Path-component invariants for spaces of positive scalar curvature metrics
aa r X i v : . [ m a t h . DG ] A p r PATH-COMPONENT INVARIANTS FOR SPACESOF POSITIVE SCALAR CURVATURE METRICSDavid J. WraithAbstract:
The Kreck-Stolz s -invariant is a classic path-component invariantfor the space of positive scalar curvature metrics on certain spin manifolds,with | s | an invariant of the path-component in the moduli space. It is anabsolute (as opposed to relative) invariant, but this strength comes at theexpense of being defined only under restrictive topological conditions. Theaim of this paper is to construct an analogous invariant for certain productmanifolds on which the s -invariant is not defined. § Given a manifold M which supports a positive scalar curvature metric, an importantbut difficult question is to determine what can be said about the topology of the space ofall positive scalar curvature metrics on M , Riem scal ≥ ( M ) . (In this paper we will alwaysassume that spaces of metrics are equipped with the smooth topology.) One can also askabout the topology of the moduli space of positive scalar curvature metrics on M . Recallthat the diffeomorphism group Diff( M ) acts on the space of all metrics Riem( M ) by pull-back, and this action preserves the property of positive scalar curvature. Thus we canform the moduli space of positive scalar curvature metricsRiem scal ≥ ( M ) / Diff( M ) . One can pose analogous questions for other curvature conditions, such as positive or non-negative Ricci curvature, negative sectional curvature etc. There has been much recentactivity in this general direction: for example see [BHSW], [BERW], [HSS], [CS], [CM],[Wa1], [Wa2], [Wr1], [Wr2], [BH], [DKT], [FO1-3] and the book [TW].In this paper we will focus on spaces of positive scalar curvature metrics. However wewish to highlight out at the outset that using the results of [Wr2], all statements involvingpositive scalar curvature can be easily modified to yield analogous statements about non-negative scalar curvature. Although these corresponding results are stronger, we havechosen to focus on positivity in order to simplify the exposition.A basic tool for studying spaces of positive scalar curvature metrics is the Kreck-Stolz s -invariant (see [KS] or [TW] for details). Under the appropriate topological conditions thisallows one to distinguish between different path components of the space of positive scalarcurvature metrics, and even between path-components of the moduli space of positive scalarcurvature metrics. As we will need to refer to these conditions regularly, for conveniencewe make the following definition: 1 efinition 0.1. A closed spin manifold M of dimension k − , k ≥ , which admits apositive scalar curvature metric and for which all real Pontrjagin classes vanish will be saidto satisfy the Kreck-Stolz conditions . It was shown in [KS] that if M satisfies the Kreck-Stolz conditions and ( M, g ) haspositive scalar curvature, then s ( M, g ) is an invariant of the path-component of of positivescalar curvature metrics containing g . Moreover, if H ( M ; Z ) = 0 (which ensures thatgiven an orientation for M the spin structure is unique), then | s | can be shown to be aninvariant of the path-component of the moduli space of positive scalar curvature metricscontaining [ g ]. This was used in [KS] to show that the moduli space of positive scalarcurvature metrics for any manifold M with H ( M ; Z ) = 0 satisfying the Kreck-Stolzconditions has infinitely many path-components. (In stark contrast, it was shown in [CM]that the moduli space of positive scalar curvature metrics on closed orientable 3-manifoldsis path-connected, provided this space is non-empty.) Using the obvious fact that metricsof positive sectional or positive Ricci curvature also have positive scalar curvature, byexamining the underlying space of positive scalar curvature metrics using the s -invariantKreck and Stolz were able to show that in dimension seven there are manifolds for whichthe moduli space of positive Ricci curvature metrics has infinitely many path-components,and also examples with positive sectional curvature for which the moduli space of suchmetrics is not path-connected. The s -invariant has subsequently been used to establishanalogous results in other contexts. For example the author showed in [Wr1] that themoduli space of Ricci positive metrics on all homotopy spheres in dimensions 4 n − ≥ n − ≥
