metrics on mani-folds with boundary. The definition of such secondary invariants paves the wayfor generalisations of the known results, such as those of [9] and [10], on the sizeof the space of positive scalar curvature metrics to manifolds with boundary.Closely related to the question of existence and classification of positivescalar curvature metrics on manifolds with boundary which are collared at theboundary, is the question of existence and classification of positive scalar cur-vature metrics on manifolds with cylindrical ends, which are collared on thecylindrical end. The usual coarse geometric approach to index theory cannotbe applied in this case, since the Roe algebras of spaces with cylindrical endstend to have vanishing K -theory. We deal with this problem by introducing avariant of Roe algebras for such spaces with more interesting K -theory. Theoperators in the new Roe algebras are required to be asymptotically invariant in2he cylindrical direction. Such operators can then be evaluated at infinity in asense to be described later. Let X be a space with cylindrical end and denote by Y ∞ its cylindrical end. Let Λ and Γ be discrete groups and ϕ : Λ → Γ a grouphomomorphism. ϕ then induces a map B Λ → B Γ of the classifying spaces of thegroups which we can assume to be injective. Given a map (
X, Y ∞ ) → ( B Γ , B Λ)of pairs we construct a long exact sequence · · · → K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) → · · · . In the above sequence e X denotes the Γ-cover of X associated to the map X → B Γ and C ∗ ( e X ) Γ , R + , Λ consists, roughly, of operators which are asymptoticallyinvariant and whose evaluation at infinity results in operators admitting Λ-invariant lifts. For a spin manifold X we associate a fundamental class tothe Dirac operator in K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ). The index of the Dirac operator onthe manifolds with cylindrical end is then defined as the image of the latterclass under the map K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ ( e X ) Γ , R + , Λ ). Given a positivescalar curvature metric on X which is collared on Y ∞ , we define a lift of thefundamental class in K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ), which proves that the nonvanishing ofthe new index is an obstruction to the existence of positive scalar metrics on X and paves the way for classification of such metrics. By removing Y ∞ we obtaina manifold with boundary, which we denote by X . We prove that there is acommutative diagram of exact sequences K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) S Γ , Λ ∗ ( X, ∂X ) K ∗ ( X, ∂X ) K ∗ ( C ∗ (Γ , Λ)) , where the lower sequence is the relative Higson-Roe sequence mentioned above.Furthermore, we show that the fundamental class of e X maps to the relativefundamental class under the middle vertical map. This shows that the relativeindex can be obtained from the new index defined in K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) andallows us to define secondary invariants in S Γ , Λ ∗ ( X, ∂X ).As another application of the machinery developed here we give a short proofthe main statement of [8]. In fact, we think that the proof of the vanishingtheorem for the relative index preseneted in this paper is much simpler andmore straightforward than those presented in [1] and [8].The paper is organised as follows. The second section is a very short reminderof the picture of K -theory for graded C ∗ -algebras due to Trout. In the thirdsection we recall basic notions from coarse geometry and the coarse geometricapproach to index theory on manifolds with and without boundary. In the fourthsection we introduce variants of Roe algebras for spaces with cylindrical ends andcylinders and define the evaluation at infinity map, which plays an importantrole in the rest of the paper. In the final sections, we define indices for Diracoperators on manifolds with cylindrical ends and discuss applications to theexistence and classification problem for metrics with positive scalar curvatureon such manifolds. This is followed by a discussion of the relationship with therelative index for manifolds with boundary and a short proof of a statement onthe relationship between the relative index and indices defined in the presenceof a positive scalar curvature metric on the boundary.3 cknowledgement . I am grateful to Thomas Schick for many inspiring dis-cussions. I would also like to thank Vito Felice Zenobi for his useful commentson an earlier draft of this paper which improved its presentation. C ∗ -algebras In this paper we will use the approach of Trout to K -theory of graded C ∗ -algebras. This description of K -theory was used by Zeidler in [11], where heproves product formulas for secondary invariants associated to positive scalarcurvature metrics. We quickly recall the basics, and refer the reader to [11,Section 2] for more details.Let H be a Real Z -graded Hilbert space and denote by K the Real C ∗ -algebra of compact operators on H . The Z -grading on H induces a Z gradingon K by declaring the even and odd parts to be the set of operators preservingand exchanging the parity of vectors respectively. The Clifford algebra Cl n,m will be the C ∗ -algebra generated by { e , . . . , e n , ǫ , . . . , ǫ m } subject to the rela-tions e i e j + e j e i = − δ ij , ǫ i ǫ j + ǫ j ǫ i = 2 δij, e i ǫ j + ǫ j e i = 0 , e ∗ i = − e i and ǫ ∗ i = ǫ i .The Real structure and the Z -grading of Cl n,m are defined by declaring thesegenerators to be real and odd. Denote by S the C ∗ -algebra C ( R ) endowedwith a Real structure given by complex conjugation and a Z -grading definedby declaring the odd and even parts to be the set of odd and even functions.Given Real, Z -graded C ∗ -algebras A and B denote by Hom( A, B ) the space of C ∗ -algebra homomorphism between A and B respecting the Real structures andthe Z -gradings, by [ A, B ] the set π (Hom( A, B )) and by A b ⊗ B their maximalgraded tensor product. The n -th K -theory group of the Real graded C ∗ -algebra A is defined to be b K n ( A ) := π n (Hom( S , A b ⊗ K ))and turns out to be isomorphic to [ S , Σ n A b ⊗ K ], where Σ n A denotes the n -thsuspension of A . Any Real graded homomorphism of C ∗ -algebras ϕ : S → A gives rise to a class [ ϕ ] := [ ϕ b ⊗ e ] ∈ b K ( A ) with e some rank one projection.Denote by S ( − ǫ, ǫ ) the Real graded C ∗ -subalgebra of S consisting of func-tions vanishing outside ( − ǫ, ǫ ). For our discussion of secondary invariants we willmake use of the fact that the inclusion S ( − ǫ, ǫ ) → S is a homotopy equivalence. Throughout this section X , Y and Z will denote locally compact metric spaceswith bounded geometry. Let Γ be a discrete group acting freely and properly on Z by isometries. Pullingback functions along the action gives rise to an action α : Γ → Aut( C ( Z )). Let( ρ, U ) be an ample covariant representation of the C ∗ -dynamical system( C ( Z ) , Γ , α )on a Hilbert space H . Here ample means that no non-zero element of C ( Z )acts as a compact operator. The space H will be referred to as a Z -module.We will also make use of Cl n -linear Z -modules which are defined analogously byreplacing the Hilbert space H with a Real, graded Hilbert Cl n -module H and by4equiring the representation ρ to be by adjointable operators. In the followingwe will denote ρ ( f ) simply by f . Definition 3.1.
An operator T ∈ L ( H ) is called locally compact if for all f ∈ C ( Z ) both T f and f T are compact. T is called a finite propagation operator if there exists R > f T g = 0 for all f, g ∈ C ( Z ) withdist(supp f, supp g ) > R . The smallest such R is called the propagation of T and is denoted by prop T . T is called Γ-equivariant if T = U ∗ γ T U γ for all γ ∈ Γ.Similarly, one defines the notions of local compactness and finite propagationfor adjointable operators on H . Definition 3.2.
The equivariant algebraic Roe algebra is the ∗ − algebra oflocally compact, finite propagation, Γ-equivariant operators on H and is denotedby R ( Z ) Γ ρ . The equivariant Roe algebra is a C ∗ -completion of the algebraic Roealgebra and is denoted by C ∗ ( d ) ( Z ) Γ ρ . Here ( d ) is a placeholder for the chosencompletion. Similarly, one defines the Cl n -linear equivariant (algebraic) Roealgebra by using finite propagation, locally compact and equivariant operatorson H . These algebras will be denoted by R ( Z ; Cl n ) Γ ρ and C ∗ ( d ) ( Z ; Cl n ) Γ ρ . Remark 3.3.
It follows from Proposition 3.9 below that the K -theory groupsof the Roe algebra are independent of the chosen ample representation. We willtherefore drop ρ form the notation. Remark 3.4.
Examples of possible completions are • the reduced completion C ∗ red ( Z ) Γ ; i.e. the closure of R ( Z ) Γ in L ( H ), • the maximal completion C ∗ max ( Z ) Γ obtained by taking the completion us-ing the universal C ∗ -norm and • the quotient completion C ∗ q ( Z ) Γ introduced in [8].In the following we will denote the Roe algebras obtained by the quotient com-pletion simply by C ∗ ( Z ) Γ and C ∗ ( Z ; Cl n ). Most of what will follow will be validfor all of the above completions, however we will state all of our results only forthe quotient completion.Later in the paper we will introduce variants of Roe algebras which aresuitable for spaces with cylindrical ends and show that the K -theory groups ofthese algebras define functors on a certain category of spaces. Our proofs of thefunctoriality of the K -theory of the new Roe algebras and their independencefrom the chosen ample modules makes use of the analogues of these results forthe classical Roe algebras. Hence, we quickly recall the latter results in thefollowing. Analogues of the results mentioned below hold for the Cl n -linearversions of the algebras introduced and we will later make use of them. Definition 3.5 (See [6, Chapter 2]) . Let X and Y be locally compact separableproper metric spaces endowed with a free and proper action of a discrete groupΓ by isometries. A map f : X → Y is called coarse if the inverse image of eachbounded set of Y under f is bounded and for each R >
S > d X ( x, x ′ ) < R implies d Y ( f ( x ) , f ( x ′ )) < S . Definition 3.6.
