Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
PPOSITIVE SCALAR CURVATURE AND ANEQUIVARIANT CALLIAS-TYPE INDEX THEOREM FORPROPER ACTIONS
HAO GUO, PETER HOCHS, AND VARGHESE MATHAI
Abstract.
For a proper action by a locally compact group G on amanifold M with a G -equivariant Spin-structure, we obtain obstruc-tions to the existence of complete G -invariant Riemannian metrics withuniformly positive scalar curvature. We focus on the case where M/G is noncompact. The obstructions follow from a Callias-type index the-orem, and relate to positive scalar curvature near hypersurfaces in M .We also deduce some other applications of this index theorem. If G is aconnected Lie group, then the obstructions to positive scalar curvaturevanish under a mild assumption on the action. In that case, we gener-alise a construction by Lawson and Yau to obtain complete G -invariantRiemannian metrics with uniformly positive scalar curvature, under anequivariant bounded geometry assumption. Contents
1. Introduction 2Results on positive scalar curvature 2A Callias-type index theorem 4Outline of this paper 5Acknowledgements 5Notation and conventions 52. Results on positive scalar curvature 62.1. Obstructions 62.2. Existence for connected Lie groups 83. An index theorem 93.1. The G -Callias-type index 93.2. Hypersurfaces and the index theorem 114. Properties of the G -Callias-type index 124.1. Sobolev modules 124.2. Vanishing 154.3. Homotopy invariance 164.4. A relative index theorem 165. Proof of the G -Callias-type index theorem 185.1. An index on the cylinder 195.2. Attaching a half-cylinder 205.3. Proof of Proposition 5.1 22 a r X i v : . [ m a t h . DG ] F e b HAO GUO, PETER HOCHS, AND VARGHESE MATHAI c -Dirac operators 317.4. Induction 327.5. Callias quantisation commutes with reduction 33References 341. Introduction
Let G be a locally compact group, acting properly on a manifold M .Suppose that M has a G -equivariant Spin-structure. The results in thispaper are about the following question. Question 1.1.
When does M admit a complete, G -invariant Riemannianmetric with uniformly positive scalar curvature? We are mainly interested in the case where
M/G is noncompact.The literature on the non-equivariant case of Question 1.1 is too vastto summarise, but important work where M is noncompact was done byGromov and Lawson [13]. A more refined perspective on the non-equivariantcase is to consider a manifold X , and let M be its universal cover, and G is fundamental group. This allows one to construct obstructions to metricsof positive scalar curvature in terms of G -equivariant index theory on M ,refining index-theoretic obstructions on X . If X is compact, this is the originof the even more refined Rosenberg index [36, 35, 37], in KO -theory of thereal maximal group C ∗ -algebra of G .More generally, if G is a discrete group not necessarily acting freely on M ,then X := M/G is an orbifold, whence Question 1.1 becomes the questionof whether X admits an orbifold metric of positive scalar curvature.We consider the case where G is not necessarily discrete, and does notnecessarily act freely. Results on this case of Question 1.1 in the case where G is an almost-connected Lie group and M/G is compact were obtained in[18, 34]. In this paper, we focus on the case where
M/G is noncompact.
Results on positive scalar curvature.
We first obtain obstructions to G -invariant Riemannian metrics with positive scalar curvature, both in the K -theory of the maximal or reduced group C ∗ -algebra of G , and in termsof numerical topological invariants generalising the ˆ A -genus. If G is a con-nected Lie group, then these obstructions vanish under a mild assumption, OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 3 as shown in [21]. In that case, we construct G -invariant Riemannian met-rics with uniformly positive scalar curvature, under an equivariant boundedgeometry assumption.Our most general obstruction result is the following. Theorem 1.2.
Let H ⊂ M be a G -invariant, cocompact hypersurface withtrivial normal bundle, that partitions M into two open sets. If M admitsa complete, G -invariant Riemannian metric with nonnegative scalar curva-ture, and positive scalar curvature in a neighbourhood of H , then index G ( D H ) = 0 ∈ K ∗ ( C ∗ ( G )) , for a Spin -Dirac operator D H on H . See Theorem 2.1. In the case where M is the universal cover of a manifold X and G is its fundamental group, this becomes Theorem A in [9].Theorem 1.2 implies topological obstructions to G -invariant Riemannianmetrics with positive scalar curvature. Let g ∈ G , and let H g ⊂ H be thefixed point set of g . Let N → H g be its normal bundle in H , and let R N bethe curvature of the Levi–Civita connection restricted to N . The g -localised ˆ A -genus of H is ˆ A g ( H ) := (cid:90) H g c g ˆ A ( H g )det(1 − ge − R N / πi ) / , for a cutoff function c g on X g . If G acts freely, then ˆ A g ( H ) = 0 if g (cid:54) = e ,and ˆ A e ( H ) is the ˆ A -genus of H/G . In general, for example in the orbifoldcase, ˆ A g ( H ) may be nonzero for different g . Corollary 1.3.
Consider the setting of Theorem 1.2. Suppose that either • G is any locally compact group and g = e ; • G is discrete and finitely generated and g is any element; or • G is a connected semisimple Lie group and g is a semisimple element.Then ˆ A g ( H ) is independent of the choice of G -invariant Riemannian metric,and ˆ A g ( H ) = 0 . See Theorem 2.4.Theorem 2 in [21] is a generalisation of Atiyah and Hirzebruch’s vanishingresult [3] to the noncompact case. It states that, if G is connected, and notevery stabiliser of its action on H is maximal compact, then index G ( D H ) =0. This implies that ˆ A g ( H ) = 0 as well. In view of Theorem 1.2 andCorollary 1.3, this makes it a natural question if M admits a G -invariantRiemannian metric with positive scalar curvature if G is a connected Liegroup. The answer, given in the present paper, turns out to be yes under acertain equivariant bounded geometry assumption.Suppose that G is a connected Lie group, and let K < G be a maximalcompact subgroup. Abels’ slice theorem [1] implies that there is a diffeo-morphism M ∼ = G × K N , for a K -invariant submanifold N ⊂ M . Consider HAO GUO, PETER HOCHS, AND VARGHESE MATHAI the infinitesimal action map ϕ : N × k → T N mapping ( y, X ) ∈ N × k to ddt (cid:12)(cid:12) t =0 exp( tX ) y . The action by K on N is said tohave no shrinking orbits with respect to a K -invariant Riemannian metricon N , if the pointwise operator norm of ϕ as a map from k to a tangentspace is uniformly positive outside a neighbourhood of the fixed point set N K . We say that N has K -bounded geometry if it has bounded geometryand no shrinking orbits. Theorem 1.4.
Suppose that K is non-abelian, and that K acts effectivelyon N with compact fixed point set. If there exists a K -invariant Riemann-ian metric on N for which N has K -bounded geometry, then the manifold G × K N admits a G -invariant metric with uniformly positive scalar curva-ture. In the compact case, the Atiyah–Hirzebruch vanishing theorem [3] im-plies that the obstruction ˆ A ( N ) to Riemannian metrics of positive scalarcurvature vanishes if K acts nontrivially on N . Then Lawson and Yau [27]constructed such metrics, under mild conditions. In a similar way, Theorem1.4 complements the vanishing result in [21]. A Callias-type index theorem.
Two effective sources of index-theoreticobstructions to metrics of positive scalar curvature on noncompact manifoldsare coarse index theory and Callias-type index theory. For some resultsinvolving coarse index theory, see for example [38] and the literature on thecoarse Novikov conjecture, in particular [12] for the equivariant setting weare interested in here. We will use Callias-type index theory.Not assuming that M is Spin for now, and letting G be any locally com-pact group, we consider a G -equivariant Dirac-type operator D on a G -equivariant vector bundle S → M . A Callias-type operator is of the form D + Φ, for a G -equivariant endomorphism Φ of S such that D + Φ is uni-formly positive outside a cocompact set. Then this operator has an indexindex G ( D + Φ) ∈ K ∗ ( C ∗ ( G )), constructed in [15]. (See Theorem 4.2 in [17]for a realisation of this index in terms of coarse geometry.) Theorem 1.5 ( G -Callias-type index theorem) . We have index G ( D + Φ) = index G ( D N ) , for a Dirac operator D N on a G -invariant, cocompact hypersurface N ⊂ M . See Theorem 3.4. Versions of this result where G is trivial were proved in[2, 5, 7, 8, 25]. Versions for operators on bundles of modules over operatoralgebras were proved in [6, 9]. Parts of our proof of Theorem 1.5 are basedon a similar strategy as the proof of the index theorem in [9].We deduce Theorem 2.1 from Theorem 1.5. This approach is an equivari-ant generalisation of the obstructions to metrics of positive scalar obtainedin [2, 7, 9]. OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 5 If g ∈ G , then under conditions, there is a subalgebra A ( G ) ⊂ C ∗ ( G ) suchthat K ∗ ( A ( G )) = K ∗ ( C ∗ ( G )), and there is a well-defined trace τ g on A ( G )given by τ g ( f ) = (cid:90) G/Z f ( hgh − ) d ( hZ ) , where Z is the centraliser of g . In various settings, including the three casesin Corollary 1.3, there are index formulas for the number τ g (index G ( D )),see [24, 41, 40]. These index formulas imply that, in the setting of Theorem1.2, τ g (index G ( D H )) = ˆ A g ( H ) . Hence Theorem 1.2 implies Corollary 1.3.Theorem 1.4 is proved via a generalisation of Lawson and Yau’s arguments[27], together with a result from [17] that allows one to induce up metricsof positive scalar curvature from N to M = G × K N .Apart from using Theorem 1.5 to prove Theorem 2.1, we obtain somefurther applications, on the image (Corollary 7.1) and cobordism invariance(Corollary 7.2) of the analytic assembly map; on the Callias-type index ofSpin c -Dirac operators (Corollary 7.3); on induction of Callias-type indicesfrom compact groups to noncompact groups (Corollary 7.4); and on theSpin c -version [31] of the quantisation commutes with reduction problem [14,28, 30, 39] for Spin c -Callias type operators (Corollary 7.5). Outline of this paper.
We state our obstruction and existence results inSection 2: Theorem 2.1, Corollary 2.4 and Theorem 2.9. In Section 3, westate the equivariant Callias-type index theorem, Theorem 3.4. Theorem 3.4is proved in Sections 4 and 5. We then deduce Theorem 2.1 and Corollary2.4 in Subsection 6.1. Theorem 2.9 is proved in Subsection 6.2. In Section7, we obtain some further applications of the Callias-type index theorem,Corollaries 7.1–7.5.
Acknowledgements.
HG was supported in part by funding from the Na-tional Science Foundation under grant no. 1564398. PH thanks Guoliang Yuand Texas A&M University for their hospitality during a research visit. VMwas supported by funding from the Australian Research Council, throughthe Australian Laureate Fellowship FL170100020.