7, there are infinitely many closed manifoldsfor which the moduli space of non-negative sectional curvature metrics has infinitely manypath-components.To provide some context, it has long been known (see [LM; IV Theorem 7.7]) that forany closed spin manifold M of dimension 4 n − ≥ scal> ( M ) has infinitely many path-components. It was pointed out in [PS; Remark2.26] that the same argument used to establish [LM; IV Theorem 7.7] can also be used toshow that the corresponding moduli space has infinitely many path-components. In theauthor’s experience this second point is not so widely known. In both cases, the argumentcan be expressed neatly using Gromov and Lawson’s relative index. This is defined forpairs of positive scalar curvature metrics g , g on M , and is given by i ( g , g ) = ind D + ( M × [0 , , g ) , where D + denotes the Dirac operator and g is any metric on M × [0 ,
1] restricting to dt + g and dt + g in a neighbourhood of the boundary components. It can be shown that thisis an invariant of the path-components of positive scalar curvature metrics to which g and g belong, and vanishes if both belong to the same component. The advantage of theKreck-Stolz s -invariant over this is that it is an absolute invariant, i.e. it only dependson a single metric. Indeed i ( g , g ) = s ( M, g ) − s ( M, g ) whenever the right-hand side is2efined. However Kreck and Stolz show ([KS; 2.16]) that it is not possible to define anabsolute invariant of this type without imposing extra topological conditions on M .The aim of this paper is to demonstrate that it is possible to make similar constructionsunder alternative topological circumstances to those in Definition 0.1. We achieve this byproviding an extension of the s -invariant to certain product manifolds. The new settingis as follows: we consider Riemannian product manifolds ( M, g M ) × ( N, g N ) , where M satisfies the Kreck-Stolz conditions, g M has positive scalar curvature, and N is a closedspin manifold of dimension 4 l , l ≥
1, with ˆ A ( N ) = 0 . For manifolds in dimensions congruent to 0 modulo 4 the ˆ A -genus is a topologicalobstruction to the existence of positive scalar curvature metrics. Nevertheless, any Rie-mannian product involving a positive scalar curvature metric on one factor can be adjustedby scaling to produce a positive scalar curvature metric. In the above product, there issome very small c > c g M + g N has positive scalar curvature.The key point here is that the s -invariant is not defined for product manifolds of thistype. To see this consider ˆ A ( N ) . The ˆ A -genus is a rational linear combination of rationalPontrjagin numbers, and hence if ˆ A ( N ) = 0 , this means that some real Pontrjagin classof N is non-zero, and in turn this means that some real Pontrjagin class of M × N is alsonon-zero. Thus the Kreck-Stolz conditions are not satisfied by the product M × N in thiscase.Let us summarise our new context in a definition: Definition 0.2.
A closed oriented Riemannian spin manifold ( X, g ) with positive scalarcurvature will be said to have a Kreck-Stolz product structure if it is orientation preservingisometric to a Riemannian product manifold ( M k − l ) − , g M ) × ( N l , g N ) , k − l ≥ , l ≥ ,where M satisfies the Kreck-Stolz conditions (Definition 0.1), ˆ A ( N ) = 0 , and H ( X ; Z ) =0 . In this case, the Kreck-Stolz product structure is a 7-tuple ( X, g, φ, M, N, g M , g N ) , where φ : ( X, g ) → ( M × N, g M + g N ) is the orientation preserving isometry. We will denote theset of all Kreck-Stolz product structures on X by K ( X ) . Remark 1:
It is implicit in the above definition that the orientations on M and N arechosen so as to be compatible with X under the isometry φ . Thus a different choice of φ might result in the orientations on M or N having to be rechosen. Now it follows fromthe K¨unneth Theorem that the condition H ( X ; Z ) = 0 forces both H ( M ; Z ) = 0 and H ( N ; Z ) = 0 , and thus a unique spin structure on X means that the spin structureson M and N are also unique for a given orientation. Consequently, the fact that φ isorientation preserving automatically means that it is spin-structure preserving. Remark 2:
One can also make certain uniqueness statements about the factors M and N appearing in a Kreck-Stolz product structure: if ( X, g ) admits a Kreck-Stolz productstructure, then this structure is uniquely determined up to isometry of the individualproduct factors. This is discussed in detail in Corollary 2.16.Notice that the diffeomorphism group of X acts on K ( X ) is a natural way: for θ ∈ Diff( X ) we set θ · ( X, g, φ, M, N, g M , g N ) = ( X, θ ∗ ( g ) , φ ◦ θ, M, N, g M , g N ) . Thus we can also consider the moduli space of such structures, K ( X ) / Diff( X ) . The main result we will establish in this paper as follows:3 heorem 0.3.