Let X and Y be as in Definition 3.5. Let H and H ′ denotean X and Y -module respectively. The support of an operator T : H → H ′ isthe complement of the union of all sets V × U ⊂ Y × X with the property that f T g = 0 for all f ∈ C ( V ) and g ∈ C ( U ). It will be denoted by Support( T ).5 efinition 3.7. Let X and Y be as in Definition 3.5. Let f : X → Y bea coarse map. Let H and H ′ denote an X and Y -module respectively. Anisometry V : H → H ′ is said to cover f if there exists an R > d Y ( f ( x ) , y ) < R for all ( y, x ) ∈ Support( T ). Lemma 3.8 ([4, Lemma 6.3.11]) . Let f, X, Y, H and H ′ be as in Definition3.7. If an isometry V covers f , then T V T V ∗ defines a map from R ( X ) Γ to R ( Y ) Γ , which extends to a map C ∗ ( X ) Γ → C ∗ ( Y ) Γ . Proposition 3.9 ([4, Proposition 6.3.12]) . Let f, X, Y, H and H ′ be as in Def-inition 3.7. There exists an isometry which covers f and thus induces a map K ∗ ( C ∗ ( X ) Γ ) → K ∗ ( C ∗ ( Y ) Γ ) . The latter map is independent of the choice ofthe isometry covering f . In particular, the K ∗ ( C ∗ ( X ) Γ ) is independent of thechoice of the X -module up to a canonical isomorphism. For the rest of the section we consider a space Z with a chosen Z -module H . In the case the action of Γ on Z is cocompact we have the following Proposition 3.10.
If the action of Γ on Z is cocompact, then K ∗ ( C ∗ ( Z ) Γ ) ∼ = K ∗ ( C ∗ q (Γ)) , where C ∗ q (Γ) is the quotient completion of the group ring of Γ asintroduced in [8] .Proof. In the proof of [4, Lemma 12.5.3] an isomorphism R ( X ) Γ ∼ = C [Γ] ⊙ K ( H ′ )is given. Here K ( H ′ ) denotes the algebra of compact operators on a suitableHilbert space H ′ . This isomorphism becomes an isometry if the left hand sideis endowed with the norm of C ∗ ( X ) Γ and the right hand side is endowed withthe norm of C ∗ q (Γ) ⊗ K ( H ′ ) and thus extends to an isomorphism of the lattertwo algebras. The claim then follows from the stability of K -theory.Given a Γ-invariant subset S ⊂ Z it will be useful to look at the ∗ -algebraof operators in R ( Z ) Γ which are supported near S in the sense of the following Definition 3.11.
Given a subset S ⊂ Z , T is said to be supported near S ifthere exists an R > T ⊂ U R ( S ) × U R ( S ). Here U R ( S ) denotes the open R -neighbourhood of S . Definition 3.12.
Let S be a Γ-invariant subset of Z . The equivariant algebraicRoe algebra of S relative to Z is the subalgebra of R ( Z ) Γ consisting of operatorssupported near S and will be denoted by R ( S ⊂ Z ) Γ . The equivariant Roealgebra of S relative to Z is the closure of R ( S ⊂ Z ) Γ in C ∗ ( Z ) Γ and is denotedby C ∗ ( S ⊂ Z ) Γ .Since S is itself a Γ-space, it has its own Roe algebra. This is related to theRoe algebra of S relative to Z by the following Proposition 3.13 ([5, Section 5, Lemma 1]) . K ∗ ( C ∗ ( S ) Γ ) ∼ = K ∗ ( C ∗ ( S ⊂ Z ) Γ ) . We will also need the notion of support of a vector in H . Definition 3.14.
Let v ∈ H . The support of v is the complement of the unionof all open subsets U with the property that f v = 0 for all f ∈ C ( U ).6 .2 Yu’s Localisation Algebras Given a C ∗ -algebra A denote by T A the C ∗ -algebra of all uniformly continuousfunctions f : [1 , ∞ ) → A endowed with the supremum norm. Definition 3.15.
The equivariant localisation algebra of Z is defined to be the C ∗ -subalgebra of T C ∗ ( Z ) Γ generated by elements f satisfying • prop f ( t ) < ∞ for all t ∈ [1 , ∞ ) • prop f ( t ) t →∞ −−−→ C ∗ L ( Z ) Γ .The K -theory of the localisation algebra provides a model for the equivariantlocally finite K -homology. Yu constructed an isomorphism Ind L : K Γ ∗ ( Z ) → K ∗ ( C ∗ L ( Z ) Γ ), where K Γ ∗ ( Z ) denotes the equivariant KK -group KK Γ ∗ ( C ( Z ) , C ). Definition 3.16.
A Γ-cover Z of a locally compact metric space M is called nice if there exists an ǫ > Z to every ǫ -ball in M is trivial. Proposition 3.17.
Let Z → M be a nice Γ -cover. Then there is an isomor-phism K ∗ ( C ∗ L ( Z ) Γ ∼ = K ∗ ( C ∗ L ( M )) induced by lifting operators on M with smallpropagation to equivariant operators on Z . In particular Ind L gives rise to anisomorphism K ∗ ( M ) ∼ = K ∗ ( C ∗ L ( Z ) Γ ) . Remark 3.18.
In the following we will assume all covers to be nice.Given a Γ-invariant subset S of Z it will be useful to define the localisationalgebra of S relative to Z . Definition 3.19.
The equivariant localisation algebra of S relative to Z is de-fined as the C ∗ -subalgebra of C ∗ L ( Z ) Γ generated by elements f with the propertythat there exists a continuous function B : [1 , ∞ ) → R vanishing at infinity suchthat prop f ( t ) < B ( t ). It will be denoted by C ∗ L ( S ⊂ Z ) Γ . Proposition 3.20 ([11, Lemma 3.7]) . K ∗ ( C ∗ L ( S ) Γ ) ∼ = K ∗ ( C ∗ L ( S ⊂ Z ) Γ ) Definition 3.21.
The equivariant structure algebra of Z is the C ∗ -subalgebraof C ∗ L ( Z ) Γ consisting of C ∗ ( Z ) Γ -valued functions f on [1 , ∞ ) with f (1) = 0. Itis denoted by C ∗ L, ( Z ) Γ .Given a Γ-cover Z → M induced by a map M → B Γ, with M compact, theindex map µ Γ : K ∗ ( M ) → K ∗ ( C ∗ (Γ)) can be defined by K ∗ ( M ) ∼ = K ∗ ( C ∗ L ( Z ) Γ ) (ev ) ∗ −−−−→ K ∗ ( C ∗ ( Z ) Γ ) ∼ = K ∗ ( C ∗ q (Γ)). Clearly, it fits into a long exact sequence . . . → S Γ ∗ ( M ) → K ∗ ( M ) → K ∗ ( C ∗ (Γ)) → . . . , where S Γ ∗ ( M ) denotes K ∗ ( C ∗ L, ( Z ) Γ ) and is called the analytic structure group .This long exact sequence is called the Higson-Roe analytic surgery sequence .7 .2.1 Fundamental Class of Dirac Operators Now suppose that Z is an n -dimensional spin manifold. We assume that Γ actsby spin structure preserving isometries. Denote by / S = P Spin ( Z ) × Spin Cl n theCl n -spinor bundle on Z . Recall that the Cl n -linear Dirac operator on Z (actingon sections of / S ) gives rise to a class in K ∗ ( Z ) Γ . Under the isomorphism of3.17, this class corresponds to the class [ /D Z ] ∈ b K ( C ∗ L ( Z ; Cl n ) Γ ) ∼ = K n ( C ∗ L ( Z ) Γ )defined by ϕ /D : S → C ∗ L ( Z ; Cl n ) Γ sending f ∈ S to ( t f ( t /D )) ∈ C ∗ L ( Z ; Cl n ) Γ . Let Λ and Γ be discrete groups and ϕ : Λ → Γ a group homomorphism. Thehomomorphism ϕ gives rise to a continuous map Bϕ : B Λ → B Γ. It alsoinduces a map ϕ : C ∗ max (Λ) → C ∗ max (Γ). We can and will assume that Bϕ isinjective. Given a compact space X , a subset Y ⊂ X and a map f : ( X, Y ) → ( B Γ , B Λ) Chang, Weinberger and Yu ([1]) define a relative index map µ Γ , Λ : K ∗ ( X, Y ) → K ∗ ( C ∗ max (Γ , Λ)). Here C ∗ max (Γ , Λ) := SC ϕ denotes the suspensionof the mapping cone of ϕ and is called the (maximal) relative group C ∗ -algebra.If X is not compact, then their construction gives rise to a relative index mapwith target the K -theory group of a relative Roe algebra. Here, we quicklyrecall the construction of the relative index map. Denote by e X and e Y the Γand Λ coverings of X and Y associated to f and f ↾ Y respectively. Denote by Y ′ the restriction of e X to Y . Using particular e X, Y ′ and e Y -modules Chang,Weinberger and Yu construct a morphism of C ∗ -algebras ψ : C ∗ max ( e Y ) Λ → C ∗ max ( Y ′ ) Λker φ ֒ → C ∗ max ( e X ) Γ . We will later discuss the morphism ψ in more detail. Applying ψ pointwise weobtain a morphism ψ L : C ∗ L, max ( e Y ) Λ → C ∗ L, max ( Y ′ ) Λker φ ֒ → C ∗ L, max ( e X ) Γ . Analogous to the absolute case, there is a map Ind rel L : K ∗ ( X, Y ) → K ∗ ( SC ψ L ). Proposition 3.22.