Notation and conventions.
All manifolds, vector bundles, group actionsand other maps between manifolds are implicitly assumed to be smooth.If a Hilbert space H is mentioned, the inner product on that space willbe denoted by ( − , − ) H , and the corresponding norm by (cid:107) · (cid:107) H . Spaces ofcontinuous sections are denoted by Γ; spaces of smooth sections by Γ ∞ .Subscripts c denote compact supports.If G is a group, and H is a subgroup acting on a set S , then we write G × H S for the quotient of G × S by the H -action given by h · ( g, s ) = ( gh − , hs ) , HAO GUO, PETER HOCHS, AND VARGHESE MATHAI for h ∈ G , g ∈ G and s ∈ S . If S is a manifold, G is Lie group and H is a closed subgroup, then this action is proper and free, and G × H S is amanifold.A continuous group action, and also the space acted on, is said to be cocompact if the quotient space is compact.2. Results on positive scalar curvature
In all parts of this paper except Subsections 2.2 and 6.2, which concern theexistence results, G will be be a locally group, acting properly on a manifold M . We do not assume that M/G is compact and are in fact interested mainlyin the case where it is not. The group G may have infinitely many connectedcomponents, and for may for example be an infinite discrete group.2.1. Obstructions.
For a proper, cocompact action by G on a manifold N , a G -equivariant elliptic differential operator D on N has an equivariantindex index G ( D ) ∈ K ∗ ( C ∗ ( G )) defined by the analytic assembly map [4].Here C ∗ ( G ) is the maximal or reduced group C ∗ -algebra of G , and the indextakes values in its even K -theory if D is odd with respect to a grading, andin odd K -theory otherwise.Our most general obstruction result is the following. Theorem 2.1.
Let M be a complete Riemannian Spin -manifold, on whicha locally compact group G acts properly and isometrically. Let H ⊂ M bea G -invariant, cocompact hypersurface with trivial normal bundle, such that M \ H = X ∪ Y for disjoint open subsets X, Y ⊂ M . If the scalar curvatureon M is nonnegative, and positive in a neighbourhood of H , then the Spin -Dirac operator D H on H , acting on sections of the restriction of the spinorbundle on M to H , satisfies index G ( D H ) = 0 ∈ K ∗ ( C ∗ ( G )) . We will deduce this result from an equivariant index theorem for Callias-type operators, Theorem 3.4, which may be of independent interest and hassome other applications as well.
Remark 2.2. If M is the universal cover of a manifold X , and G is thefundamental group of X acting on M in the usual way, then Theorem 2.1reduces to Theorem A in [9], by the Miˇsˇcenko–Fomenko realisation of theanalytic assembly map in that case [29].Theorem 2.1 implies a set of topological obstructions to G -invariant pos-itive scalar curvature metrics on M . Let X be any Riemannian manifold onwhich G acts properly, isometrically and cocompactly. Let g ∈ G , and let X g ⊂ X be its fixed point set. (Properness of the action implies that X g = ∅ if g is not contained in a compact subset of G .) Let N → X g be the normalbundle to X g in X . The connected components of X g are submanifolds of X of possibly different dimensions, so the rank of N may jump between OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 7 these components. In what follows, we implicitly apply all constructions tothe connected components of X g and add the results together.By a cutoff function we will mean a smooth function c : M → [0 ,
1] suchthat supp( c ) has compact intersection with each G -orbit, and for each x ∈ M we have (cid:90) G c ( g − x ) dg = 1 . We will also use cutoff functions for other group actions, which are definedanalogously.Let R N be the curvature of the Levi–Civita connection restricted to N .Let ˆ A ( X g ) be the ˆ A -class of X g . Let Z G ( g ) < G be the centraliser of g . Let c g be a cutoff function for the action by Z G ( g ) on X g . Definition 2.3.
The g -localised ˆ A -genus of X isˆ A g ( X ) := (cid:90) X g c g ˆ A ( X g )det(1 − ge − R N / πi ) / . If g = e , then ˆ A e ( X ) = (cid:90) X c e ˆ A ( X )is the L - ˆ A genus of X used in [41]. If G is discrete and acts properly andfreely on X , then ˆ A e ( X ) = ˆ A ( X/G ). Corollary 2.4.
Suppose that either • G is any locally compact group and g = e ; • G is discrete and finitely generated and g is any element; • G is a connected semisimple Lie group and g is a semisimple element.Let M be a manifold on which G acts properly and that admits a G -equivariant Spin -structure. Let H ⊂ M be a G -invariant, cocompact hypersurface suchthat M \ H = X ∪ Y for disjoint open subsets X, Y ⊂ M . The localised ˆ A -genus ˆ A g ( H ) is independent of the choice of a Riemannian metric. If M admits a complete, G -invariant Riemannian metric whose scalar curvatureis nonnegative, and positive in a neighbourhood of H , then ˆ A g ( H ) = 0 . Theorem 2 in [21] is a generalisation of Atiyah and Hirzebruch’s vanish-ing theorem [3] to actions by noncompact groups. It states that if G is aconnected Lie group, and not all stabilisers of the action by G on H are max-imal compact, then index G ( D H ) = 0. So in this setting, the obstructionsin Theorem 2.1 and Corollary 2.4 vanish. This makes it a natural questionwhether Riemannian metrics as in Theorem 2.1 exist if G is a connected Liegroup. A partial affirmative answer to that question is given in Subsection2.2.This also means that the natural place to look for examples and appli-cations where Theorem 2.1 and Corollary 2.4 yield nontrivial obstructionsis the setting where G has infinitely many connected components. (Thevanishing result generalises directly to the case where G has finitely many HAO GUO, PETER HOCHS, AND VARGHESE MATHAI connected components.) As noted in Remark 2.2, Theorem 2.1 implies The-orem A in [9], in the case where G is the fundamental group of M/G and M is its universal cover. More generally, if G is discrete, then M/G is anorbifold. Then Theorem 2.1 and Corollary 2.4 lead to obstructions to orb-ifold metrics on
M/G with nonnegative scalar curvature, and positive scalarcurvature near the sub-orbifold
N/G . If G acts freely, then ˆ A g ( H ) is zeroif g (cid:54) = e (and ˆ A e ( H ) = ˆ A ( H/G )), but in the orbifold case the localisedˆ A -genera for nontrivial elements g are additional obstructions to positivescalar curvature.2.2. Existence for connected Lie groups.
In this subsection, we supposethat G is a connected Lie group. As pointed out at the end of the previ-ous subsection, for such groups the obstructions to G -invariant Riemannianmetrics with positive scalar curvature in Theorem 2.1 and Corollary 2.4 van-ish under a mild assumption on the action, so it is natural to investigateexistence of such metrics.If G is connected, Abels’ global slice theorem [1] implies we have a dif-feomorphism M ∼ = G × K N , for a K -invariant submanifold N ⊂ M . Ourexistence result, Theorem 2.9, supposes that such a slice N has K -boundedgeometry , a notion introduced in Definition 2.8.Suppose a compact, connected Lie group K acts isometrically on a com-plete Riemannian manifold ( N, g N ). Let b denote a bi-invariant Riemannianmetric on K . For each y ∈ N we have a linear map(2.1) ϕ y : k → T y N defined by ϕ y ( X ) := ddt (cid:12)(cid:12) t =0 exp( tX ) y , for X ∈ k . Define a pointwise normfunction(2.2) (cid:107) ϕ (cid:107) : N → R , y (cid:55)→ (cid:107) ϕ y (cid:107) , where (cid:107) · (cid:107) denotes the linear operator norm with respect to b and g N . Definition 2.5.
We say that the action of K on N has no shrinking orbits if, for any neighbourhood U of the fixed point set N K , there exists a constant C U > y ∈ N \ U we have (cid:107) ϕ ( y ) (cid:107) ≥ C U , where the norm function is taken with respect to the Riemannian metric g .We remark that the condition of the action having no shrinking orbits isindependent of b . Example 2.6.
Suppose that N = R , on which K = SO(2) acts in the nat-ural way. Let ψ ∈ C ∞ ( R ) be positive and rotation-invariant, and considerthe Riemannian metric on R equal to ψ times the Euclidean metric. Thenfor all y ∈ R and X ∈ R ∼ = so (2), ϕ y ( X ) = X (cid:18) −
11 0 (cid:19) y. OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 9 So (cid:107) ϕ (cid:107) ( y ) = ψ ( y ) (cid:107) y (cid:107) , where (cid:107) y (cid:107) is the Euclidean norm of y . Hence theaction has no shrinking orbits if and only if the function y (cid:55)→ ψ ( y ) (cid:107) y (cid:107) has apositive lower bound outside a neighbourhood of ( R ) SO(2) = { } .Now let us define the notion of K -bounded geometry, which is a strength-ening of the standard notion of bounded geometry. Definition 2.7.
A Riemannian manifold has bounded geometry if • its injectivity radius is positive; • for each l ≥ C l > (cid:107)∇ l R (cid:107) ∞ ≤ C l , where R is the Riemann curvature tensor. Definition 2.8.
The action of K on N is said to have K -bounded geometry if it has no shrinking orbits and N has bounded geometry.Our main existence result, proved in Subsection 6.2, is the following. Theorem 2.9.
Let G be a connected Lie group, and K < G a maximalcompact subgroup with non-abelian identity component. Let N be a mani-fold admitting an effective action by K with compact fixed point set. If thereexists a Riemannian metric on N such that the K -action has K -boundedgeometry, then the manifold G × K N admits a G -invariant metric with uni-formly positive scalar curvature. This result may be viewed as a strengthening of the vanishing of theobstructions to G -invariant metrics of positive scalar curvature in Theorem2.1 and Corollary 2.4 in the case of connected Lie groups, by the result in[21], in the same way that Lawson and Yau’s [27] construction of metricsof positive scalar curvature strengthens the vanishing of the ˆ A -genus as inAtiyah and Hirzebruch’s vanishing theorem [3] in the compact case. (Seethe diagram on page 233 of [27].) Remark 2.10.
By Abels’ slice theorem [1], every manifold with a properaction by a connected Lie group G is of the form G × K N in Theorem2.9. The condition that N K is compact is equivalent to the condition thatthe points in G × K N whose stabilisers in G are maximal compact form acocompact set. 3. An index theorem
We will deduce Theorem 2.1, and hence Corollary 2.4, from an equivariantindex theorem for Callias-type operators, Theorem 3.4. This is based onequivariant index theory for such operators with respect to proper actions,developed in [15]. The proof of the index theorem involves several argumentsanalogous to those in [9].3.1.