Consider a Riemannian manifold ( X, g ) which has a Kreck-Stolz productstructure ( X, g, φ, M, N, g M , g N ) . Then there is a function ˜ s : K ( X ) → Q such that givenany other positive scalar curvature metric g ′ on X for which ( X, g ′ ) has a Kreck-Stolzproduct structure ( X, g ′ , φ, M, N, g ′ M , g ′ N ) , (so the diffeomorphism φ : X → M × N isshared by both structures), the following statements hold.(i) The relative index i ( g, g ′ ) = ˜ s ( X, g ) − ˜ s ( X, g ′ ) , so in particular if g and g ′ belong tothe same path-component of the space of positive scalar curvature metrics on X, then ˜ s ( X, g ) = ˜ s ( X, g ′ ) . (ii) Riem scal ≥ ( X ) has infinitely many path-components of positive scalar curvature met-rics distinguished by ˜ s. (iii) If [ g ] denotes the class of g in the moduli space of positive scalar curvature metrics on X, then for any h ∈ [ g ] , | ˜ s ( X, h ) | = | ˜ s ( X, g ) | .(iv) | ˜ s | descends to give a Q -valued function on the moduli space of Kreck-Stolz structures K ( X ) / Diff ( X ) . Notational remark:
It might appear that ˜ s depends on the whole Kreck-Stolz productstructure, however this is not the case. By Remark 2 above, the Kreck-Stolz structure onlydepends on the pair ( X, g ) up to orientation preserving isometry of the factors (
M, g M )and ( N, g N ) , and we will see in due course that ˜ s is invariant under such isometries. Henceour notation ˜ s ( X, g ) is justified.Note that if we allowed the degenerate case l = 0, i.e. the case where N is a point, ˜ s would reduce to s .To illustrate Theorem 0.3, we will present some explicit examples. These will take theform of product manifolds, for which we can take the diffeomorphism in the Kreck-Stolzstructure to be the identity map.Recall that a K3 surface K satisfies ˆ A ( K ) = −
2, and so this simply-connected spinmanifold cannot support a metric of positive scalar curvature. Similarly there is a simply-connected spin ‘Bott manifold’ B for which ˆ A ( B ) = 1 , so this too does not admit apositive scalar curvature metric. (The Bott manifold can be constructed by forming theboundary connected sum of 28 copies of the manifold constructed by plumbing the tangentdisk bundle of S to itself according to the E -graph. The resulting object has boundary S , and this can then be made into a smooth closed manifold B by gluing in a disc D .Together with H P , B generates Ω spin ∼ = Z ⊕ Z . ) We note that despite the fact thatneither K nor B admit positive scalar curvature, both are known to admit Ricci flatmetrics. Using Theorem 0.3 together with the definition of ˜ s (Definition 2.8) and resultsfrom [Wr1; page 2014] we immediately obtain: Theorem 0.4. If K denotes the K3 surface, B the Bott manifold, and Σ n − is anyhomotopy n -sphere ( n ≥ ) which bounds a parallelisable manifold, then there is a sequence g j of Ricci positive metrics on Σ such that given Ricci flat metrics g K on K and g B on B we have ˜ s (Σ × K , g j + g K ) = − j | bP n | + q n − (2 n − −
1) ;˜ s (Σ × B , g j + g B ) = j | bP n | + q n − (2 n − − , here q = q (Σ) is an integer depending on Σ , and where bP n denotes the group of dif-feomorphism classes of homotopy spheres bounding a parallelisable manifold of dimension n . In particular, for different j the metrics g j + g K respectively g j + g B belong to differentpath-components of the space of positive scalar curvature metrics on Σ × K respectively Σ × B . Theorem 0.4 should be compared with Theorem 0.7 in [Wr2]. In fact, combiningTheorem 0.3 with the results in [Wr2] it is not difficult to show that the condition of positivescalar curvature in Theorem 0.4 can be replaced with non-negative Ricci curvature.We should also point out that products involving the Bott manifold appear in the sta-ble Gromov-Lawson-Rosenberg conjecture (see for example [S2; § Rosenberg index of a connected closed spin manifold M ofdimension at least five vanishes if and only if for some k ≥ , the manifold M × ( B ) k admits a positive scalar curvature metric. Here ( B ) k denotes the k -fold product.Whereas the modulus of the Kreck-Stolz s -invariant gives a well-defined invariant onthe path-components of the moduli space of positive scalar curvature metrics, we are notin a position to assert that the same is true for ˜ s. The essential difference is that metricssupporting Kreck-Stolz product