Ind rel L is an isomorphism. If, furthermore, X is compact,then K ∗ ( SC ψ ) ∼ = K ∗ ( C ∗ max (Γ , Λ)) . Evaluation at 1 gives rise to morphisms ev : C ∗ L, max ( e Y ) Λ → C ∗ max ( e Y ) Λ andev : C ∗ L, max ( e X ) Γ → C ∗ max ( e X ) Γ . The diagram C ∗ L, max ( e Y ) Λ C ∗ max ( e Y ) Λ C ∗ L, max ( e X ) Γ C ∗ max ( e X ) Γev ψ L ψ ev is commutative. Hence, the evaluation at 1 maps give rise to a morphism SC ψ L → SC ψ , which we also denote by ev . Definition 3.23.
The relative index map µ Γ , Λ is defined to be the composition K ∗ ( X, Y ) Ind rel L −−−→ K ∗ ( SC ψ L ) (ev ) ∗ −−−−→ K ∗ ( SC ψ ) . emark 3.24. If X is compact, the isomorphism K ∗ ( SC ψ ) ∼ = K ∗ ( C ∗ max (Γ , Λ))allows us to consider µ Γ , Λ as a map with values in the K -theory of the relativegroup C ∗ -algebra. Remark 3.25.
Instead of the maximal completion of the group rings and theRoe algebras, one can consider the quotient completion introduced in [8] andobtain a similar relative index map. If the group homomorphism ϕ : Λ → Γ isinjective, then one can also use the reduced completion of the group rings andRoe algebras.Analogous to the absolute case the relative index map fits into a long exactsequence. The map ψ L gives rise, by restriction, to a map ψ L, : C ∗ L, ( e Y ) Λ → C ∗ L, ( e X ) Γ . We have a short exact sequence of C ∗ -algebras0 → SC ψ L, → SC ψ L ev −−→ SC ψ → , which gives rise to a long exact sequence of K -theory groups · · · → K ∗ ( SC ψ L, ) → K ∗ ( X, Y ) µ Γ , Λ −−−→ K ∗ ( SC ψ ) → · · · . Remark 3.26.
Similarly, one defines maps C ∗ ( L ) ( e Y ; Cl n ) Λ → C ∗ ( L ) ( e X ; Cl n ) Γ ,which we will also denote by ψ ( L ) . Given a compact spin manifold M with boundary N with a metric on M whichis collared at the boundary, consider the manifold M ∞ obtained by attaching N ∞ := N × [0 , ∞ ) to M along N . Extend the metric on M to a metric on M ∞ using the product metric on the half-cylinder (the metric on R + is theusual one). Denote by [ /D M ∞ ] the fundamental class of the Dirac operator on M ∞ in K ∗ ( M ∞ ). Given a map f : ( M, N ) → ( B Γ , B Λ), the construction ofthe previous section gives rise to a relative index map µ Γ , Λ : K ∗ ( M, N ) → K ∗ ( C ∗ (Γ , Λ)).
Definition 3.27.
The relative index of the Dirac operator on M is defined tobe the image of [ /D M ∞ ] under the composition K ∗ ( M ∞ ) −→ K ∗ ( M ∞ , N ∞ ) ∼ = −→ K ∗ ( M, N ) µ Γ , Λ −−−→ K ∗ ( C ∗ (Γ , Λ)) . where the isomorphism K ∗ ( M ∞ , N ∞ ) ∼ = −→ K ∗ ( M, N ) is given by excision.The nonvanishing of the relative index obstructs the existence of positivescalar curvature metrics on M . Proposition 3.28 ([1, Proposition 2.18],[8, Theorem 5.1],[2, Theorem 4.12]) . If there exists a positive scalar curvature metric on M which is collared at theboundary, then the relative index of the Dirac operator on M vanishes. Coarse Spaces with Cylindrical Ends
Let X be a locally compact metric space with a free and proper action of adiscrete group Γ by isometries. For a Γ-invariant subset Y of X we can endow Y × R with a Γ-action by setting γ ( y, t ) = ( γy, t ). Definition 4.1.
Let X and Y be as above. The space X is said to have acylindrical end with base Y if there exists a Γ-equivariant isometry ι : Y × [0 , ∞ ) → X satisfying • ι (( y, y • lim R →∞ dist( ι ( Y × [ R, ∞ )) , X − Y ∞ ) = ∞ Here Y ∞ denotes ι ( Y × [0 , ∞ )) and Y × [0 , ∞ ) is endowed with the productmetric. Definition 4.2.
Let (
X, Y, ι ) and ( X ′ , Y ′ , ι ′ ) be spaces with cylindrical ends. amap f : X → X ′ is called a coarse map of spaces with cylindrical ends if it is acoarse map and satisfies • f ( X \ Y ∞ ) ⊂ X ′ \ Y ′∞ and • f ( ι ( y, t )) = ι ′ ( g ( y ) , t ) with g := f ↾ Y . Using the isometry ι one can define an action of R + on C ( Y ∞ ) by setting L s ( f )( ι (( y, t ))) = f ( ι ( y, t − s )) for s ∈ R + . We would like to define a variant ofRoe algebras for spaces with cylindrical ends. In order to do this we use moduleswhich are equipped with an action of R + by partial isometries, which is compat-ible with the action of R + on C ( Y ∞ ). Before making this precise we introducesome notation. Let H Y be a Y -module. The Hilbert space L ( R + ; H Y ) can beendowed with the structure of Y ∞ -module in a natural way. On L ( R + ; H Y )one can define a family of partial isometries P st s by P st s ( f )( t ) = f ( t − s ) for t ≥ s and P st s ( f )( t ) = 0 otherwise. Definition 4.3.
Let (
X, Y, ι ) be a space with cylindrical end. A Hilbert space iscalled an X -module tailored to the end if there is a tuple ( ρ, U, { P s } ) satisfyingthe following properties: • ( ρ, U ) is a covariant ample representation of C ( X ) on H . • P s is a strongly continuous family of partial isometries on H satisfying – P − s = P ∗ s – P ∗ s P s = e ρ ( χ ι ( Y × [0 , ∞ )) ) for all s > – P s P ∗ s = e ρ ( χ ι ( Y × [ s, ∞ )) ) for all s > – ρ ( f ) P s = P s ρ ( L s ( f )) for all f ∈ C ( Y ∞ ). • For some Y -module H Y , there is a Γ-equivariant unitary: W : χ Y ∞ H → L ( R + ; H Y ) which covers the identity and satisfies W P s = P st s W .10ere, the tuple ( ρ, U, { P s } ) is part of the structure of the X -module and e ρ isthe extension of the representation ρ to the bounded Borel functions.Similarly, one can define Cl n -linear modules tailored to the end. The fol-lowing definitions generalise in an obvious manner to the Cl n -linear context. Inthe rest of the section ( X, Y, ι ) will be a space with cylindrical end (endowedwith a Γ-action) and H will denote an X -module tailored to the end. We willconstruct a variant of Roe algebras for spaces with cylindrical ends. Since H is in particular an X -module, it can be used to construct the usual equivariantalgebraic Roe algebra R ( X ) Γ . Definition 4.4.
An operator T ∈ L ( H ) is called asymptotically R + -invariant if lim R →∞ sup s> || ( P − s T P s − T ) χ ι ( Y × [ R, ∞ )) || = 0 . Lemma 4.5.