The G -Callias-type index. From now on, M will be a complete Rie-mannian manifold on which G acts properly and isometrically. Let S → M denote a Z / G -equivariant Clifford module over M , and D an odd-graded Dirac operator on Γ ∞ ( S ), associated to a G -invariant Clifford con-nection on S via the Clifford action by T M on S .Let Φ be an odd, G -equivariant, fibrewise Hermitian vector bundle endo-morphism of S . Definition 3.1.
The endomorphism Φ is admissible for D if • the operator D Φ + Φ D on Γ ∞ ( S ) is a vector bundle endomorphism;and • there are a cocompact subset Z ⊂ M and a constant C > ≥ (cid:107) D Φ + Φ D (cid:107) + C on M \ Z .In this setting the operator D + Φ is called a G -Callias-type operator .In the rest of the paper, we will use the following notation. Let µ denotethe modular function on G . Let C ∗ ( G ) be either the reduced or maximalgroup C ∗ -algebra of G . Let G act on sections of the bundle S by( gs )( x ) := g ( s ( g − x )) , for s a section, g ∈ G and x ∈ M .Equip the space Γ c ( S ) with a right C c ( G )-action defined by(3.2) ( s · b )( x ) := (cid:90) G ( gs )( x ) · b ( g − ) µ ( g ) − / dg and a C c ( G )-valued inner product defined by(3.3) ( s , s ) C ∗ ( G ) ( g ) := µ ( g ) − / ( s , gs ) L ( S ) , for s , s ∈ Γ ∞ c ( S ), b ∈ C c ( G ), g ∈ G , and x ∈ M . Let E be the Hilbert C ∗ ( G )-module completion of Γ ∞ c ( S ) with respect to this structure.The definition of the equivariant index of G -Callias-type operators isbased on the following result, Theorem 4.19 in [15]. Theorem 3.2.
There is a continuous G -invariant cocompactly supportedfunction f on M such that (3.4) F := ( D + Φ) (cid:0) ( D + Φ) + f (cid:1) − / , is a well-defined, adjointable operator on E , such that ( E , F ) is a Kasparov ( C , C ∗ ( G )) -cycle. Its class in KK ( C , C ∗ ( G )) is independent of the function f chosen. For details about the definition of the operator F , we refer to Definition4.11 in [15]. Definition 3.3.
The G -index of the G -Callias-type operator D + Φ is theclass index G ( D + Φ) := [ E , F ] ∈ K ( C ∗ ( G )) = KK ( C , C ∗ ( G ))as in Theorem 3.2. OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 11
Hypersurfaces and the index theorem.
In the setting of the pre-vious subsection, we now suppose that S = S ⊕ S for an ungraded, G -equivariant Clifford module S over M , where the first copy of S is theeven part of S , and the second copy is the odd part. Suppose that(3.5) D = (cid:18) D D (cid:19) for a Dirac operator D on S , and that(3.6) Φ = (cid:18) i Φ − i Φ (cid:19) for a Hermitian endomorphism Φ of S . (The conditions on D Φ + Φ D thenbecome conditions on [ D , Φ ].)Let Z be as in Definition 3.1. Let M − ⊂ M be a G -invariant, cocompactsubset containing Z in its interior, such that N := ∂M − is a smooth sub-manifold of M . Let M + be the closure of the complement of M − , so that N = M − ∩ M + and M = M − ∪ M + . In this and similar settings, we write M = M − ∪ N M + . By (3.1), the restriction of Φ to N is fibrewise invertible. Let S N + → N and S N − → N be its positive and negative eigenbundles. (These arevector bundles, even though eigenbundles for single eigenvalues may notbe.) Clifford multiplication by the unit normal vector field ˆ n to N pointinginto M + , times − i , defines G -invariant gradings on S N ± .Let ∇ S be the Clifford connection on S used to define D . By restrictionand projection, it defines connections ∇ S N ± on S N ± . The Clifford action by T M | N on S | N preserves S N ± by the first condition in Definition 3.1; see alsoRemark 1.2 in [2]. Hence the connections ∇ S N ± define Dirac operators D S N ± on Γ ∞ ( S N ± ). These operators are odd-graded. Because N is cocompact, D S N + has an equivariant indexindex G ( D S N + ) ∈ K ( C ∗ ( G ))defined by the analytic assembly map [4]. Theorem 3.4 ( G -Callias-type index theorem) . We have (3.7) index G ( D + Φ) = index G ( D S N + ) ∈ K ( C ∗ ( G )) . Versions of this result where G is trivial were proved in [2, 5, 7, 8, 25].Versions for operators on bundles of modules over operator algebras areproved in [6, 9].There are various index theorems for the the image of the right handside of (3.7) under traces [24, 41, 40] or pairings with higher cyclic cocycles[23, 33, 34]. Via these results, Theorem 3.4 yields topological expressionsfor the corresponding images of the left hand side of (3.7). The results in[24, 41, 40] will be used to deduce Corollary 2.4 from Theorem 2.1. Properties of the G -Callias-type index To prove Theorem 3.4, we will make use of the properties of the index ofDefinition 3.3 that we describe below.4.1.
Sobolev modules.
We start by recalling the definition of SobolevHilbert C ∗ ( G )-modules from [15]. Let M , G , S and D be as in Subsection3.1. Definition 4.1.
For each nonnegative integer j , define Γ ∞ ,jc ( S ) to be thepre-Hilbert C c ( G )-module whose underlying vector space is Γ ∞ c ( S ), equippedwith the right C c ( G )-action defined by (3.2), and C c ( G )-valued inner prod-uct defined by (cid:104) e , e (cid:105) E j = j (cid:88) k =0 ( D k e , D k e ) C ∗ ( G ) , where e , e ∈ Γ ∞ c ( S ) and ( − , − ) C ∗ ( G ) is as in (3.3). Here we set D equal tothe identity operator. Denote by E j ( S ) = E j the vector space completion ofΓ ∞ ,jc ( S ) with respect to the norm induced by (cid:104)− , −(cid:105) E j , and extend naturallythe C ∞ c ( G )-action to a C ∗ ( G )-action, and (cid:104)− , −(cid:105) E j to a C ∗ ( G )-valued innerproduct on E j , to give it the structure of a Hilbert C ∗ ( G )-module. We call E j the j -th G -Sobolev module with respect to D .The module E defined above Theorem 3.2 equals E . The following versionof the Rellich lemma holds for Sobolev modules (Theorem 3.12 in [15]). Theorem 4.2.
Let f be a continuous G -invariant cocompactly supportedfunction on M . Then multiplication by f defines an element of K ( E s , E t ) whenever s > t . We will state and prove a homotopy invariance result, Proposition 4.9,for the index in Definition 3.3, that will be of use later. A hypothesis inthis result is that a certain vector bundle endomorphism defines adjointableoperators on the Sobolev modules E and E . In order to check this conditionin some geometric situations relevant to us, we will need Propositions 4.3and 4.4 below. Proposition 4.3.
A smooth, G -invariant, uniformly bounded bundle endo-morphism of S defines an element of L ( E ) .Proof. Let Ψ be a smooth, G -invariant, uniformly bounded bundle endo-morphism of S . Since Ψ is uniformly bounded, it defines a bounded oper-ator on L ( S ). Let (cid:107) Ψ (cid:107) denote its operator norm, let c be a cutoff func-tion on M , and let Ψ ∗ be the pointwise adjoint of Ψ. Since the operatorΨ := Ψ ∗ Ψ − (cid:107) Ψ (cid:107) is positive on L ( S ), it has a positive square root Q thatone observes is G -invariant. For a fixed e ∈ Γ ∞ c ( S ), the function g (cid:55)→ ( c Ψ ( ge ) , ge ) L ( S ) = ( √ cQ ( ge ) , √ cQ ( ge )) L ( S ) has compact support in G , by properness of the G -action. Thus the map G → L ( S ) defined by g (cid:55)→ √ cQ ( ge ) has compact support in G . OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 13
It follows that for any unitary representation π of G on a Hilbert space H and h ∈ H , v := (cid:90) G µ ( g ) − / √ cQ ( ge ) ⊗ π ( g ) h dg is a well-defined vector in L ( S ) ⊗ H . By computations similar to those inthe proof of Proposition 5.4 in [15], one sees that (cid:107) v (cid:107) L ( S ) ⊗ H equals (cid:90) G (cid:90) G (cid:10) gc Ψ g − e, e (cid:11) E ( g (cid:48) ) dg · ( π ( g (cid:48) ) h, h ) H dg (cid:48) . Thus, for any unitary representation π of G , π (cid:18)(cid:90) G (cid:10) gc Ψ g − e, e (cid:11) E dg (cid:19) = π ( (cid:104) Ψ e, e (cid:105) E )is a positive operator, where we let f ∈ C c ( G ) act on H by π ( f ) h := (cid:90) G f ( g ) π ( g ) h dg. It follows that the element (cid:104) Ψ e, e (cid:105) E = (cid:10)(cid:0) Ψ ∗ Ψ − (cid:107) Ψ (cid:107) (cid:1) e, e (cid:11) E = (cid:104) Ψ e, Ψ e (cid:105) E − (cid:107) Ψ (cid:107) (cid:104) e, e (cid:105) E . is positive in C ∗ ( G ). Hence Ψ extends to an operator on all of E . Similarly,Ψ ∗ defines an operator on all of E that one checks is the adjoint of Ψ. (cid:3) Proposition 4.4.
Suppose that there are a G -invariant, cocompact subset K ⊂ M and a G -invariant, cocompact hypersurface N ⊂ M such that thereis a G -equivariant isometry M \ K ∼ = N × (0 , ∞ ) , and a G -equivariant vectorbundle isomorphism S | M \ K ∼ = S | N × (0 , ∞ ) . Let Ψ be a G -equivariant vectorbundle endomorphism of S . Suppose that, on M \ K , Ψ and D are constantin the factor (0 , ∞ ) of M \ K ∼ = N × (0 , ∞ ) . Then Ψ defines an element of L ( E ) . The proof uses the next lemma. To state it, let H ( S ) be the completionof Γ ∞ c ( S ) with respect to the inner product( − , − ) H ( S ) = ( − , − ) L ( S ) + ( D − , D − ) L ( S ) . Lemma 4.5.
Let M and S be as in Proposition 4.4. Let Θ be a bounded,positive operator on H ( S ) such that • Θ preserves the subspace Γ ∞ c ( S ) ; • for any e ∈ Γ ∞ c ( S ) , the function a : G → R given by g (cid:55)→ (Θ( ge ) , ge ) H ( S ) has compact support in G .Then (cid:90) G (cid:10) g Θ g − e, e (cid:11) E dg is a positive element of C ∗ ( G ) . Proof.