structures will typically be distributed discretely througha path component of positive scalar curvature metrics, and we can only make a directcomparison between those which share the same spin-structure preserving isometry to aproduct, up to a product isometry of the individual factors. It could be that two suchmetrics lie in different path-components of the space of positive scalar curvature metricsand yield different | ˜ s | values, but some isometric copy of one of the metrics lies in the samepath-component as the other. In this situation, both metrics give rise to equivalance classesin the moduli space of positive scalar curvature metrics which belong to the same pathcomponent, though their | ˜ s | -values disagree. This just serves to highlight the difficultiesassociated with trying to extend path-component invariants beyond the classical results.Theorem 0.3 raises the obvious question as to whether ˜ s is a path-component invariantfor at least some Kreck-Stolz product manifolds M × N. It turns out that this questionis intimately related to the question of which manifolds are ˆ A -multiplicative fibres, inthe sense of [HSS; Definition 1.8]. Of particular relevance here is whether or not suchproducts can be ˆ A -multiplicative fibres in degree 0 , which means that for any fibre bundle F → E → S where the fibre F = M × N, we have ˆ A ( E ) = 0 . (In [HSS; Proposition1.9] it is shown that the vanishing of all rational Pontrjagin classes is sufficient to satisfythis condition, however this does not help for Kreck-Stolz products.) If F is such a fibre,then for any diffeomorphism φ : F → F and any positive scalar curvature metric g on F , the Gromov-Lawson relative index i ( g, φ ∗ g ) = 0 . This is a consequence of the the factthat the Atiyah-Patodi-Singer index formula is well-behaved under gluing manifolds: if( V , g ) , ( V , g ) are Riemannian spin manifolds with boundary, and ψ : ∂V → − ∂V is aspin-structure preserving isometry, then ind D + V + ind D + V = ˆ A ( V ∪ ψ V ) . The argumentis as follows. Consider a metric on F × [0 ,
1] interpolating between g on one boundarycomponent and φ ∗ g on the other. Assume as usual that the interpolating metric is aproduct near each boundary. Now form the mapping torus T φ of the diffeomorphism φ : F → F, noticing that our metric on F × [0 ,
1] descends to give a well-defined metric¯ g on T φ . If we remove a small piece of the form F × [0 , ǫ ] from this torus, where we are5ssuming that ǫ is so small that the metric on F × [0 ,
1] is a product throughout F × [0 , ǫ ] , then clearly the relative index for the boundary metrics is i ( g, g ) = 0 . The relative indexof the remaining part of the mapping torus is i ( g, φ ∗ g ) , but by the above gluing formula, i ( g, φ ∗ g ) = i ( g, g ) + i ( g, φ ∗ g )= ind D + ( F × [0 , ǫ ] , dt + g ) + ind D + ( T φ \ ( F × [0 , ǫ ]) , ¯ g )= ˆ A ( T φ ) . Now if F is an ˆ A -multiplicative fibre in degree 0, then ˆ A ( T φ ) = 0 , and the relative indexclaim follows. (Alternatively, we could argue from [HSS; Proposition 2.2].)The significance of the equation i ( g, φ ∗ g ) = 0 for ˜ s is as follows. Suppose that g, g ′ are positive scalar curvature metrics on a manifold X which belong to the same path-component of positive scalar curvature metrics, and for which both ( X, g ) and (
X, g ′ ) haveKreck-Stolz product structures with respect to isometries ψ, θ : X → M × N respectively.Thus ( X, ψ ∗ ( θ − ) ∗ g ′ ) has a Kreck-Stolz structure with isometry ψ. By Theorem 0.3(i) wededuce that i ( g, ψ ∗ ( θ − ) ∗ g ′ ) = ˜ s ( X, g ) − ˜ s ( X, ψ ∗ ( θ − ) ∗ g ′ ) , but this relative index vanishes if X is an ˆ A -multiplicative fibre. By Theorem 0.3(iii)we also have that ˜ s ( X, g ′ ) = ˜ s ( X, ψ ∗ ( θ − ) ∗ g ′ ) , and consequently ˜ s ( X, g ) = ˜ s ( X, g ′ ) . Al-lowing for the fact that diffeomorphisms might reverse orientation and change the signof ˜ s (see Lemma 2.11), in the case of an ˆ A -multiplicative fibre, we deduce that | ˜ s | is apath-component invariant for the moduli space of positive scalar curvature metrics on X. Thus we are motivated to pose the following problem, in part to find examples ofmanifolds on which | ˜ s | is a path-component invariant for the moduli space of positivescalar curvature metrics, and in part to extend the scope of [HSS; Proposition 1.9]: Problem 0.5.