The set of operators in R ( X ) Γ , which are asymptotically R + -invariant is a ∗ -subalgebra.Proof. Let
S, T ∈ R ( X ) Γ be asymptotically R + -invariant. Set R := prop T . Inthe following χ R will denote χ ι ( Y × [ R, ∞ )) . Since P s P − s = χ s (for all s >
0) andelements in the image of
T P s χ R are supported in ι ( Y × [ R − R + s, ∞ )) wehave for R > R ( P − s ST P s ) χ R = ( P − s SP s P − s T P s ) χ R . Furthermore, since elements in the image of ( P − s T P s ) χ R are supported in ι ( Y × [ R − R , ∞ )) we have( P − s SP s P − s T P s ) χ R = ( P − s SP s χ R − R P − s T P s ) χ R . From the asymptotic R + -invariance, it follows that P − s SP s χ R − R = Sχ R − R + E R − R ,s ( S ) and P − s T P s χ R = T χ R + E R,s ( T ) withlim R →∞ sup s> || E R − R ,s ( S ) || = 0 = lim R →∞ sup s> || E R,s ( T ) || . ( ∗ )Therefore ( P − s ST P s − ST ) χ R is equal to Sχ R − R T χ R + Sχ R − R E R,s ( T )+ E R − R ,s ( S ) T χ R + E R − R ,s ( S ) E R,s ( T ) − ST χ R = Sχ R − R E R,s ( T ) + E R − R ,s ( S ) T χ R + E R − R ,s ( S ) E R,s ( T ) . The latter equality and ( ∗ ) imply that ST is asymptotically R + -invariant. Wenow show that T ∗ is also asymptotically R + -invariant. We have( P − s T ∗ P s − T ∗ ) χ R = ( χ R ( P − s T P s − T )) ∗ . Furthermore, since the propagation of T is R the right hand side is equal to( χ R ( P − s T P s − T ) χ R − R ) ∗ = ( χ R E R − R ,s ( T )) ∗ . This shows that T ∗ is asymp-totically R + -invariant. The fact that the set of asymptotically R + -invariantoperators is closed under addition is clear.11 efinition 4.6. The equivariant algebraic Roe algebra of X tailored to the endis the ∗ -subalgebra of R ( X ) Γ consisting of asymptotically R + -invariant opera-tors. It will be denoted by R ( X ) Γ , R + . The Roe algebra of X tailored to theend is the closure of R ( X ) Γ , R + in C ∗ ( d ) ( X ) Γ and will be denoted by C ∗ ( d ) ( X ) Γ , R + .Similarly, using a Cl n -module tailored to the end, one defines R ( X ; Cl n ) Γ , R + and C ∗ ( d ) ( X ; Cl n ) Γ , R + . Remark 4.7.
Note that the algebraic and C ∗ -algebraic Roe algebras definedabove depend, a priori, on the chosen modules tailored to the end. We willsee later, that the K -theory groups of the C ∗ -algebras defined using differentmodules are canonically isomorphic. Remark 4.8.
The equivariant Roe algebra of X tailored to the end obtainedby using the quotient completion will simply be denoted by C ∗ ( X ) Γ , R + . In thefollowing we will only make use of the quotient completion; however, most ofthe results are also valid for the reduced and maximal completions.Let ( X ′ , Y ′ , ι ′ ) be another space with a cylindrical end and H ′ an X ′ -moduletailored to the end given by the data ( ρ ′ , U ′ , { P ′ s } ). Definition 4.9.
Let f : X → X ′ be a map of spaces with cylindrical ends. Anisometry V : H → H ′ is said to cover f if it covers f in the sense of [4, Definition6.3.9] and satisfies V P s = P ′ s V . Lemma 4.10.
Let f and V be as in Definition 3.7. Then T → V T V ∗ definesa map C ∗ ( X ) Γ , R + → C ∗ ( X ′ ) Γ , R + .Proof. The fact that conjugation by V gives a map C ∗ ( X ) Γ → C ∗ ( X ′ ) Γ is thecontent of [4, Lemma 6.3.11]. We show that if T ∈ R ( X ) Γ is asymptotically R + -invariant, then so is V T V ∗ . In the following e ρ and e ρ ′ will denote the extensionof ρ and ρ ′ to the bounded Borel functions on X and X ′ respectively. Using thefact that V intertwines the families { P s } and { P ′ s } we get( P ′− s V T V ∗ P ′ s − V T V ∗ ) ρ ′ ( χ R ) = V ( P − s T P s − T ) V ∗ ρ ′ ( χ R ) = V ( P − s T P s − T ) V ∗ P ′ s P ′− s = V ( P − s T P s − T ) P s P − s V ∗ = V ( P − s T P s − T ) e ρ ( χ R ) V ∗ , which proves the claim. Proposition 4.11.
Let f : X → X ′ be a map of spaces with cylindrical ends.Then there is an isometry V : H → H ′ which covers f . Conjugation by V induces a homomorphism K ∗ ( C ∗ ( X ) Γ , R + ) → K ∗ ( C ∗ ( X ′ ) Γ , R + ) which does notdepend on the choice of the covering isometry V . In particular, K ∗ ( C ∗ ( X ) Γ , R + ) does not depend on the choice of the X -module tailored to the end up to acanonical isomorphism.Proof. We prove the existence of an isometry covering f . The proof that theinduced map on the K -theory groups by conjugation with V does not depend onthe choice of V is the same as that of [4, Lemma 5.2.4]. We have H ∼ = χ X \ Y ∞ H ⊕ χ Y ∞ H ∼ = χ X \ Y ∞ H ⊕ ( H Y ⊗ L ( R + )). Similarly H ′ ∼ = χ X ′ \ Y ′∞ H ′ ⊕ ( H ′ Y ′ ⊗ L ( R + )). By Proposition 3.9, there are isometries V : χ X \ Y ∞ H → χ X \ Y ∞ H ′ and V : H Y → H ′ Y ′ covering the restrictions of f to X \ Y ∞ and Y respectively.We use the above decompositions of H and H ′ and set V = V ⊕ ( V ⊗ Id).12ince the isomorphisms χ Y ∞ H ∼ = H Y ⊗ L ( R + ) and χ Y ′∞ H ′ ∼ = H ′ Y ′ ⊗ L ( R + )cover the identity maps on Y ∞ and Y ′∞ respectively, V , seen as an isometryfrom H to H ′ , covers f in the sense of Definition 3.7. Furthermore, the latterisomorphisms intertwine the families { P s } and { P ′ s } with the standard familiesof partial isometries { P st s } on H Y ⊗ L ( R + ) and H Y ′ ⊗ L ( R + ), which impliesthat V intertwines { P s } and { P ′ s } . Thus, V covers f in the sense of Definition4.9.One can also define localisation and structure algebras tailored to the end. Definition 4.12.
The equivariant localisation algebra of X tailored to the end isdefined to be the C ∗ -algebra of T C ∗ ( X ) Γ , R + generated by elements f satisfying • prop f ( t ) < ∞ for all t ∈ [1 , ∞ ) • prop f ( t ) t →∞ −−−→ C ∗ L ( X ) Γ , R + . The equivariant structure algebra of X isdefined to be the subalgebra of C ∗ L ( X ) Γ , R + generated by f which further satisfy f (1) = 0. It will be denoted by C ∗ L, ( X ) Γ , R + . Remark 4.13.
One can also prove the existence of families of isometries cov-ering a given map in a suitable sense and inducing maps between localisationand structure algebras tailored to the end. One can then deduce an analogueof Proposition 4.11 for structure and localisation algebras tailored to the end.These statements can be proved by using the approach of the proof of Propo-sition 4.11 and slight modifications of the proofs for the classical structure andlocalisation algebras.
One of our main goals in the following is to evaluate asymptotically R + -invariantoperators on a space ( X, Y, ι ) with cylindrical end and obtain R -invariant opera-tors on the cylinder over Y . In this section we define a Roe algebra for cylinderswhich will be the target of the aforementioned ”evaluation at infinity map”. Inthe following Y will denote a locally compact separable metric space endowedwith a free and proper action of a discrete group Γ by isometries. Endow Y × R with the product metric. Furthermore, L ′ s ( f )( y, t ) = f ( y, t − s ) defines an actionof R on C ( Y × R ). Let H Y be a Y -module. The space L ( R , H Y ) can thenbe endowed with the structure of a Y × R -module. There is a family { Q st s } ofunitaries on L ( R , H Y ) given by the shift of functions in the R -direction Definition 4.14.
A Hilbert space H is called a cylindrical Y × R -module ifthere is a tuple ( ρ, U, { Q s } ) satisfying the following properties: • ( ρ, U ) is a covariant ample representation of C ( Y × R ) on H . • { Q s } is a strongly continuous group of unitaries commuting with the rep-resentation U of Γ on H and satisfying ρ ( f ) Q s = Q s ρ ( L ′ s ( f )). • For some Y -module H Y , there is a unitary isomorphism W : H → L ( R , H Y )which covers the identity map of Y × R in the sense of Definition 3.7, inter-twines the families { Q s } and { Q st s } and which does not shift the supportof vectors in the R -direction. 13 cylindrical Y × R -modules is in particular a Y × R -module and allows usto define the usual Roe algebras R ( Y × R ) Γ and C ∗ ( Y × R ) Γ Definition 4.15.
An operator T ∈ R ( Y × R ) Γ is called R -invariant if Q − s T Q s − T = 0for all s ∈ R . The closure of the ∗ -algebra of such elements in C ∗ ( Y × R ) Γ willbe denoted by C ∗ ( Y × R ) Γ × R . Similarly, using a cylindrical Y × R -Cl n -module,one defines C ∗ ( Y × R ; Cl n ) Γ , R + .Now let Y ′ be another space. Let f : Y × R → Y ′ × R be a coarse map,which is the suspension of a map g : Y → Y ′ . Let H and H ′ be cylindrical Y × R and Y ′ × R -modules respectively. A slight modification of the proof ofProposition 4.11, proves the following Proposition 4.16.