Let Q be the positive square root of Θ in B ( H ( S )). Since a hascompact support, and (Θ( ge ) , ge ) H ( S ) = ( Q ( ge ) , Q ( ge )) H ( S ) , the map G → H ( S ) defined by g (cid:55)→ Q ( ge ) has compact support in G . As in the proof ofProposition 4.3, one finds that for any unitary representation π of G on aHilbert space H and h ∈ H , v := (cid:90) G µ ( g ) − / Q ( ge ) ⊗ π ( g ) h dg is a well-defined vector in H ( S ) ⊗ H , and that (cid:90) G (cid:90) G (cid:10) gQ g − e, e (cid:11) E ( g (cid:48) ) dg · ( π ( g (cid:48) ) h, h ) H dg (cid:48) = (cid:107) v (cid:107) H ( S ) ⊗ H ≥ . Similarly to the proof of Proposition 4.3, we deduce that (cid:82) G (cid:104) g Θ g − e, e (cid:105) E dg is a positive element of C ∗ ( G ). (cid:3) Proof of Proposition 4.4.
Because of the forms of M and S , there is a canon-ical (up to equivalence) first Sobolev norm (cid:107) · (cid:107) on sections of S that is G -invariant, and invariant under the relevant class of translations in thefactor (0 , ∞ ) of N × (0 , ∞ ). Because Ψ is an order zero differential operatorconstant on the factor (0 , ∞ ), it defines a bounded operator with respect to (cid:107) · (cid:107) . Due to the form of D , the norm on H ( S ) is equivalent to (cid:107) · (cid:107) , andso Ψ defines a bounded operator on H ( S ).Let π N : M \ K ∼ = N × (0 , ∞ ) → N be the natural projection. Let c be acutoff function on M such that c | M \ K = π ∗ N c N for a cutoff function c N on N . Let Ψ ∗ and c ∗ denote the respective adjointsof Ψ and c in B ( H ( S )). Then the operatorΨ := c Ψ ∗ Ψ + Ψ ∗ Ψ c ∗ H ( S ) with norm at most (cid:107) Ψ (cid:107) (cid:107) c (cid:107) , wherethe norms are taken in B ( H ( S )).Let c (cid:48) be a smooth, nonnegative function on M that is identically 1 onthe support of c , and such that c (cid:48) | M \ K = π ∗ N c (cid:48) N for a compactly supported function c (cid:48) N on N . Consider the endomorphismΨ := ( c (cid:48) ) ∗ c (cid:48) (cid:107) Ψ (cid:107) (cid:107) c (cid:107) − Ψ of S . For the same reasons as for Ψ, it definesa bounded operator on H ( S ). Fix e ∈ Γ ∞ c ( S ). Because Ψ is a positivebounded operator on H ( S ), we may apply Lemma 4.5 with Θ = Ψ toconclude that(4.1) (cid:90) G (cid:10)(cid:0) g Ψ g − (cid:1) e, e (cid:11) E dg = (cid:90) G (cid:10)(cid:0) g (( c (cid:48) ) ∗ c (cid:48) ) g − (cid:107) Ψ (cid:107) (cid:107) c (cid:107) (cid:1) e, e (cid:11) E dg −(cid:104) Ψ ∗ Ψ e, e (cid:105) E OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 15 is positive in C ∗ ( G ). Define b : M × G → R by b ( x, g ) := b x ( g ) := c (cid:48) ( g − x ) . By construction of c (cid:48) , the quantities C ( x ) := (cid:90) G c (cid:48) ( g − x ) dg, C ( x ) := (cid:107) dc (cid:48) (cid:107) ∞ · vol (supp G ( b x ))are bounded as functions of x ∈ M (see also Remark 4.6 below). A directcalculation shows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) G (cid:10)(cid:0) g (( c (cid:48) ) ∗ c (cid:48) ) g − (cid:107) Ψ (cid:107) (cid:107) c (cid:107) (cid:1) e, e (cid:11) E dg (cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( G ) = (cid:107) Ψ (cid:107) (cid:107) c (cid:107) (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) G (cid:10) c (cid:48) g − e, c (cid:48) g − e (cid:11) E dg (cid:13)(cid:13)(cid:13)(cid:13) C ∗ ( G ) ≤ (cid:107) Ψ (cid:107) (cid:107) c (cid:107) (cid:0) (cid:107) C (cid:107) ∞ (cid:107) e (cid:107) E + (cid:107) C (cid:107) ∞ (cid:107) e (cid:107) E (cid:1) ≤ C (cid:107) Ψ (cid:107) (cid:107) e (cid:107) E , for some constant C . Together with positivity of (4.1), this implies that (cid:107) Ψ e (cid:107) E = (cid:107)(cid:104) e, Ψ ∗ Ψ e (cid:105) E (cid:107) C ∗ ( G ) ≤ C (cid:107) Ψ (cid:107) (cid:107) e (cid:107) E , so that Ψ extends to an operator on all of E . Similarly, the H ( S )-adjointΨ ∗ defines an operator on E that one checks is the adjoint of Ψ. (cid:3) Remark 4.6.
As can be seen from the proof, the conclusion of Proposition4.4 holds more broadly for any M on which the functions C and C on M are bounded.4.2. Vanishing.
Two cases where the index of Definition 3.3 vanishes arestraightforward to prove, but we state them here because they will be usedin various places.
Lemma 4.7. If (3.1) holds on all of M , then index G ( D + Φ) = 0 .Proof. In this setting, the operator F in (3.4) is invertible. This impliesthat the KK -cycle ( E , F ) is operator homotopic to the degenerate cycle( E , F ( F ∗ F ) − / ). (cid:3) Lemma 4.8. If M/G is compact, and D and Φ are of the forms (3.5) and (3.6) , then index G ( D ) = 0 .Proof. In this setting, Φ is bounded, and the cycle ( E , F ) is operator homo-topic to (cid:16) E , D √ D +1 (cid:17) .In general, let A be a C ∗ -algebra, let E be a Hilbert A -module, and set E := E ⊕ E . and an adjointable operator F on E such that ( E , F ) is aKasparov ( C , A )-cycle, and F is of the form(4.2) F = (cid:18) F F (cid:19) , for a (necessarily self-adjoint) F ∈ L ( E ). Then [ E , F ] = 0 ∈ K ( A ).Because D √ D +1 is of the form (4.2), the claim follows. (cid:3) Homotopy invariance.
The index of Definition 3.3 has a homotopyinvariance property analogous to Proposition 4.1 in [9]. This homotopyinvariance applies in a more general setting than Callias-type operators.Let P be an odd, G -equivariant Dirac-type operator on a Z / S , and let Ψ be an odd, smooth G -equivariant, uniformlybounded Hermitian vector bundle endomorphism of S . Fix t < t ∈ R . For t ∈ [ t , t ], consider the operator P t := P + t Ψ. Proposition 4.9 (Homotopy invariance) . Suppose that (1) for j = 0 , , the endomorphism Ψ defines an adjointable operator onthe Sobolev module E j of Definition 4.1; (2) there is a nonnegative, G -invariant, cocompactly supported function f ∈ C ∞ ( M ) such that for all t ∈ [ t , t ] , the operator P t + f : E →E is invertible, with inverse in L ( E , E ) .Then index G ( P t ) ∈ K ( C ∗ ( G )) is independent of t ∈ [ t , t ] .Proof. The proof proceeds identically to the proof of Proposition 4.1 in [9],with the exceptions that Theorem 4.2 should be substituted for Lemma 4.2in [9], and Lemmas 4.4 and 4.6(a) in [15] should be substituted for Lemmas1.4 and 1.5 in [7], respectively. (cid:3)
Corollary 4.10. If D + Φ is a G -Callias-type operator on S → M , and Ψ is a G -equivariant, odd vector bundle endomorphism of S that equals zerooutside a cocompact set, then index G ( D + Φ) = index G ( D + Φ + Ψ) .Proof. We set P = D +Φ and apply Proposition 4.9. Since Ψ is cocompactlysupported, the first condition in Proposition 4.9 holds by Proposition 3.5 in[15]. For the same reason, Φ + t Ψ is a Callias-type potential for all t ∈ R ,so by Theorem 5.6 in [15], the second condition in Proposition 4.9 holds for t ∈ [0 , f may depend on t . But since Ψ iszero outside a cocompact set, we can choose f independent of t . The claimthen follows from Proposition 4.9. (cid:3) Remark 4.11.
In Proposition 4.9, it is not assumed that Ψ is a Callias-typepotential in the sense of Definition 3.1. We will use Proposition 4.9 in thisgreater generality in the proof of Lemma 5.2.
Remark 4.12.
Corollary 4.10 can be used to give an alternative proof ofLemma 4.7: this corollary implies that in the setting of that lemma,index G ( D + Φ) = index G ( D − Φ) = index G (cid:0) ( D + Φ) ∗ (cid:1) = − index G ( D + Φ) . See also Corollary 4.9 in [9].4.4.
A relative index theorem.
We will use an analogue of Bunke’s rel-ative index theorem, Theorem 1.2 in [7]. For j = 1 ,
2, let M j , S j , D j andΦ j , respectively, be as M , S , D and Φ were before. Suppose that there area G -invariant hypersurface N j ⊂ M j and a G -invariant tubular neighbour-hood U j of N j , and that there is a G -equivariant isometry ψ : U → U suchthat OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 17 • ψ ( N ) = N ; • ψ ∗ ( S | U ) ∼ = S | U ; • ψ ∗ ( ∇ S | U ) = ∇ S | U , where ∇ S j is the Clifford connection used todefine D j ; and • Φ | U corresponds to Φ | U via ψ .Suppose that M j = X j ∪ N j Y j for closed, G -invariant subsets X j , Y j ⊂ M j .We identify N and N via ψ and write N for this manifold when we do notwant to distinguish between N and N . Using the map ϕ , form M := X ∪ N Y ; M := X ∪ N Y . For j = 3 ,
4, let S j , D j and Φ j be obtained from the corresponding data on M and M by cutting and gluing along U ∼ = U via ψ . Theorem 4.13.
In the above situation, index G ( D + Φ ) + index G ( D + Φ ) = index G ( D + Φ ) + index G ( D + Φ ) . Proof.