Find examples of Kreck-Stolz product manifolds M × N which are ˆ A -multiplicative fibres in degree 0. In relation to the above problem, it is worth remarking that either such examples arecommonplace, which would be interesting from the point-of-view of ˜ s, or they are not. Inthe latter case, this means that there must be rich classes of bundles in which the ˆ A -genusdoes not behave multiplicatively. To the best of the author’s knowledge, only one familyof bundles in which ˆ A is not multiplicative is known at present, namely that constructedin [HSS; Theorem 1.4].As ˜ s is only defined for metrics isometric to a product, this naturally leads to thefollowing question, which we believe is of independent interest: Question 0.6.
Consider a product manifold M × N which admits a positive scalar cur-vature metric. Can one find conditions on M and N under which every path-componentof positive scalar curvature metrics contains a product metric? Note that there are situations in which M × N admits positive scalar curvature metricsbut no product metric with positive scalar curvature. For example consider the case where M is a simply-connected spin 4-manifold with ˆ A ( M ) = 0 which does not admit a positive6calar curvature metric (see [R; Counterexample 1.13]), and where N is a K3 surface. Asnoted above, a K3 surface is a simply-connected spin manifold with non-zero ˆ A -genus, andtherefore does not support a metric of positive scalar curvature. The product M × N isthen a simply-connected spin 8-manifold with ˆ A ( M × N ) = ˆ A ( M ) ˆ A ( N ) = 0 . By Gromov-Lawson [GL], all simply-connected spin 8-manifolds with vanishing ˆ A -genus admit positivescalar curvature metrics. The author is grateful to Boris Botvinnik for pointing out thisexample. Of course, dimension four manifolds are somewhat special from a positive scalarcurvature point of view. For an example involving higher-dimensional factors, one couldstart with the the manifold M described by Thomas Schick in [S1], which is a counter-example to the (unstable) Gromov-Lawson-Rosenberg conjecture. (See for example [RS; §
4] for a general discussion on this.) Now M does not admit a positive scalar curvaturemetric, however for some k ≥ M × ( B ) k does admit such a metric (where B is the Bott manifold as above), as the stable Gromov-Lawson-Rosenberg conjecture isknown to hold for M . Of course M × ( B ) n cannot admit a product metric with positivescalar curvature as none of the factors individually support such a metric.In a slightly different direction we note that product metrics play a crucial role in therecent paper [TWi], which investiagtes the moduli space of non-negative Ricci curvaturemetrics on certain manifolds, and will similarly be a central feature in a forthcoming paperof Boris Botvinnik and the author on the same topic.This paper is laid out as follows. In § s -invariant, as this provides the blueprint for establishing Theorem 0.3. The constructionof ˜ s and the proof of Theorem 0.3 are contained in § § s -invariant We begin by recalling the index theorem of Atiyah-Patodi-Singer:
Theorem 1.1. ([APS]) Let ( W, g W ) be a compact even dimensional Riemannian spinmanifold with non-empty boundary M , where the metric g W is a product dt + g M ina neighbourhood of the boundary. Consider the Atiyah-Singer Dirac operator D + on W acting on the subspace of spinor bundle sections for which the restriction to M belongs tothe span of the negative eigenspaces of the operator induced on M . Then the index of this(restricted domain) Dirac operator on W is given byind D + ( W, g W ) = Z W ˆ A ( p ∗ ( W, g W )) − h ( M, g M ) + η ( M, g M )2 , where ˆ A denotes the ˆ A -polynomial in the Pontrjagin forms of the metric, h is the dimensionof the space of harmonic spinors on the boundary M , and η is the eta-invariant of the Diracoperator on M . W and the metric on W . However it is not difficult to see that only the metric in aneighbourhood of the boundary actually influences the value of the integral. (The argumentbehind this is detailed in Lemma 2.5.) In the light of this observation it is natural to ask:can we separate out the topological