Let f, H and H ′ be as above. There exists an isome-try V : H → H ′ which covers f in the sense of Definition 3.7 and inter-twines the families { Q s } and { Q ′ s } . Conjugation by V induces a homomor-phism K ∗ ( C ∗ ( Y × R ) Γ × R ) → K ∗ ( C ∗ ( Y ′ × R ) Γ × R ) . The latter homomorphismis independent of the choice of the isometry V satisfying the above properties.In particular, K ∗ ( C ∗ ( Y × R ) Γ × R ) does not depend on the chosen cylindrical Y × R -module. Let (
X, Y, ι ) be a space with cylindrical end on which Γ acts as above. Asymp-totically R + -invariant operators can be “evaluated at infinity” in the sense ofPropositions 4.19 and 4.20 to give R -invariant operators on Y × R . In order todo this we first introduce the notion of ( X, Y, ι ) modules, which is given by a pairconsisting of an X -module tailored to the end and a cylindrical Y × R -modulewhich are related in a special way. Definition 4.17.
Let (
X, Y, ι ) be a space with cylindrical end. A pair (
H, H ′ ) ofHilbert spaces is called a ( X, Y, ι ) -module , if there is a tuple ( ρ, ρ ′ , U, U ′ , { P s } , { Q s } , i )satisfying the following properties: • ( ρ, U, { P s } ) and ( ρ ′ , U ′ , { Q s } ) endow H and H ′ with the structure of an X -module tailored to the end and a cylindrical Y × R -module respectively. • i is a unitary χ Y ∞ H → χ Y × R + H ′ intertwining the Γ-representations andthe representations of C ( Y ∞ ) and C ( Y × R + ) on χ Y ∞ H and χ Y × R + H ′ respectively. • Q s ◦ i = i ◦ P s ↾ χ Y ∞ H for all s > Remark 4.18.
Note that ρ ′ ( f ) Q s = Q s ρ ′ ( L ′ s ( f )) in particular implies that Q s applied to vectors in H ′ which are supported in Y × [ R, ∞ ), results in vectorswith support in Y × [ R + s, ∞ ). This observation will be used in the proof ofProposition 4.19.In the following we will call an element v ∈ H ′ compactly supported if itssupport in the sense of Definition 3.14 is a compact subset of Y × R . Thenondegeneracy of ρ ′ implies that compactly supported vectors are dense in H ′ .14 roposition 4.19. For T ∈ R ( e X ) Γ , R + and a compactly supported vector v ∈ H ′ the limit T ∞ v := lim s →∞ Q − s iT i ∗ Q s v exists in H ′ and the mapping v T ∞ v extends to a continuous linear map T ∞ on H ′ . Furthermore, the operator T ∞ defined in this way is an element of R ( Y × R ) Γ × R .Proof. In the following χ R will denote χ ι ( Y × [ R, ∞ )) and will be seen as an oper-ator on H . χ ′ R will denote χ Y × [ R, ∞ ) and will act as on operator on H ′ . • The limit exists: for ǫ > e R such that sup s> || ( P − s T P s − T ) χ R || <ǫ for all R ≥ e R . Let e s be such that Q s ( v ) is supported on Y × R + for all s ≥ e s . Set s = e R + e s . Then we have || Q − s + s iT i ∗ Q s + s v − Q − s iT i ∗ Q s v || = || Q − s ( Q − s iT i ∗ Q s − iT i ∗ ) Q s v || . Note that ( Q − s iT i ∗ Q s − iT i ∗ ) Q s v = i ( P − s T P s − T ) χ e R i ∗ Q s v ; hence || Q − s ( Q − s iT i ∗ Q s − iT i ∗ ) Q s v || < || ( P − s T P s − T ) χ e R |||| v || , where we use that Q s is a unitary. The latter inequality shows that { Q − s iT i ∗ Q s v } s ≥ e s is a Cauchy net and thus has a limit. • T ∞ is a bounded operator on H ′ : we clearly have || T ∞ v || ≤ || T |||| v || for all compactly supported v which shows that v T ∞ v is a boundedoperator on the dense subspace of compactly supported vectors in H ′ andthus extends to a bounded operator on H ′ . • T ∞ is an R and Γ-invariant operator: for t ∈ R we have Q − t T ∞ Q t v = Q − t ( lim s →∞ Q − s iT i ∗ Q s Q t v ) = lim s →∞ Q − s − t iT i ∗ Q s + t v =lim s →∞ Q − s iT i ∗ Q s v = T ∞ v for all compactly supported v . Therefore Q − t T ∞ Q t = T ∞ . A similarcomputation and the fact that the R -action and the Γ-action on H ′ com-mute proves the Γ-invariance. • T ∞ is locally compact: We show that for ψ ∈ C c ( Y × R ) ψT ∞ is compact.The proof of the compactness of T ∞ ψ is similar and even more straight-forward. There exists M > ψ is containedin Y × [ − M, ∞ ). Set R := prop T . If v is compactly supported withsupport in Y × ( −∞ , − M − R ), then ψQ − s iT i ∗ Q s v = 0. We thus havea commutative diagram H ′ H ′ χ ′− M − R H ′ χ ′− M − R H ′ . ψT ∞ χ ′− M − R ψT ∞ Therefore, it suffices to show that the restriction of ψT ∞ to χ ′− M − R H ′ iscompact. First we show that { χ ′− M − R Q − s iT i ∗ Q s χ ′− M − R } s ≥ M + R is anorm convergent net of operators on χ ′− M − R H ′ . Set s := e R + M + R .Then, similar to the above computation, we have || χ ′− M − R Q − s + s iT i ∗ Q s + s χ ′− M − R − χ ′− M − R Q − s iT i ∗ Q s χ ′− M − R || =15 | χ ′− M − R Q − s ( Q − s iT i ∗ Q s − iT i ∗ ) Q s χ ′− M − R || . Furthermore,( Q − s iT i ∗ Q s − iT i ∗ ) Q s χ ′− M − R = i ( P − s T P s − T ) χ e R i ∗ Q s χ ′− M − R , which implies || χ ′− M − R Q − s ( Q − s iT i ∗ Q s − iT i ∗ ) Q s χ ′− M − R || < ǫ. Hence, { ψχ ′− M − R Q − s iT i ∗ Q s χ ′− M − R } s ≥ M + R is norm convergent andconverges strongly to ψT ∞ in L ( χ ′− M − R H ′ ). Thus ψT ∞ restricted to χ ′− M − R H ′ is actually the norm limit of ψχ ′− M − R Q − s iT i ∗ Q s χ ′− M − R = χ ′− M − R Q − s L ′ s ( ψ ) iT i ∗ Q s χ ′− M − R = χ ′− M − R Q − s iL s ( ψ ) T i ∗ Q s χ ′− M − R as s tends to infinity. The compactness of ψT ∞ ↾ χ ′− M − R H ′ then followsfrom that of L s ( ψ ) T . Proposition 4.20.