This proof is an adaptation of the proof of Theorem 1.14 in [7], withsome results from [15] added. For j = 1 , , ,
4, let E j and F j be as E and F above and in Theorem 3.2, for the data indexed by j . Using superscriptsop to denote opposite gradings, we write E := E ⊕ E ⊕ E op3 ⊕ E op4 and F := F ⊕ F ⊕ F ⊕ F . We will show that(4.3) [ E , F ] = 0 ∈ KK ( C , C ∗ ( G )) , which is equivalent to the theorem.For j = 1 ,
2, let χ X j , χ Y j ∈ C ∞ ( M j ) be real-valued functions such that • supp( χ X j ) ⊂ X j ∪ U j and supp( χ Y j ) ⊂ Y j ∪ U j ; • ψ ∗ ( χ X | U ) = χ X | U and ψ ∗ ( χ Y | U ) = χ Y | U ; and • χ X j + χ Y j = 1.We view pointwise multiplication by these functions as operators χ X : E → E ; χ X : E → E ; χ Y : E → E ; χ Y : E → E . (4.4)The adjoints of these operators map in the opposite directions, and are alsogiven by pointwise multiplication by the respective functions. Using thesemultiplication operators, and the grading operator γ , we form the operator X := γ − χ ∗ X χ ∗ X − χ ∗ Y χ ∗ Y χ X χ Y χ X − χ Y on E . Then X is an odd, self-adjoint, adjointable operator on E such that X = 1. As such, it generates a Clifford algebra Cl.We claim that XF + F X is a compact operator. This is based on theRellich lemma for Hilbert C ∗ ( G )-modules, Theorem 4.2. Let χ be one ofthe functions χ X j or χ Y j , viewed as an operator from E k to E l as in (4.4).Let f j ∈ C ∞ ( M ) be as in Theorem 3.2, for the operator D j + Φ j . For j = 1 , , , λ ∈ R , the operator ( D j + f j + λ ) on E j is invertible byLemma 4.6 in [15], and we denote its inverse by R j ( λ ).Then, as in the proof of Theorem 1.14 in [7], and using Proposition 4.12in [15], we find that the operator(4.5) χ ∗ ◦ F l − F k ◦ χ ∗ : E l → E k equals(4.6) 2 π (cid:90) R (cid:16) − grad( χ ) R l ( λ ) + D k R k ( λ ) grad( χ ) R l ( λ )+ D k R k ( λ ) grad( f k ) R k ( λ ) grad( χ ) R l ( λ ) + D k R k ( λ ) grad( χ ) D l R l ( λ ) (cid:17) dλ. Theorem 4.2, together with Lemma 4.6(a) in [15], implies that for all cocom-pactly supported continuous functions ϕ on M j , the compositions ϕD nj R j ( λ ), D nj ϕR j ( λ ) and D nj R j ( λ ) ϕ are compact operators on E j if n = 0 ,
1, and ad-jointable operators if n = 1. So all the terms in the integrand in (4.6) arecompact operators. By Lemmas 4.6 and 4.8 in [15], the norm of the in-tegrand in (4.6) is bounded by a ( b + λ ) − for constants a, b >
0. So theintegral converges in the operator norm on L ( E ), and we conclude that (4.5)is a compact operator on E . This implies that XF + F X is a compactoperator.Because X generates Cl and X anticommutes with F modulo compacts,the pair ( E , F ) is a Kasparov (Cl , C ∗ ( G ))-cycle. Its class in KK (Cl , C ∗ ( G ))is mapped to the left-hand aside of (4.3) by the pullback along the inclusionmap C (cid:44) → Cl. That map is zero by Lemma 1.15 in [7], so (4.3) follows. (cid:3)
Theorem 4.13 implies the following version of Proposition 5.9 in [9].
Corollary 4.14.
In the setting of Theorem 4.13, suppose that for j = 1 , ,the set X j is cocompact, and contains a set Z j for Φ j as in Definition 3.1.Then index G ( D + Φ ) = index G ( D + Φ ) . Proof.
This fact can be deduced from Theorem 4.13 in exactly the same wayProposition 5.9 in [9] is deduced from Theorem 5.7 in [9]. Compared to thatproof in [9], references to Corollaries 3.4 and 4.9 and Theorem 5.7 in thatpaper should be replaced by references to Lemmas 4.7 and 4.8 and Theorem4.13, respectively, in the present paper. (cid:3)
The crucial assumption in Corollary 4.14 is that all data near N can beidentified with the corresponding data near N .5. Proof of the G -Callias-type index theorem The first and most important step in the proof of Theorem 3.4 is Propo-sition 5.1, which states that index G ( D + Φ) equals the index of a G -Callias-type operator on the manifold N × R , which we will call the cylinder on N . OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 19
Figure 1.
The cylinder N × R See Figure 1. Such an approach is taken in proofs of various other indextheorems for Callias-type operators; see for example [2, 6, 7, 9].In this section, we consider the setting of Subsection 3.2. In particular, D and Φ are assumed to be of the forms (3.5) and (3.6).5.1. An index on the cylinder.
Let S N × R ± → N × R be the pullbacksof the bundles of S N ± → N defined in Subsection 3.2 along the projection N × R → N . They are Clifford modules, with Clifford actionsˆ c ( v, t ) = c ( v + t ˆ n ) , for v ∈ T N and t ∈ R , where c is the Clifford action by T M on S (whichpreserves S N + as pointed out in Subsection 3.2), and ˆ n is the normal vectorfield to N in the direction of M + . Let D S N × R + be the Dirac operator onΓ ∞ ( S N × R + ) defined by this Clifford action, and the pullback to N × R of therestriction to N of the Clifford connection ∇ S N + used to define D S N + .Let χ ∈ C ∞ ( R ) be an odd function such that χ ( t ) = t for all t ≥
2. Wealso denote its pullback to N × R by χ . Then pointwise multiplication by χ is an admissible endomorphism for D S N × R + . Whenever a Dirac operatorwith a subscript 0 is given, we will remove that subscript to denote the cor-responding Dirac operator on two copies of the Clifford module in question,as in (3.5). In the current setting, this gives us the Dirac operator D S N × R + = D S N × R + D S N × R + on Γ ∞ ( S N × R + ⊕ S N × R + ). We also consider the admissible endomorphism χ N × R = (cid:18) iχ − iχ (cid:19) of S N × R + ⊕ S N × R + . Figure 2.
The manifold M Proposition 5.1.
We have (5.1) index G ( D + Φ) = index G ( D S N × R + + χ N × R ) . The proof of Proposition 5.1 that we give below is an analogue of theproof of Theorem 5.4 in [9]. We give this proof in Subsections 5.2 and 5.3,referring to [9] for details in some places, and using results from [15] andfrom Section 4.5.2.
Attaching a half-cylinder.
Let S N × R → N × R be the pullback of S | N → N . We choose U small enough so that S | U ∼ = S N × R | U .Because the sets X j are cocompact in Corollary 4.14, we initially com-pare the left-hand side of (5.1) to an index on a manifold where only M + isreplaced by a half-cylinder N × [1 , ∞ ). To be more precise, index G ( D + Φ)is invariant under changes in the Riemannian metric on cocompact setsbecause the Kasparov ( C , C ∗ ( G ))-cycles corresponding to two G -invariantRiemannian metrics differing only a cocompact set are homotopic by con-vexity of the space of G -invariant Riemannian metrics. We choose a metricsuch that there is a neighbourhood U of N that is isometric to N × (1 / , / D + Φ does not change if we change Φ in a cocompact set. So we may assume that Φ is constant in the directionnormal to N inside U ; i.e. for all n ∈ N and t ∈ (1 / , / ( n, t ) = Φ N ( n ),for an endomorphism Φ N of S | N . We further choose U such that a set Z as in Definition 3.1 is contained in M − \ U .Let ∇ S N × R be the pullback of ∇ S | N to a connection on S N × R . Wechoose the Clifford connection ∇ S to define D so that on U , it equals therestriction of ∇ S N × R to N × (1 / , / N , we can form the Riemannian manifold M C := M − ∪ N [1 , ∞ ) (see Figure 3), and define the Clifford module S C → M C such that it equals S on M − and S N × R on N × (1 / , ∞ ). Let ∇ S C be theClifford connection on S C corresponding to ∇ S on M − and to ∇ S N × R on N × (1 / , ∞ ). Let D C be the resulting Dirac operator. OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 21
Figure 3.
The manifold M C We define an endomorphism Φ C of S C that is equal to Φ on M − and tothe pullback of Φ N on N × (1 / , ∞ ). Recall that by removing the subscript0 from D C we refer to the construction (3.5). Similarly, when we remove thesubscript 0 from Φ C , we will be referring to the endomorphism Φ C definedby Φ C as in (3.6). Then Corollary 4.14 immediately implies that(5.2) index G ( D + Φ) = index G ( D C + Φ C ) . The connection ∇ S N × R , and therefore the corresponding Dirac operator,does not preserve the decomposition S N × R = S N × R + ⊕ S N × R − . With respectto this decomposition, that Dirac operator has the form(5.3) D S N × R + AB D S N × R − , for vector bundle homomorphisms A : S N × R − → S N × R + and B : S N × R + → S N × R − . (See Section 5.16 in [6], or use the fact that the difference of twoconnections is an endomorphism-valued one-form.) The Dirac operator D C equals this operator on N × (1 / , ∞ ). Let ∇ S N × R ± be the pullback of ∇ S N ± toa connection on S N × R ± . Consider a Clifford connection ∇ S C on S C that isequal to the direct sum of ∇ S N × R + and ∇ S N × R − on N × (1 / , ∞ ) and to ∇ S on M − \ U . Then the corresponding Dirac operator ˜ D C is equal to D S N × R + D S N × R − on N × (1 / , ∞ ) and to D on M − \ U . Lemma 5.2.
There exists a λ ≥ such that λ Φ C is admissible for ˜ D C . Forsuch λ , (5.4) index G ( D C + Φ C ) = index G ( ˜ D C + λ Φ C ) . Proof.
Existence of λ with the desired property can be established as in theproof of Lemma 5.13 in [9]. The equality (5.4) can be proved via a linear ho-motopy; again see the proof of Lemma 5.13 in [9] for details, where referencesto Proposition 4.1 in [9] should be replaced by references to Proposition 4.9 in the current paper, and one uses Propositions 4.3 and 4.4 to check thatthe first condition in Proposition 4.9 is satisfied.To be explicit, an application of Proposition 4.9 shows thatindex G ( D C + λ Φ C ) = index G ( ˜ D C + λ Φ C ) . This follows from Proposition 4.9 by setting P = D C + λ Φ C , Ψ = (cid:18) AB , (cid:19) ,with A and B as in (5.3), t = − t = 0. Note that although Ψ is not aCallias-type potential here, Proposition 4.9 still applies, since Ψ defines anelement of L ( E ) by Proposition 4.3 and an element of L ( E ) by Proposition4.4. (The conditions of Proposition 4.4 hold because of the forms of D C ,Φ C and Ψ.) A more straightforward application of Proposition 4.9 yieldsindex G ( D C + Φ C ) = index G ( D C + λ Φ C ). (cid:3) Let χ ∈ C ∞ ( R ) be an odd function such that χ ( t ) = − / ≤ t ≤ / / ≤ t ≤ / t if t ≥ . (The property that χ is unbounded in a way that makes it proper is onlyused in the proof of Lemma 5.6; see Lemma 5.5.) Such functions form asubset of the set of functions χ in Subsection 5.1, but Proposition 5.1 forgeneral χ follows from the case for this class of functions, because the indexon the right-hand side of (5.1) does not change if we modify χ in a cocompactset. (And at any rate, to prove Theorem 3.4, we only need Proposition 5.1to hold for one such function χ .)Let γ N be the grading operator on S | N that equals ± S N ± . Let γ N × R be its pullback to S N × R . Let Φ χ be the endomorphism of S C equalto χγ N × R on N × (1 / , ∞ ) and equal to zero on the rest of M C . Lemma 5.3.