dependence on W from the metric dependence nearthe boundary? The answer to this is a qualified yes: the integral of any summand in theintegrand can be rewritten in this way provided it is decomposable (in the sense that itinvolves a product of forms), and provided that M has vanishing real Pontrjagin classes.From now on let us assume that M does indeed have vanishing real Pontrjagin classes.Let α, β denote Pontrjagin forms or products of Pontrjagin forms on W . As a consequenceof the above assumption together with the product structure of the metric g W near theboundary, following the notation in [KS; 2.8] we can define a form d − ( α ∧ β ) on M bysetting d − ( α ∧ β ) = ˆ α ∧ ( β | M ) , where ˆ α satisfies d ˆ α = α | M . A simple Stokes’ Theoremargument then shows that Z W α ∧ β = Z M d − ( α ∧ β ) + h j − [ α ] ∪ j − [ β ] , [ W, M ] i , where j : H ∗ ( W, M ; R ) → H ∗ ( W ; R ) is the map induced by inclusion, and the angledbrackets denote evaluation on the fundamental homology class. By the cohomology longexact sequence of the pair ( W, M ) it is easy to see that we need M to have vanishing realPontrjagin classes in order for the required pre-images under j to be defined.Suppose now that W has dimension 4 k . The top-dimensional term of the ˆ A -polynomialhas all its summands decomposable except for the term in p k ( W, g W ) . In order to deal withthis, a linear combination of the ˆ A and L -polynomials is formed which has zero p k term.Specifically, Theorem 1.1 is applied to ˆ A + a k L where a k = 1 / (2 k +1 (2 k − − . Theresulting index formula is given byind D + ( W, g W ) = Z M d − ( ˆ A + a k L )( p ∗ ( M, g M )) − h ( M, g M ) + η ( M, g M )2 − a k η ( B ( M, g M )) − t ( W ) , where η ( B ( M, g M )) is the eta-invariant of the signature operator B on M , and t ( W ) isthe topological term t ( W ) = −h ( ˆ A + a k L )( j − p ∗ ( W )) , [ W, M ] i + a k σ ( W )where the p ∗ ( W ) are the Pontrjagin classes of W (as opposed to forms), and σ ( W ) is thesignature. If we assume that the scalar curvature of M is positive, this forces h ( M, g M ) = 0 . The idea behind the s -invariant is to collect together all the terms depending on theboundary ( M, g M ): 8 efinition 1.3. Given a closed spin manifold M k − with positive scalar curvature andvanishing real Pontrjagin classes, the s -invariant is given by s ( M, g ) = − η ( M, g M ) − a k η ( B ( M, g M )) + Z M d − ( ˆ A + a k L )( p ∗ ( M, g M )) . With this definition we see immediately that s ( M, g M ) = ind D + ( W, g W ) + t ( W ) . Moreover if the metric g W also has positive scalar curvature then the index term abovevanishes, leaving s ( M, g M ) = t ( W ) . In this situation s is completely determined by thetopology of the bounding manifold W .Note that Definition 1.3 does not require M to be the boundary of a suitable manifold W . The key properties of the s -invariant can be proved by applying the above analysis tothe case W = M × I for any interval I , after showing that s is additive across disjoint unionsand is sign-sensitive to orientation. Specifically it can be shown (as already mentioned in §
0) that s is a path-component invariant for the space of positive scalar curvature metrics.Furthermore if H ( M ; Z ) = 0 then | s ( M, g ) | ∈ Q is an invariant of the path-componentof the moduli space of positive scalar curvature metrics on M containing g . (See [KS;Proposition 2.13].) § s -invariant Let W k − l ) ( k > l ≥
1) and N l be compact oriented spin manifolds. We supposethat W has boundary M (possibly disconnected), and that N is a closed manifold. If π W (respectively π N ) denote the projections of W × N onto W (respectively N ), thenby the Whitney formula the total rational or real Pontrjagin class satisfies p ( W × N ) = p ( π ∗ W ( T W )) p ( π ∗ N ( T N )) since T ( W × N ) ∼ = π ∗ W ( T W ) ⊕ π ∗ N ( T N ) . By the multiplicativeproperty of the ˆ A -polynomial we obtain the equality of polynomials ˆ A ( p ( W × N )) =ˆ A ( p ( π ∗ W ( T W ))) ˆ A ( p ( π ∗ N ( T N ))) . By the naturality of the Pontrjagin classes we can thenwrite ˆ A ( p ( W × N )) = π ∗ W ( ˆ A ( p ( M )) π ∗ N ( ˆ A ( p ( N )) . Lemma 2.1.