The map ev ∞ : R ( e X ) Γ , R + → R ( Y × R ) Γ × R given by T T ∞ is continuous if the domain and target space are endowed with the normsof C ∗ ( e X ) Γ , R + and C ∗ ( Y × R ) Γ × R respectively. Thus it gives rise to a morphismof C ∗ -algebras ev ∞ : C ∗ ( e X ) Γ , R + → C ∗ ( Y × R ) Γ × R .Proof. If we endow R ( e X ) Γ , R + with the reduced norm, the continuity of the mapev ∞ : R ( e X ) Γ , R + → C ∗ red ( Y × R ) Γ × R follows from the proof of the previous propo-sition. Indeed, we already saw that this map is a contraction. The continuity ofthis map for the quotient completion follows from its continuity for the reducedcompletion, the commutativity of the the diagram R ( X ) Γ , R + R ( Y × R ) Γ × R R ( X/N ) Γ /N, R + R ( Y /N × R ) Γ /N × R ev ∞ ev ∞ for all normal subgroups N of Γ and the definition of the quotient completionin [8, Section 4]. It remains to show that it is a morphism of ∗ -algebras. Let S and T be in R ( e X ) Γ , R + and let v ∈ H ′ be compactly supported. We have( T S ) ∞ v = lim s Q − s iT Si ∗ Q s v = lim s Q − s iT i ∗ Q s Q − s iSi ∗ Q s v =lim s Q − s iT i ∗ Q s ( S ∞ v + E ( s )) = T ∞ ( S ∞ v ) . The last equality follows from the fact that || Q − s iT i ∗ Q s ( E ( s )) || ≤ || T |||| E ( s ) || .The rest is clear.Thus an ( X, Y, ι )-module allows us to define an evaluation at infinity map.Next we will prove a functoriality result, which in particular shows that theinduced map on K -theory is independent of the chosen ( X, Y, ι )-module. Let( b X, b Y , b ι ) be another space with cylindrical end (and a Γ-action). Let ( b H, b H ′ )be an ( b X, b Y , b ι )-module. Let f : ( X, Y, ι ) → ( b X, b Y , b ι ) be a map of spaces withcylindrical ends. In particular, the suspension of the restriction of f to Y definesa map Y × R → b Y × R . In this situation we have the following16 roposition 4.21. There are isometries V : H → b H and V ′ : H ′ → c H ′ whichsatisfy the conditions of Definition 4.9 and Proposition 4.16 respectively andwhich make the diagram C ∗ ( X ) Γ , R + C ∗ ( Y × R ) Γ × R C ∗ ( b X ) Γ , R + C ∗ ( b Y × R ) Γ × R ev ∞ Ad V Ad V ′ ev ∞ commutative. In particular, the map ( ev ∞ ) ∗ : K ∗ ( C ∗ ( X ) Γ , R + ) → K ∗ ( C ∗ ( Y × R ) Γ × R ) does not depend on the choice of the ( X, Y, ι ) -module up to the usualcanonical isomorphisms.Proof. Let V ′ : H ′ → c H ′ satisfy the conditions of Proposition 4.16 and suchthat V ′ and ( V ′ ) ∗ map vectors which are supported in Y × R + and b Y × R + to vectors which are supported in b Y × R + and Y × R + respectively. We havedecompositions H ∼ = χ X \ Y ∞ H ⊕ χ Y ∞ H and b H = χ b X \ b Y ∞ b H ⊕ χ b Y ∞ b H . Using thesedecompositions we define V to be the isometry (cid:18) V b i ∗ V ′ i (cid:19) , where V : χ X \ Y ∞ H ⊕ χ Y ∞ H → χ b X \ b Y ∞ b H is any isometry covering the restric-tion of f to X \ Y ∞ and b i is the unitary from the definition of an ( b X, b Y , b ι )-module identifying χ b Y ∞ b H and χ b Y × R + b H ′ . Now we show that for T ∈ R ( X ) Γ , R + ,(Ad V ′ ◦ ev ∞ )( T ) = (ev ∞ ◦ Ad V )( T ). This then finishes the proof of the propo-sition. Let v ∈ b H ′ be compactly supported. We have(Ad V ′ ◦ ev ∞ )( T ) v = V ′ lim s Q − s iT i ∗ Q s V ′∗ v = lim s b Q − s V ′ iT i ∗ V ′∗ b Q s v. On the other hand (ev ∞ ◦ Ad V )( T ) v = lim s b Q − s b iV T V ∗ b i ∗ b Q s v . Set R = prop T .For s sufficiently large b Q s v is supported in Y × ( R , ∞ ). Thereforelim s b Q − s b iV T V ∗ b i ∗ b Q s v = lim s b Q − s b i b i ∗ V ′ iT i ∗ V ′∗ b i b i ∗ b Q s v = lim s b Q − s V ′ iT i ∗ V ′∗ b Q s v. Hence, (Ad V ′ ◦ ev ∞ )( T ) = (ev ∞ ◦ Ad V )( T ). (Γ , Λ) -equivariant Roe Algebras Let (
X, Y, ι ) be a space with cylindrical end. We do not assume the existence ofan action of Γ on X . Let Λ , Γ and ϕ be as in Section 3.3. Suppose there existsa map of pairs η : ( X, Y ∞ := ι ( Y × R + )) → ( B Γ , B Λ) satisfying η ( ι (( y, t ))) = η ( ι (( y, t ∈ R + . This allows us to define Γ-coverings e X, Y ′ ( ∞ ) of X, Y ( ∞ ) and a Λ-covering ] Y ( ∞ ) of Y ( ∞ ) . We obtain in this way new spaces withcylindrical ends ( e X, Y ′ , ι ′ ) and ( e Y ∞ , e Y , e ι ). In this section the Roe algebras willbe constructed using fixed ( e X, Y ′ , ι ′ ) and ( f Y ∞ , e Y , e ι )-modules. The constructionof the previous section gives rise to evaluation at infinity maps C ∗ ( f Y ∞ ) Λ , R + → C ∗ ( e Y × R ) Λ × R and C ∗ ( e X ) Γ , R + → C ∗ ( Y ′ × R ) Γ × R . Chang, Weinberger and Yu17onstructed a map C ∗ ( e Y × R ) Λ → C ∗ ( Y ′ × R ) Γ 1 . It is easy to see that this maprespects the R -invariance and asymptotic R + -invariance of operators. Thus weget, by restriction, a map ϕ : C ∗ ( e Y × R ) Λ × R → C ∗ ( Y ′ × R ) Γ × R . Definition 4.22. T ∈ C ∗ ( e X ) Γ , R + is called asymptotically Λ -invariant if ev ∞ ( T )is contained in the image of ϕ . The pullback of C ∗ ( e X ) Γ , R + and C ∗ ( e Y × R ) Λ × R along ev ∞ and ϕ , is called the (Γ , Λ) -equivariant Roe algebra of X and will bedenoted by C ∗ ( e X ) Γ , R + , Λ . Remark 4.23.
By definition, we have a commutative diagram C ∗ ( e X ) Γ , R + , Λ C ∗ ( e Y × R ) Λ × R C ∗ ( e X ) Γ , R + C ∗ ( Y ′ × R ) Γ × R ϕ ev ∞ and elements of C ∗ ( e X ) Γ , R + , Λ are given by pairs ( S, T ) with S ∈ C ∗ ( e X ) Γ , R + , T ∈ C ∗ ( e Y × R ) Λ × R with ev ∞ ( S ) = ϕ ( T ). Definition 4.24.
The (Γ , Λ) -equivariant localisation algebra ((Γ , Λ) -equivariantstructure algebra ) of X is defined to be the pullback of the following diagram C ∗ L, (0) ( e Y × R ) Λ × R C ∗ L, (0) ( e X ) Γ , R + C ∗ L, (0) ( Y ′ × R ) Γ × R ϕ ev ∞ It will be denoted by C ∗ L, (0) ( e X ) Γ , R + , Λ .We obtain an analogue of the Higson-Roe sequence for spaces with cylindricalends: the short exact sequence0 → C ∗ L, ( e X ) Γ , R + , Λ → C ∗ L ( e X ) Γ , R + , Λ → C ∗ ( e X ) Γ , R + , Λ → · · · → K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) → · · · . Let X be an n -dimensional spin manifold with a cylindrical end with base Y .By this we mean that ( X, Y, ι ) is a space with cylindrical end, ι is smoothand X \ ι ( Y × (0 , ∞ )) is a smooth codimension zero submanifold with bound-ary Y . We fix a map η : ( X, Y ∞ := ι ( Y × R + )) → ( B Γ , B Λ) satisfying η ( ι (( y, t ))) = η ( ι (( y, t ∈ R + which gives rise to certain covers They constructed the map between the maximal Roe algebras. In [8] the quotient com-pletion was introduced and it was shown, that one has a similar map between the quotientcompletions of the equivariant algebraic Roe algebras. X and Y , which we will denote as in the previous section. Denote by L ( / S e X ) , L ( / S Y ′ × R ) , L ( / S g Y ∞ ) and L ( / S e Y × R ) the square integrable sections ofthe Cl n -spinor bundles on e X, Y ′ × R , f Y ∞ and e Y × R respectively. The pairs( L ( / S e X ) , L ( / S Y ′ × R )) and ( L ( / S g Y ∞ ) , L ( / S e Y × R )) can be given the structure ofan ( e X, Y ′ , ι ′ ) Cl n -module and an ( f Y ∞ , e Y , e ι ) Cl n -module in the natural way re-spectively. In particular, the families of unitaries on L ( / S Y ′ × R ) and L ( / S e Y × R )needed in the definition of cylindrical Y ′ × R and e Y × R -modules will be givenby the shift of sections in the R -direction and will be denoted by { Q ′ s } and { e Q s } respectively. We will use these modules to construct the relevant C ∗ -algebras inthe following section. As in Section 3.2.1, we obtain classes [ /D e X ] and [ /D e Y × R ]in b K ( C ∗ L ( e X ; Cl n ) Γ ) and b K ( C ∗ L ( e Y × R ; Cl n ) Λ ) respectively. Note that e Y × R is a manifold with cylindrical end with base e Y . In the following we will definea fundamental class for the Dirac operators on X and its cylindrical end inthe K -theory groups of the (Γ , Λ)-equivariant localisation algebra and discussindices and secondary invariants obtained from it. We will need the following
Lemma 5.1.