We have index G ( ˜ D C + λ Φ C ) = index G ( ˜ D C + Φ χ ) . Proof.
This can be proved via a linear homotopy between λ Φ C and Φ χ . Thedetails are precisely as in the proof of Lemma 5.15 in [9], with referencesto Propositions 4.1 and 5.9 in [9] replaced by references to Proposition 4.9(combined with Propositions 4.3 and 4.4) and Corollary 4.14, respectively,in the present paper. (cid:3) Proof of Proposition 5.1.
Let M −− be the manifold M − with reversedorientation. Form the manifold M − C := ( N × ( −∞ , ∪ N M −− . See Figure 4. (The notation is motivated by the fact that M C with reversedorientation is naturally equal to ( N × ( −∞ , − ∪ N M −− , which can beidentified with M − C via a shift over a distance 2.) OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 23
Figure 4.
The manifold M − C Let S − → M −− be equal to the vector bundle S | M − , but with the oppositeClifford action (where v ∈ T M −− acts as c ( − v )). Let S C, − → M − C be theClifford module that is equal to S N × R on N × ( −∞ , /
4] and to S − on M −− .From the Clifford connections ∇ S N × R ± on S N × R ± and a Clifford connection ∇ S − on S − , construct a Clifford connection ∇ S C, − on S C, − by ∇ S C, − := (cid:40) ∇ S N × R + ⊕ ∇ S N × R − on N × ( −∞ , / ∇ S − on M −− . Using this connection, we obtain the Dirac operator D C, − on Γ ∞ ( S C, − ).Then D C, − equals ˜ D C on N × (1 / , / χ equals 1 on (3 / , / χ and (cid:18) χ − (cid:19) are equal to γ N × R . (Here we use 2 × S N × R = S N × R + ⊕ S N × R − .) So we can define theendomorphism Φ C, − of S C, − by setting it equal to Φ χ on ( N × (3 / , ∪ N M −− and equal to(5.5) (cid:18) χ − (cid:19) on N × ( −∞ , / S C, − = S N × R = S N × R + ⊕ S N × R − . Lemma 5.4.
We have index G ( D C, − + Φ C, − ) = 0 . Proof.
Because χ = − −∞ , − / C, − of S C, − by setting it equal to Φ C, − on N × ( −∞ , − /
4) (where it equals(5.5)) and equal to − N × ( − , ∪ N M −− . For that endomorphism,the estimate (3.1) holds on all of M . Therefore by Lemma 4.7,index G ( D C, − + ˜Φ C, − ) = 0 . The claim now follows from Corollary 4.10. (cid:3)
Proof of Proposition 5.1.
Consider the cylinder N × R as in Figure 1. Thedata ( M C , S C , Φ χ ) and ( M − C , S C, − , Φ C, − ) coincide in a neighbourhood of Figure 5.
The manifold M − ∪ N M −− N × { } . By cutting along N × { } and gluing, we obtain the correspondingdata ( N × R , S N × R , Φ N × R ) and ( M − ∪ N M −− , S M − ∪ N M −− , Φ M − ∪ N M −− ). SeeFigure 5. To be explicit,(5.6) Φ N × R = (cid:18) χ − (cid:19) on S N × R + ⊕ S N × R − .Theorem 4.13 implies thatindex G ( ˜ D C + Φ χ ) + index G ( D C, − + Φ C, − )= index G ( D N × R + Φ N × R ) + index G ( D M − ∪ N M −− + Φ M − ∪ N M −− ) . By Lemmas 4.8 and 5.4, this implies thatindex G ( ˜ D C + Φ χ ) = index G ( D N × R + Φ N × R ) . The connection ˜ ∇ S N × R on S N × R obtained from cutting and gluing theconnections ∇ S C and ∇ S C, − is the direct sum connection ∇ S N × R + ⊕ ∇ S N × R + .So the corresponding Dirac operator D S N × R equals D S N × R = D S N × R + D S N × R + . By the explicit form (5.6) of Φ N × R , the operator D S N × R ± i Φ N × R on Γ ∞ ( S N × R )is the direct sum of the operators D S N × R + ± iχ on Γ ∞ ( S N × R + ) and D S N × R − ∓ i on Γ ∞ ( S N × R − ). Soindex G ( D N × R +Φ N × R ) = index G ( D S N × R + + χ N × R )+index G (cid:18) D S N × R − + (cid:18) − ii (cid:19)(cid:19) . Lemma 4.7 then implies that the second term on the right-hand side is zero,whence index G ( ˜ D C + Φ χ ) = index G ( D S N × R + + χ N × R ) . OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 25
The claim now follows from (5.2) in conjunction with Lemmas 5.2 and5.3. As pointed out above Lemma 5.3, the case for the class of functions χ we have used implies the case of the more general functions χ allowed inProposition 5.1. (cid:3) Proof of Theorem 3.4.
Let E N × R be the Hilbert C ∗ ( G )-module con-structed from Γ c ( S N × R + ) as in Subsection 3.1. We write D χ := D S N × R + + χ N × R for brevity. Lemma 5.5.
For all a > the operator ( D χ + a ) − on E N × R is compact.Proof. This is (a special case of) an analogue of Theorem 2.40 in [11]. Theproof proceeds in the same way, with the difference that the operator D χ canonly be bounded below by a function h that is G -proper , in the sense thatthe inverse image of a compact set is cocompact instead of compact. Onechooses the bump functions g n in the proof of Theorem 2.40 in [11] to be G -invariant and cocompactly supported. Theorem 4.2, and Lemma 4.6(a)in [15], imply that g n ( D χ + a ) − is a compact operator on E N × R . And as inthe proof of Theorem 2.40 in [11], one shows that g n ( D χ + a ) − convergesto ( D χ + a ) − in the operator norm on L ( E N × R ). (cid:3) We will use an analogue of Theorem 6.6 in [9].
Lemma 5.6.
The operator (5.7) D χ (cid:0) D χ + f (cid:1) − / − D χ (cid:0) D χ + 1 (cid:1) − / lies in K ( E N × R ) .Proof. By Proposition 4.12 in [15], and as in (6.6) in [9], the operator (5.7)equals(5.8) 2 π (cid:90) ∞ D χ ( D χ + f + λ ) − ( f − D χ + 1 + λ ) − dλ. The operator ( D χ + 1 + λ ) − is compact by Lemma 5.5. Lemmas 4.6 and4.8 in [15] imply that the integrand in (5.8) is bounded by a ( b + λ ) − forcertain a, b >
0, so the integral converges in operator norm. It thereforedefines a compact operator. (cid:3)
Theorem 3.4 follows from Proposition 5.1 and the following fact.
Proposition 5.7.
In the setting of Proposition 5.1, index G ( D χ ) = index G ( D S N + ) . Proof.
Let E N be the Hilbert C ∗ ( G )-module constructed from Γ c ( S N + ) as inSubsection 3.1. Then E N × R = E N ⊗ L ( R ). So Lemma 5.6 implies thatindex G ( D χ ) is represented by the unbounded Kasparov cycle(5.9) (cid:0) E N ⊗ L ( R ) ⊗ C , D χ (cid:1) . The Callias-type operator D χ onΓ ∞ ( S N × R + ⊕ S N × R + ) = Γ ∞ ( S N + ) ⊗ C ∞ ( R ) ⊗ C is equal to(5.10) (cid:32) D S N + D S N + (cid:33) ⊗ C ∞ ( R ) + γ S N + ⊗ (cid:18) i ddt i ddt (cid:19) +1 Γ ∞ c ( S N + ) ⊗ (cid:18) − iχiχ (cid:19) , where γ S N + is the grading operator on S N + , equal to − i times Clifford multi-plication by the unit normal vector field ˆ n on N pointing into M + (so that γ S N + ⊗ i ddt = c (ˆ n ) ⊗ ddt ). Let E ± N be the even and odd-graded parts of E N ,and let D S N + ± be the restriction of D S N + to even and odd-graded sections of S N + , respectively. With respect to the decomposition(5.11) E N ⊗ L ( R ) ⊗ C = ( E + N ⊗ L ( R )) ⊕ ( E − N ⊗ L ( R )) ⊕ ( E + N ⊗ L ( R )) ⊕ ( E − N ⊗ L ( R )) , the operator (5.10) equals(5.12) D χ = E + N ⊗ ( i ddt − iχ ) D S N + − ⊗ L ( R ) D S N + + ⊗ L ( R ) E − N ⊗ ( − i ddt − iχ )1 E + N ⊗ ( i ddt + iχ ) D S N + − ⊗ L ( R ) D S N + + ⊗ L ( R ) E − N ⊗ ( − i ddt + iχ ) 0 0 The kernel of i ddt ± iχ in C ∞ ( R ) is one-dimensional, and spanned by thefunction f ± ( t ) = e ∓ (cid:82) t χ ( u ) du . By the properties of χ , f + ∈ L ( R ), whereas f − (cid:54)∈ L ( R ). It follows that i ddt − iχ is invertible on the appropriate domain, while i ddt + iχ is zero on C f + and invertible on f ⊥ + .Consider the submodules E := ( E + N ⊗ C f + ) ⊕ ⊕ ⊕ ( E − N ⊗ C f + ); E := E ⊥ = ( E + N ⊗ f ⊥ + ) ⊕ ( E − N ⊗ L ( R )) ⊕ ( E + N ⊗ L ( R )) ⊕ ( E − N ⊗ f ⊥ + )of (5.11). These are preserved by the operator D χ . (For E , this is immediatefrom (5.12); for E , this follows from the facts that D χ is symmetric andpreserves E .) We find that the cycle (5.9) decomposes as(5.13) E , D S N + − ⊗ C f + D S N + + ⊗ C f + ⊕ ( E , D χ | E ) . The operator D χ | E is essentially self-adjoint on the initial domain ofcompactly supported smooth sections by Proposition 5.5 in [15], and its OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 27 square has a positive lower bound in the Hilbert C ∗ ( G )-module sense. Thusits self-adjoint closure is invertible, so that the second term in (5.13) ishomotopic to a degenerate cycle. The first term represents index G ( D S N + ). (cid:3) Remark 5.8.
A similar argument in the case where G is trivial is hintedat below Lemma 4.1 in [25].6. Proofs of results on positive scalar curvature
Obstruction results.