With W , N as above, suppose we choose a product metric g W + g N on W × N . Then for the Pontrjagin forms corresponding to this metric we have p i ( W × N ; g W + g N ) = X j + k = i π ∗ W p j ( W ; g W ) ∧ π ∗ N p k ( N ; g N ) . Proof.
The Pontrjagin forms are symmetric polynomials in the curvature form for thegiven metric. Recall that given a local tangent frame field { s i } for a Riemannian n -manifold, the curvature form Ω is an ( n × n )-matrix of 2-forms (Ω ij ) with entries definedby R ( X, Y )( s j ) = n X i =1 Ω ij ( X, Y ) s i . g W + g N and frame fields s , ..., s k − l ) ∈ Γ( T W ⊕ ⊂ T ( W × N ) and s k − l )+1 , ..., s k ∈ Γ(0 ⊕ T N ) , the curvature 2-form satisfiesΩ = (cid:18) Ω W
00 Ω N (cid:19) where Ω W and Ω N are the pull-backs of the curvature forms of ( W, g W ) respectively( N, g N ) . (See [Mo] page 208.) The total Pontrjagin form is then given bydet (cid:16) I − πi Ω (cid:17) = det (cid:16) I − πi Ω W (cid:17) ∧ det (cid:16) I − πi Ω N (cid:17) , where the determinants on the right-hand side are the pull-backs (to W × N ) of the totalPontrjagin forms of ( W, g W ) and ( N, g N ) . The lemma then follows by expanding thesetotal classes into their individual terms. ⊓⊔ Corollary 2.2.
With the set-up of Lemma 2.1 we have the following decomposition of ˆ A -polynomials into Pontrjagin forms: ˆ A ( p ∗ ( W × N ; g W + g N )) = π ∗ W ˆ A ( p ∗ ( W ; g W )) ∧ π ∗ N ˆ A ( p ∗ ( N ; g N )) . In particular for the top-dimensional forms we have ˆ A k ( p ∗ ( W × N ; g W + g N )) = π ∗ W ˆ A k − l ( p ∗ ( W ; g W )) ∧ π ∗ N ˆ A l ( p ∗ ( N ; g N )) . Proof.
As discussed above, the equivalent formula to the first statement holds for Pon-trjagin classes, and follows from the decomposition of those classes on product manifolds.By Lemma 2.1 Pontrjagin forms for product manifolds equipped with product metrics de-compose into terms involving the individual factor manifolds in exactly the same way asPontrjagin classes. The result follows immediately. For the second statement we simplynote that a top dimensional form on W × N can only be formed from a product of topdimensional forms on the factors, since any higher degree form must be zero. ⊓⊔ Lemma 2.3.
Given top-dimensional differential forms α , β on oriented manifolds X re-spectively Y , we have Z X × Y π ∗ X α ∧ π ∗ Y β = (cid:16)Z X α (cid:17)(cid:16)Z Y β (cid:17) . Proof.
This equation holds as it holds locally in any coordinate neighbourhood which is aproduct of coordinate neighbourhoods for X and Y individually. Using such a coordinatesystem the calculation reduces to showing that for appropriate functions a and b : Z U × V a ( x , ..., x r ) b ( y , ..., y s ) dx ...dx r dy ...dy s = Z U a ( x , ..., x r ) dx ...dx r Z V b ( y , ..., y s ) dy ...dy s , ⊓⊔ Corollary 2.4.