The following diagrams are commutative S C ∗ ( e X ; Cl n ) Γ , R + C ∗ ( Y ′ × R ; Cl n ) Γ × R ev ∞ S C ∗ ( e Y × R ; Cl n ) Λ × R C ∗ ( Y ′ × R ; Cl n ) Γ × R . ϕ Here
S → C ∗ ( e X ; Cl n ) Γ , R + , S → C ∗ ( Y ′ × R ; Cl n ) Γ × R and S → C ∗ ( e Y × R ; Cl n ) Λ × R denote the functional calculi for /D e X , /D Y ′ × R and /D e Y × R respectively.Proof. First note that the isometry ι ′ allows us to identify the Cl n -spinor bundlesover Y ′ × R + and Y ′∞ , which in turn gives rise to the unitary i ′ : χ Y ′∞ L ( / S e X ) → χ Y × R + L ( / S Y ′ × R ). Let v ∈ L ( / S Y ′ × R ) be compactly supported. For f ∈ S whose Fourier transform is supported in ( − r, r ), it is well known that f ( /D e X )and f ( /D Y ′ × R ) have propagation less than r and depend on the r -local geom-etry in the sense that f ( /D e X ) w and f ( /D Y ′ × R ) v depend only on the Rieman-nian metric in the r -neighbourhood of the supports of w and v respectively.For v ∈ L ( / S Y ′ × R ) with compact support pick s such that Q ′ s v is supportedin Y ′ × [2 r, ∞ ) for all s > s . The previous observation then implies that if ( /D e X ) i ∗ Q ′ s v = f ( /D Y ′ × R ) Q ′ s v for all s > s . Hencelim s Q ′− s if ( /D e X ) i ∗ Q ′ s v = lim s Q ′− s f ( /D Y ′ × R ) Q ′ s v. However, because the Riemannian metric on Y ′ × R is R -invariant, Q ′ s commuteswith the Dirac operator and its functions. This implies that Q ′− s f ( /D Y ′ × R ) Q ′ s v = f ( /D Y ′ × R ) v and shows that for f with compactly supported Fourier transformev ∞ ( f ( /D e X )) = f ( /D Y ′ × R ). The commutativity of the left diagram then fol-lows from the fact that the functions in S with compactly supported Fouriertransform form a dense subset.Now we show the commutativity of the right diagram. First we need torecall one of the main properties of the map ϕ : C ∗ ( e Y × R ; Cl n ) Λ → C ∗ ( Y ′ × R ; Cl n ) Γ . Since all the covers are assumed to be nice one has bijections C ∗ ( e Y × R ; Cl n ) Λ × R ǫ ∼ = C ∗ ( Y × R ; Cl n ) R ǫ and C ∗ ( Y ′ × R ; Cl n ) Γ × R ǫ ∼ = C ∗ ( Y × R ; Cl n ) R ǫ ,19here C ∗ ( Y × R ; Cl n ) R is constructed using L ( / S Y × R ) as the Y × R -module, ǫ is a sufficiently small positive real number, and C ∗ ( · ) · ǫ denotes the set ofelements in the corresponding Roe algebra which have propagation less than ǫ .The bijections are given by pushdowns and lifts of operators on different covers.Furthermore, ϕ makes the diagram C ∗ ( e Y × R ; Cl n ) Λ × R ǫ C ∗ ( Y ′ × R ; Cl n ) Γ × R ǫ C ∗ ( Y × R ; Cl n ) R ǫ . ∼ = ϕ ∼ = commutative. Let f ∈ S have a Fourier transform which is supported in ( − ǫ ′ , ǫ ′ ),with ǫ ′ sufficiently small. The observation that f applied to the different Diracoperators depends only on the ǫ ′ -local geometry and the niceness of covers implythat f ( /D e Y × R ) , f ( /D Y × R ) and f ( /D Y ′ × R ) correspond to each other under thepushdown/lift maps. The commutativity of the latter diagram then implies thatfor f with the above property ϕ ( f ( /D e Y × R )) = f ( /D Y ′ × R ). The commutativity ofthe right diagram in the claim of the lemma then follows from the fact, that the C ∗ -subalgebra of S generated by functions whose Fourier transform is supportedin a fixed interval ( − C, C ) is the whole of S , since it separates points.Lemma 5.1 allows us to make the following Definition 5.2.
The (Γ , Λ)-fundamental class of the X is the class [ /D e X, e Y ] ∈ b K ( C ∗ L ( e X ; Cl n ) Γ , R + , Λ ) ∼ = K n ( C ∗ L ( e X ) Γ , R + , Λ ) defined by ϕ /D f X, e Y : S → C ∗ L ( e X ; Cl n ) Γ , R + , Λ , f ( t ( f ( 1 t /D e X ) , f ( 1 t /D e Y × R )) . The (Γ , Λ)-index of the Dirac operator associated to the map η : ( X, Y ∞ := ι ( Y × R + )) → ( B Γ , B Λ) as above is defined to be the image of [ /D e X, e Y ] underthe map (ev ) ∗ : K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ ( e X ) Γ , R + , Λ ). Suppose that the scalar curvature of the metric g on X is bounded from belowby ǫ . The same then holds for the lifts of g to various covers of X and Y ( ∞ ).This implies that the spectra of the various Dirac operators considered heredo not intersect the interval ( − √ ǫ , √ ǫ ). Let ψ be a homotopy inverse to theinclusion S ( − √ ǫ , √ ǫ ) → S . Definition 5.3.
Let g be as above. The (Γ , Λ)-rho-invariant of g is the class in K ( C ∗ L, ( e X ; Cl n ) Γ , R + , Λ ) ∼ = K n ( C ∗ L ( e X ) Γ , R + , Λ ) defined by the morphism ϕ /D f X, e Y ◦ ψ : S → C ∗ L, ( e X ) Γ , R + , Λ and will be denoted by ρ Γ , Λ ( g ). 20learly, ρ Γ , Λ ( g ) lifts [ D e X, e Y ] and by the exactness of the sequence · · · → K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) → · · · we have the following Proposition 5.4.
If the metric on X has positive scalar curvature then the (Γ , Λ) -index of the Dirac operator vanishes. One can define a notion of concordance for positive scalar curvature metricson manifolds with cylindrical ends. Let g and g ′ be such metrics on X . Theyare called concordant if there exist a positive scalar curvature metric G on X × R and a map j : Y × R × R + → Y ∞ × R which makes ( X × R , Y × R , j )a manifold with cylindrical end and such that G restricted to X × (1 , ∞ ) is g + dt and restricted to X × ( −∞ ,
0) is g ′ + dt . Using the strategy of Zeidler in[11] and by replacing the usual Roe, localisation and structure algebras by their(Γ , Λ)-invariant counterparts one can without much difficulty prove a partitionedmanifold index theorem for secondary invariants for manifolds with cylindricalends and prove the concordance invariance of the (Γ , Λ)-rho-invariant. Howeverwe refrain from discussing this, since it does not entail any novelties.More generally following the approach of [11] we define partial (Γ , Λ)-rho-invariants associated to metrics having positive scalar curvature outside of agiven subset Z of X . Denote by Z ′ and and Z ′′ the preimages of Z and( Z ∩ ι ( Y × { } )) × R under the covering maps e X → X and e Y × R → Y × R respec-tively. Denote by C ∗ ( Z ′ ⊂ e X ) Γ , R + , Λ the C ∗ -subalgebra of C ∗ ( e X ) Γ , R + , Λ consist-ing of elements ( T , T ) with T ∈ C ∗ ( Z ′ ⊂ e X ) Γ and T ∈ C ∗ ( Z ′′ ⊂ e Y × R ) Λ .Denote by C ∗ L,Z ′ ( e X ) Γ , R + , Λ the preimage of C ∗ ( Z ′ ⊂ e X ) Γ , R + , Λ under the eval-uation at 1 map. The justification for the following definition is provided by[7, Lemma 2.3]. Definition 5.5.
Given a metric g on X which is collared at the boundary whosescalar curvature is bounded below by ǫ > Z define the class ρ Γ , Λ Z ( g ) by the morphism ϕ /D f X, e Y ◦ ψ : S → C ∗ L,Z ′ ( e X ) Γ , R + , Λ . Another higher index theoretic notion which has been successfully used toobtain information about the size of the space of positive scalar curvature met-rics on closed manifolds is the higher index difference, which gives rise to amap from the space of positive scalar curvature metrics to the K -theory of thegroup C ∗ -algebra of the manifold. We now show that one can easily definea (Γ , Λ)-index difference of two positive scalar curvature metrics for manifoldswith cylindrical ends. This becomes particularly interesting after we discussthe application of the above machinery to relative higher index theory in thenext section. Let g and g be two metrics on X with scalar curvature boundedbelow by ǫ > G on X × R which restricts to g ⊕ dt and g ⊕ dt on X × [0 , ∞ ) and X × ( −∞ , − X -direction. Definition 5.6.
Let g , g and G be as above. The (Γ , Λ)-index difference of g and g is the image of ρ Γ , Λ X × [0 , ( G ) under the composition K n +1 ( C ∗ L, e X × [0 , ( e X ) Γ , R + , Λ ) (ev ) ∗ −−−−→ K n +1 ( C ∗ ( e X × [0 , ⊂ e X × R ) Γ , R + , Λ )21 K n +1 ( C ∗ ( e X ) Γ , R + , Λ ) , where the last map is induced by projection on e X . It will be denoted byind Γ , Λ ( g , g ). As mentioned above, the relative index map of Chang, Weinberger and Yufor manifolds with boundary takes values in mapping cones of equivariant Roealgebras. Note that given a manifold (
X, Y, ι ) with cylindrical end, X := X \ ι ( Y × (0 , ∞ )) is a manifold with boundary Y . By restriction, we obtain a map η : ( X, Y ) → ( B Γ , B Λ). Let ψ ( L, (0)) : C ∗ ( L, (0)) ( e Y ) Λ → C ∗ ( L, (0)) ( e X ) Γ denote themap introduced in Section 3.3. In the following, we will see that there exists acommutative diagram of exact sequences K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) K ∗ ( SC ψ L, ) K ∗ ( SC ψ L ) K ∗ ( SC ψ ) . Proposition 5.7.