We now deduce Theorem 2.1 from Theorem3.4, and Corollary 2.4 from Theorem 2.1 and index theorems in [24, 40, 41].
Proof of Theorem 2.1.
This proof is an adaptation of the proof of Theorem2.1 in [2].First suppose that M is odd-dimensional. Let κ denote scalar curvature.Let K ⊂ X be a cocompact subset of M such that N ⊂ K , κ > K , and the distance from X \ K to Y is positive. Let χ ∈ C ∞ ( M ) G be afunction such that χ ( x ) = 1 for all x ∈ Y and χ ( x ) = − x ∈ X \ K .Consider the operator D as in (3.5), where D is the Spin-Dirac operator on M , and the admissible endomorphism Φ as in (3.6), where Φ is pointwisemultiplication by χ . Then the set M − in Subsection 3.2 can be chosen tobe cocompact as required, and so that N = N − ∪ H , where f | N − = −
1. Inthis setting,(6.1) S N + = S | H , where S → M is the spinor bundle. (This is consistent with Corollary 7.3.)For any λ ∈ R , Lichnerowicz’ formula implies that( D ± iλχ ) = D ± iλc ( dχ ) + λ χ ≥ κ/ − λ (cid:107) dχ (cid:107) + λ χ . On M \ K , the function on the right-hand side equals κ/ λ ≥ λ . Since κ is G -invariant, and positive on the cocompact set K , it has a positive lowerbound on that set. Further, dχ is G -invariant and cocompactly supported,hence bounded. So we can choose λ > κ/ − λ (cid:107) dχ (cid:107) > K . It follows that ( D ± iλχ ) has a positive lower bound. This impliesthat D + Φ is invertible, so index G ( D + Φ) = 0. The claim then follows byTheorem 3.4 and (6.1).If M is even-dimensional, then the claim follows by applying the result inthe odd-dimensional case to the manifold M × S . (cid:3) Proof of Corollary 2.4.
This follows from Theorem 2.1 and index formulasfor traces defined by orbital integrals applied to index G ( D H ). These indexformulas are: • Theorem 6.10 in [41] if G is any locally compact group and g = e ; • Theorem 6.1 in [40] if G is discrete and finitely generated and g isany element; • Proposition 4.11 in [24] if G is a connected semisimple Lie group and g is a semisimple element.These results imply that in the setting of Corollary 2.4, τ g (index G ( D H )) = ˆ A g ( H ) , for a trace τ g . As index G ( D H ) is independent of the choice of Riemannianmetric, so is ˆ A g ( H ). Finally, vanishing of index G ( D H ) implies vanishing ofˆ A g ( H ) for all g as above. (cid:3) Existence result.
In the remainder of this section, we prove Theorem2.9 by generalising a construction by Lawson and Yau [27]. We first provein this subsection an extension of Theorem 3.8 in [27], namely Proposition6.1.To prepare, let us recall the steps in the construction of Lawson and Yau’spositive scalar curvature metrics [27], which we denote by ˜ g t , on a compactmanifold N .Let K be a compact Lie group acting on N . Consider the principal K -bundle defined by the map p : K × N → N, ( k, y ) (cid:55)→ k − y. Take a K -invariant Riemannian metric g N on N . Let b be a bi-invariantRiemannian metric on K . Let ˆ g denote the lift of g N to the orthogonalcomplement to ker( T p ) in T ( K × N ) with respect to the product metric b ⊕ g N on K × N .For each t >
0, let ˆ b t be the lift of the metric t b on K to T K × N ⊂ T ( K × N ). Then g t := ˆ g ⊕ ˆ b t is a Riemannian metric on the total space K × N . One can check that, foreach t , g t is invariant under the left K -action on K × N defined by l · ( k, y ) = ( kl − , y ) . Thus g t descends, via the projection onto the second factor, π : K × N → N, ( k, y ) (cid:55)→ y, to a K -invariant metric ˜ g t on N . Further, one sees that π is a Riemanniansubmersion with respect to the metrics g t and ˜ g t on the total space and baserespectively.The following proposition shows that, under the conditions stated, for allsufficiently small t , ˜ g t has positive scalar curvature outside a neighbourhoodof the fixed point set. It is an adaptation of the proof of Theorem 3.8 in[27] to the more general setting when N is non-compact but has K -boundedgeometry; see Definition 2.8. Proposition 6.1.
Let N be a manifold with an action by a non-abelian,compact, connected Lie group K . Fix a bi-invariant metric on K . If g N isa K -invariant Riemannian metric on N with K -bounded geometry, then for OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 29 any neighbourhood U of the fixed point set N K , there exists t U > such thatfor all t ≤ t U , the metric ˜ g t constructed above has uniform positive scalarcurvature on N \ U .Proof. We will follow the steps in the proof of Theorem 3.8 in [27] andshow where the assumptions of bounded geometry and no shrinking orbits(Definition 2.5) are needed to obtain the conclusion.For each y ∈ N \ N K , there is an orthogonal splitting k = k y ⊕ p y , where k y is the Lie subalgebra of the isotropy subgroup K y of y . The map ϕ y from(2.1) restricts to an injection on p y . Denote the orthogonal complement of ϕ y ( p y ) in T y N by V y . Then T ( e,y ) ( K × N ) ∼ = k y ⊕ p y ⊕ ϕ y ( p y ) ⊕ V y . For each y ∈ N \ N K , choose an orthonormal basis { e ( y ) , . . . , e l y ( y ) } of p y with respect to b such that for all j, k = 1 , . . . , l y , g (cid:0) ϕ y ( e j ( y )) , ϕ y ( e k ( y )) (cid:1) = σ j ( y ) δ jk for some continuous, positive functions σ j . For each j = 1 , . . . , l y , define afunction λ j : N \ N K → (0 , ∞ ) by λ j ( y ) := σ j ( y )(1 + σ j ( y ) ) . By the calculations in the proofs of Propositions 3.6 and 3.7 in [27] andthe assumption that g N has bounded geometry, for any neighbourhood U of N K , the scalar curvature of ˜ g t at any y ∈ N \ U is bounded below by(6.2) l y (cid:88) j,k =1 t λ j ( y ) λ k ( y ) ( t + λ j ( y ) )( t + λ k ( y ) ) (cid:107) [ e j ( y ) , e k ( y )] (cid:107) b + O (1)as t →
0, where the O (1) term is independent of y . Since the K -actionhas no shrinking orbits with respect to g N , there exists c U > λ j ( y ) > c U for each y ∈ N \ U and j = 1 , . . . , l y . In particular, for t ≤ c U ,the expression (6.2) is bounded below by(6.3) l y (cid:88) j,k =1 t (cid:107) [ e j ( y ) , e k ( y )] (cid:107) b + O (1) . Now, without loss of generality we may assume that K = SU(2) or K =SO(3), as any compact, connected, non-abelian Lie group has such a sub-group. Since K has no subgroups of codimension 1, we have l y = dim p y ≥ y ∈ N \ N K . For all j and k , (cid:107) [ e j ( y ) , e k ( y )] (cid:107) b is 4 times the sectionalcurvature of the plane spanned by e j ( y ) and e k ( y ) with respect to the metric b , and this is constant in y , and positive for K = SU(2) or K = SO(3). Thusfor any neighbourhood U of N K , there exists t U > t ≤ t U ,the expression (6.3), and hence also (6.2), is uniformly positive outside U .It follows that for all such t , the scalar curvature of ˜ g t is uniformly positiveoutside U . (cid:3) We now deduce Theorem 2.9 from the following noncompact generalisa-tion of the main result in [27].
Theorem 6.2.
Let N be a manifold that admits an effective action by acompact, connected, non-abelian Lie group K , such that the fixed point set N K is compact. If there exists a K -invariant Riemannian metric on N suchthat the K -action has K -bounded geometry, then N admits a K -invariantmetric with uniformly positive scalar curvature.Proof. Since N K is compact and the action is effective, by Section 4 of[27] there exists t >
0, a K -invariant neighbourhood U of K with compactclosure, and a K -invariant Riemannian metric g (cid:48) on N such that each metric˜ g (cid:48) t , constructed from g (cid:48) as in subsection 6.2, has positive scalar curvature on U for 0 < t < t .Fix a bi-invariant metric b on K . Let g (cid:48)(cid:48) be a K -invariant metric on N for which the K -action has K -bounded geometry. Let { f , f } be a smooth, K -invariant partition of unity on N such that f ≡ U and f ≡ N \ U (cid:48) , where U (cid:48) is a relatively compact neighbourhood of N K containingthe closure of U . Then g N := f g (cid:48) + f g (cid:48)(cid:48) is a K -invariant Riemannian metric on N . Applying the prescription inSubsection 6.2 to g N , we obtain a family { ˜ g t } t> of K -invariant metrics on N . We claim that for sufficiently small t , ˜ g t has uniformly positive scalarcurvature on N .To see this, let (cid:107) ϕ (cid:107) be the norm function (2.2) associated to the metric g N . Since g (cid:48)(cid:48) and g N coincide on N \ U (cid:48) , and g (cid:48)(cid:48) has K -bounded geometry,there exists C U (cid:48) > (cid:107) ϕ (cid:107) ( y ) ≥ C U (cid:48) for all y ∈ N \ U (cid:48) . One seesthat g N has K -bounded geometry, and so by Proposition 6.1, there exists t = ( t ) U > t ≤ t , ˜ g t has uniformly positive scalarcurvature on N \ U . It follows that for all t ≤ min { t , t } , ˜ g t has uniformlypositive scalar curvature on N . (cid:3) Proof of Theorem 2.9.
In the setting of Theorem 2.9, Theorem 6.2 impliesthat N admits a K -invariant metric with uniformly positive scalar curvature.By Theorem 4.6 in [17] (see also Theorem 58 in [18]), this metric inducesa G -invariant Riemannian metric on G × K N of uniformly positive scalarcurvature. (cid:3) Further applications of the Callias-type index theorem
We used Theorem 3.4 to prove Theorem 2.1 in Subsection 6.1. We givesome other applications of Theorem 3.4 here.7.1.
The image of the assembly map. If M/G is noncompact, and G isnot known to satisfy Baum–Connes surjectivity, then it is a priori unclearif index G ( D + Φ) lies in the image of the Baum–Connes assembly map [4];see the question raised on page 3 of [15]. Theorem 3.4 implies that this isin fact the case for G -Callias-type operators as defined above: OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 31
Corollary 7.1.
The Callias-index of D + Φ lies in the image of the Baum–Connes assembly map. Cobordism invariance of the assembly map.
Theorem 3.4 leadsto a perspective on cobordism invariance of the analytic assembly map.
Corollary 7.2.