The following is a commutative diagram of short exact se-quences C ∗ ( e Y ⊂ e Y ∞ ) Λ C ∗ ( e Y ∞ ) Λ , R + C ∗ ( e Y × R ) Λ × R C ∗ ( e X ⊂ e X ) Γ C ∗ ( e X ) Γ , R + , Λ C ∗ ( e Y × R ) Λ × R . ϕ ev ∞ ( ϕ, ev ∞ ) ( ϕ, id) Analogous diagrams exist when C ∗ is replaced by C ∗ L and C ∗ L, .Proof. We first show that the first row is exact. It follows immediately fromthe definition of ev ∞ that R ( e Y ⊂ e Y ∞ ) Λ is in its kernel. By continuity we getthat C ∗ ( e Y ⊂ e Y ∞ ) is in the kernel of ev ∞ . Furthermore, [3, Lemma 3.12] impliesthat the kernel of ev ∞ is exactly C ∗ ( e Y ⊂ e Y ∞ ) Λ . It remains to show that ev ∞ is surjective. For T ∈ R ( e Y × R ) Λ × R , the operator χ e Y × R + T χ e Y × R + maps to T under ev ∞ . The surjectivity then follows from the fact that the image ofa homomorphism of C ∗ -algebras is closed. The exactness of the second rowcan be proven using similar arguments. However we note that the exactness inthe middle uses the fact that lim R →∞ dist( ι ( Y ′ × [ R, ∞ )) , e X − Y ′∞ ) = ∞ (seeDefinition4.1). As for the commutativity of the diagram we note that ϕ andev ∞ commute. Proposition 5.8.
The inclusion C C ∗ ( e Y ⊂ g Y ∞ ) Λ → C ∗ ( e X ⊂ e X ) Γ → C C ∗ ( g Y ∞ ) Λ , R + → C ∗ ( e X ) Γ , R + , Λ gives rise to isomorphisms of K -theory groups. Analogous statements hold when C ∗ is replaced by C ∗ L and C ∗ L, . roof. Note that the mapping cone of the identity map on C ∗ ( e Y × R ) Λ × R iscontractible and thus has trivial K -theory. The statement then follows froman application of the five-lemma to the long exact sequence of K -theory groupsassociated to the short exact sequence of mapping cones0 → C C ∗ ( e Y ⊂ g Y ∞ ) Λ → C ∗ ( e X ⊂ e X ) Γ → C ( ϕ, ev ∞ ) → C id → Proposition 5.9.
There is a commutative diagram of long exact sequences K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) K ∗ ( SC ψ L, ) K ∗ ( SC ψ L ) K ∗ ( SC ψ ) , where the vertical maps are given by the compositions K ∗ ( C ∗ ( L, (0)) ( e X ) Γ , R + , Λ ) → K ∗ ( SC C ∗ ( L, (0)) ( g Y ∞ ) Λ , R + → C ∗ ( L, (0)) ( e X ) Γ , R + , Λ ) ∼ = K ∗ ( SC C ∗ ( L, (0)) ( e Y ⊂ g Y ∞ ) Λ → C ∗ ( L, (0)) ( e X ⊂ e X ) Γ ) ∼ = K ∗ ( SC ψ ( L, (0)) ) . Proof.
The diagram in the claim of the proposition is obtained by composingthe diagrams K ∗ ( C ∗ L, ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) K ∗ ( SC ( ϕ L, , ev ∞ L, ) ) K ∗ ( SC ( ϕ L , ev ∞ L ) ) K ∗ ( SC ( ϕ, ev ∞ ) ) , and K ∗ ( SC ( ϕ L, , ev ∞ L, ) ) K ∗ ( SC ( ϕ L , ev ∞ L ) ) K ∗ ( SC ( ϕ, ev ∞ ) ) K ∗ ( SC ψ L, ) K ∗ ( SC ψ L ) K ∗ ( SC ψ ) , ∼ = ∼ = ∼ = where ( ϕ ( L, (0)) , ev ∞ ( L, (0)) ) denotes the map C ∗ ( L, (0)) ( f Y ∞ ) Λ , R + → C ∗ ( L, (0)) ( e X ) Γ , R + , Λ of Proposition 5.7. The commutativity of the first diagram is due to the natu-rality of the mapping cone exact sequence and the commutativity of the seconddiagram is clear.Denote the image of the fundamental class of the Dirac operator on e X underthe composition K ∗ ( C ∗ L ( e X ) Γ ) → K ∗ ( SC C ∗ L ( g Y ∞ ) Λ → C ∗ L ( e X ) Γ ) ∼ = K ∗ ( SC C ∗ L ( e Y ⊂ g Y ∞ ) Λ → C ∗ L ( e X ⊂ e X ) Γ )by [ D e X, e Y ]. 23 emma 5.10. The class [ D e X, e Y ] maps to [ D e X, e Y ] under the map K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) → K ∗ ( SC ψ L ) of Proposition 5.9.Proof. We first note that the commutativity of the diagram K ∗ ( C ∗ L ( f Y ∞ ) Λ , R + ) K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ L ( f Y ∞ ) Λ ) K ∗ ( C ∗ L ( e X ) Γ ) , where the second vertical map is given by the composition of the projectiononto the C ∗ L ( e X ) Γ , R + component followed by the inclusion, implies that of K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( SC C ∗ L ( g Y ∞ ) Λ , R + → C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ L ( e X ) Γ ) K ∗ ( SC C ∗ L ( g Y ∞ ) Λ → C ∗ L ( e X ) Γ ) . Furthermore, the diagram K ∗ ( SC C ∗ L ( e Y ⊂ g Y ∞ ) Λ → C ∗ L ( e X ⊂ e X ) Γ ) K ∗ ( SC C ∗ L ( g Y ∞ ) Λ , R + → C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( SC C ∗ L ( g Y ∞ ) Λ → C ∗ L ( e X ) Γ ) , where all the arrows are isomorphisms, is commutative. The claim then followsfrom the commutativity of the latter two diagrams and the fact that [ D e X, e Y ]lifts the fundamental class of e X Corollary 5.11.
The (Γ , Λ)-index of the Dirac operator associated to (
X, Y, ι )maps to the relative index of the Dirac operator on X under K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) → K ∗ ( SC ψ ) defined in Proposition 5.9.Combining Lemma 5.10 and Proposition 5.4 gives a new (and very natural)proof of the following Proposition 5.12.
The nonvanishing of the relative index of the Dirac operatoron a manifold with boundary is an obstruction to the existence of a positive scalarmetric which is collared at the boundary.
Given a metric g on X which has positive scalar curvature outside X , onecan define a localised coarse index in C ∗ ( e X ) Γ ). In [8] it was shown that thisindex maps to the relative index of X . We quickly recall the construction ofthe localised index and use the machinery developed previously to give a shortproof of the latter statement. Definition 5.13.
Denote by C ∗ L, e X ( e X ) Γ the preimage of C ∗ ( e X ⊂ e X ) Γ underev : C ∗ L ( e X ) Γ → C ∗ ( e X ) Γ . 24uppose that the scalar curvature of the metric restricted to the complementof X is bounded from below by ǫ >
0. The following proposition is well-known.As in [11] one can define a partial ρ -invariant ρ Γ X ( g ) ∈ K n ( C ∗ L, e X ( e X ) Γ ) using themorphism ϕ D f X ◦ ψ : S → C ∗ L, e X ( e X ; Cl n ) Γ . Definition 5.14.
The localised coarse index ind Γ e X ( g ) is the image of ρ Γ X ( g )under (ev ) ∗ : K n ( C ∗ L, e X ( e X ) Γ ) → K n ( C ∗ ( e X ⊂ e X ) Γ ). Remark 5.15.
Note that in the above situation we can also define ρ Γ , Λ X ( g ).Furthermore, we note that the commutativity of the diagram K ∗ ( C ∗ L, e X ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ L, e X ( e X ) Γ ) K ∗ ( C ∗ ( e X ⊂ e X ) Γ ) , (ev ) ∗ and the fact that ρ Γ , Λ X ( g ) is a lift of ρ Γ X ( g ) under the horisontal map implythat ind Γ e X ( g ) is the image of ρ Γ , Λ X ( g ) under the map K ∗ ( C ∗ L, e X ( e X ) Γ , R + , Λ ) → K ∗ ( C ∗ ( e X ⊂ e X ) Γ ).The following lemma is a simple observation Lemma 5.16.
The following diagram is commutative K ∗ ( C ∗ L, e X ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ ( e X ⊂ e X ) Γ ) K ∗ ( SC C ∗ ( e Y ⊂ g Y ∞ ) Λ → C ∗ ( e X ⊂ e X ) Γ ) K ∗ ( C ∗ L ( e X ) Γ , R + , Λ ) K ∗ ( C ∗ ( e X ) Γ , R + , Λ ) K ∗ ( SC C ∗ ( g Y ∞ ) Λ , R + → C ∗ ( e X ) Γ , R + , Λ ) . Suppose X is compact. Then K ∗ ( C ∗ ( e X ⊂ e X ) Γ ) ∼ = K ∗ ( C ∗ (Γ)). Using theprevious remark and lemma we obtain the following corollary, which was one ofthe main statements of [8]. Corollary 5.17.
Suppose X is compact. Then ind Γ e X ( g ) maps to the relativeindex of Chang, Weinberger and Yu under the map K ∗ ( C ∗ (Γ)) → K ∗ ( C ∗ (Γ , Λ)).
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2, 3.20, 5.1, 5.32, 3.20, 5.1, 5.3