Let X be an odd-dimensional Riemannian manifold withboundary N , on which G acts properly and isometrically, preserving N ,such that M/G is compact. Suppose that a neighbourhood U of N is G -equivariantly isometric to N × [0 , ε ) , for some ε > . Let S X → X be a G -equivariant Clifford module, and consider the Clifford module S X | N → N ,graded by i times Clifford multiplication by the inward-pointing unit normal.Suppose that S X | U ∼ = S X | N × [0 , ε ) . Let D N be the Dirac operator on S X | N associated to a G -invariant Clifford connection ∇ N for the Clifford actionby T N on S X | N . Then index G ( D N ) = 0 .Proof. Form the manifold M by attaching the cylinder N × (0 , ∞ ) to X along U . Extend S X and the Clifford action to M in the natural way. Wewrite S for the extension of S X to M . The connection ∇ N pulls backto a Clifford connection on S | N × (0 , ∞ ) . Because the G -invariant Cliffordconnections form an affine space, we can extend this pulled-back connectionto all of M using a G -invariant Clifford connection on X \ N and a partitionof unity. Let D be the associated Dirac operator.Let Φ be the identity endomorphism on S . Then Φ, as in (3.6) isadmissible for D as in (3.5), and (3.1) holds on all of M . By Lemma 4.7,this implies that index G ( D + Φ) = 0. In this case, S N + = S X | N , so byTheorem 3.4, index G ( D N ) = 0. (cid:3) c -Dirac operators.Corollary 7.3. Consider the setting of Theorem 3.4, and assume that M is odd-dimensional and D is a Spin c -Dirac operator. Then there is a union N + of connected components of N and a Spin c -structure on N + with spinorbundle S | N + such that (7.1) index G ( D + Φ) = index G ( D S | N + ) ∈ K ( C ∗ ( G )) , where D S | N + is a Spin c -Dirac operator on the spinor bundle S | N + → N + .If N is connected, then index G ( D + Φ) = 0 .Proof. Since S is an irreducible Clifford module, and S N + ⊂ S | N is invariantunder the Clifford action of T M | N , over each connected component X of N the bundle S N + | X is either zero or S | X . Since M is odd-dimensional, S | X is the spinor bundle of the Spin c -structure of X that it inherits from M . So(7.1) follows.If N is connected, then we either have N + = ∅ , in which case index G ( D +Φ) = 0 because S N + is the zero bundle, or N + = N , in which case (7.1) and Lemma 4.7 imply thatindex G ( D + Φ) = index G (cid:18) D + (cid:18) i − i (cid:19)(cid:19) = 0 . (cid:3) There is a converse to Corollary 7.3 in the following sense. Let N + ⊂ N beany union of connected components; there is a finite number of such subsetsof N since N/G is compact. We can define an admissible endomorphism Φ such that (7.1) holds, by taking Φ to be multiplication by a G -invariantfunction on M that equals 1 on N + and − N \ N + , and is constant 1or − N bounding acocompact set, and any set N + of connected components of N , we have anindex(7.2) index N + G ( D ) := index G ( D + Φ) , with Φ and Φ related as in (3.6), independent of the choice of Φ with theproperty that Φ is positive definite on N + and negative definite on N \ N + .Versions of the index (7.2) are sometimes used in applications of Callias-type index theorems to obstructions to positive scalar curvature for Spin-manifolds, see [2, 9] and the proof of Theorem 2.1 in Subsection 6.1.7.4. Induction.
Suppose that G is an almost connected, reductive Liegroup, and let K < G be maximal compact. In [18, 19, 24] some re-sults were proved relating G -equivariant indices to K -equivariant indicesvia Dirac induction. Such results allow one to deduce results in equivariantindex theory for actions by noncompact groups from corresponding resultsfor compact groups. This was applied to obtain results in geometric quan-tisation [19, 20, 22] and geometry of group actions [18, 21]. Corollary 7.4below is a version of this idea for the index of Definition 3.3.To state this corollary, we consider the setting of Subsection 3.2. UsingAbels’ slice theorem, we write M = G × K Y , for a K -invariant submanifold Y ⊂ M , and S = G × K S | Y . Let g = k ⊕ p be a Cartan decomposition.Then T M ∼ = G × K ( T Y ⊕ p ). We assume that the G -invariant Riemannianmetric on M is induced by a K -invariant Riemannian metric on Y and anAd( K )-invariant inner product on p via this identification.We assume for simplicity that the adjoint representation Ad : K → SO( p )lifts to the double cover Spin( p ) of SO( p ). (This is true for a double cover of G .) Then the standard Spin representation S p of Spin( p ) may be viewed asa representation of K . We assume that S | Y = S Y ⊗ S p for a Clifford module S Y → Y . The Clifford action by T M | Y on S | Y equals c Y ⊗ S p + 1 S Y ⊗ c p ,for a Clifford action c Y by T Y on S Y , and the Clifford action c p of p on S p .We choose the G -invariant connection ∇ S so that ∇ S | Y = ∇ S Y ⊗ S p +1 S Y ⊗ ∇ S p , for Clifford connections ∇ S Y on S Y and ∇ S p on Y × S p Since Φ is G -equivariant, it is determined by its restriction to Y , whichis a K -equivariant endomorphism of S | Y . We assume that Φ | Y = Φ Y ⊗ p , OSITIVE SCALAR CURVATURE AND A CALLIAS INDEX THEOREM 33 for a K -equivariant endomorphism Φ Y of S Y . (What follows remains trueif Φ = Φ Y ⊗ p + 1 ⊗ Φ p for an Ad( K )-invariant endomorphism of S p , butthis requires a small extra argument that we omit here.)Consider the Dirac operator D Y := c Y ◦ ∇ S Y on Γ ∞ ( S Y ). Form D Y from D Y as in (3.5) and Φ Y from Φ Y as in (3.6). Let R ( K ) be the representationring of K and(7.3) D-Ind GK : R ( K ) → K ∗ ( C ∗ ( G ))be the Dirac induction map [4]. Corollary 7.4.
The operator D Y + Φ Y is a K -equivariant Callias-type op-erator, and index G ( D + Φ) = D-Ind GK (index K ( D Y + Φ Y )) . Proof.
Theorem 3.4 implies that index G ( D + Φ) = index G ( D S N + ). Write Y N := Y ∩ N , so that N = G × K Y N . Then Y N is a compact manifold. De-fine D S Y N + analogously to D Y . The induction result for cocompact actions,Theorem, 4.5 in [19], Theorem 5.3 in [24] or Theorem 46 in [18], implies thatindex G ( D S N + ) = D-Ind GK (index K ( D S Y N + )) . Another application of Theorem 3.4, now with G replaced by K , or Theorem1.5 in [2] with a compact group action added, shows that index K ( D S Y N + ) =index K ( D Y + Φ Y ). (cid:3) Callias quantisation commutes with reduction.
Theorem 3.11in [16] is a quantisation commutes with reduction result for the equivariantindex of Spin c -Callias-type operators. This result applies to reduction at thetrivial representation of G ; i.e. to an index defined in terms of G -invariantsections of S . Using Theorem 3.4, we can generalise this result to reductionat more general representations, or more precisely, at arbitrary generatorsof K ( C ∗ r ( G )). Furthermore, this result is ‘exact’ rather than asymptotic asTheorem 3.11 in [16], in the sense that one does not need to consider highpowers of a line bundle.In the setting of Subsection 3.2, we now assume that M is odd-dimensional,and that S is the spinor bundle for a G -equivariant Spin c -structure. Let D be the Spin c -Dirac operator on Γ( S ), defined by the Clifford connectioncorresponding to a connection ∇ L on the determinant line bundle L .The Spin c -moment map associated to ∇ L is the map µ : M → g ∗ suchthat for all X ∈ g ,2 πi (cid:104) µ, X (cid:105) = L X − ∇ LX M , ∈ End( L ) = C ∞ ( M, C ) , where L X denotes the Lie derivative with respect to X , and X M is thevector field induced by X . The reduced space at an element ξ ∈ g ∗ is definedas M ξ := µ − ( ξ ) /G ξ , where G ξ is the stabiliser of ξ . This reduced spaceis noncompact in general, and may not be smooth. But the reduced space N ξ := ( µ − ( ξ ) ∩ N ) /G ξ is compact. It is not always a smooth manifold, butif it is, and ξ ∈ k ∗ , then we have an identification N ξ ∼ = Y Nξ := ( µ − ( ξ ) ∩ Y ∩ N ) /K ξ , with Y and Y N as in Subsection 7.4, including Spin c -structures.See Propositions 3.13 and 3.14 in [22]. Let N + ⊂ N be as in Corollary 7.3.Then we similarly have N + ξ ∼ = Y N + ξ := ( µ − ( ξ ) ∩ Y ∩ N + ) /K ξ in the smoothcase, including Spin c -structures.There is a nontrivial way to define a Spin c -quantisation Q Spin c ( Y N + ξ ) ∈ Z ,even when Y N + ξ is not smooth, described in detail in Section 5.1 of [31].Motivated by the identification N + ξ ∼ = Y N + ξ in the smooth case, we define Q Spin c ( N + ξ ) := Q Spin c ( Y N + ξ ) for ξ ∈ k ∗ .Let T < K be a maximal torus, and fix a positive root system for (
K, T ).Let V ∈ ˆ K have highest weight λ ∈ i t ∗ (cid:44) → k ∗ (cid:44) → g ∗ . (The first inclusionis defined by t ∗ ∼ = ( k ∗ ) Ad ∗ ( T ) , the second by the Cartan decomposition.)Following [31, 32], we call an element ξ ∈ k ∗ an ancestor of V if the coadjointorbit Ad ∗ ( K ) ξ is admissible in the sense of [32], and its K -equivariant Spin c -quantisation is V . There exists a finite set A ( V ) of ancestors representingall different such coadjoint orbits.Let C ∗ r ( G ) be the reduced group C ∗ -algebra of G and D-Ind GK the Diracinduction map (7.3). By the Connes–Kasparov conjecture, proved in [10,26, 42], the abelian group K ∗ ( C ∗ r ( G )) is free, with generators D-Ind GK [ V ],where V runs over ˆ K .Recall the definition of the Callias index of Spin c -Dirac operators (7.2). Corollary 7.5 (Callias quantisation commutes with reduction) . We have (7.4) index N + G ( D ) = (cid:77) V ∈ ˆ K (cid:88) ξ ∈ A ( V ) Q Spin c ( N + ξ ) D-Ind GK [ V ] ∈ K ∗ ( C ∗ r ( G )) . Proof.
By Corollary 7.3, index N + G ( D ) = index G ( D S | N + ), where now D S | N + is a Spin c -Dirac operator on N + . Theorem 4.6 in [22] implies that index G ( D S | N + )equals (cid:77) V ∈ ˆ K (cid:88) ξ ∈ A ( V ) Q Spin c ( N + ξ ) D-Ind GK [ V ] . (cid:3) Remark 7.6.
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