With W , N and metrics as above we have Z W × N ˆ A k ( p ∗ ( W × N ; g W + g N )) = ˆ A ( N ) Z W ˆ A k − l ( p ∗ ( W ; g W )) , where ˆ A ( N ) is the ˆ A -genus of N . Proof.
This follows immediately from Corollary 2.2, Lemma 2.3 and the fact that ˆ A ( N ) = R N ˆ A l ( p ∗ ( N ; g N )) . From now on we will assume that the product metric g W + g N takes the form dt + g M + g N near the boundary. Since the Pontrjagin forms are defined using the Levi-Civitaconnection of the metric, if we replace the g W + g N by another metric g which takes thesame product form dt + g M + g N near the boundary we will change the Pontrjagin forms,however this will not change the value of the integral over W × N : Lemma 2.5.
Consider an oriented manifold X n with non-empty connected boundary Y .Let φ be a top dimensional Pontrjagin form (respectively a top dimensional wedge productof Pontrjagin forms) on X corresponding to a Riemannian metric g X , which is a product dt + g Y near the boundary. If φ ′ is the top dimensional Pontrjagin form (respectivelythe corresponding wedge product of Pontrjagin forms) arising from a metric g ′ X which alsotakes the form dt + g Y near the boundary, then Z X φ = Z X φ ′ . Proof.
We consider the metric g X ∪ g ′ X on the oriented double of X , X ∪ ( − X ) . This metricis smooth as the individual metrics agree near the common boundary. Now all orienteddouble manifolds are oriented boundaries, and hence all Pontrjagin numbers of X ∪ ( − X )must vanish. Thus if we let ψ be the top dimensional Pontrjagin form (respectively wedgeproduct of Pontrjagin forms) on X ∪ ( − X ) arising from g X ∪ g ′ X , then Z X ∪ ( − X ) ψ = 0 . But Z X ∪ ( − X ) ψ = Z X φ + Z − X φ ′ = Z X φ − Z X φ ′ . Thus R X φ = R X φ ′ as claimed. ⊓⊔ From Corollary 2.4 and Lemma 2.5 we obtain:11 orollary 2.6.
Let W , N , g W and g N be as before, with g W taking the form dt + g M near ∂W = M, and let g be any metric on W × N which takes the same product form dt + g M + g N as g W + g N near the boundary. Then Z W × N ˆ A k ( p ∗ ( W × N ; g )) = ˆ A ( N ) Z W ˆ A k − l ( p ∗ ( W ; g W )) , and applying the Atiyah-Patodi-Singer index theorem to ( W × N ; g ) we obtainind D + ( W × N ; g ) = ˆ A ( N ) Z W ˆ A k − l ( p ∗ ( W ; g W )) − h + η M × N ; g M + g N ) . Following [KS], from now on we will make the assumption that the real Pontrjaginclasses of M = ∂W vanish . This assumption allows us to re-write the above integral.Following the argument and notation in [KS] as outlined in § Proposition 2.7.
With W , N and g as above, and assuming the real Pontrjagin classesof M vanish,ind D + ( W × N ; g ) = ˆ A ( N ) hZ M d − ( ˆ A + a k − l L )( p ∗ ( M ; g M )) − a k − l η ( B ( M, g M )) − t ( W ) i − h + η M × N ; g M + g N ) , where L is the Hirzebruch L -polynomial, B denotes the signature operator, a n := 1 / (2 n +1 (2 n − − , and the topological term t ( W ) is given by t ( W ) = − D ( ˆ A + a k − l L )( j − p ∗ ( W )) , [ W, M ] E + a k − l σ ( W ) where j denotes the inclusion map j : H ∗ ( W, M ; R ) → H ∗ ( W ; R ) and σ ( W ) is the signatureof W . Proof.
It follows from Theorem 1.1 and the Atiyah-Patodi-Singer index theorem appliedto the signature operator ([APS; 4.14]) that Z W ( ˆ A + a k − l L )( p ∗ ( W ; g W )) = Z W ˆ A ( p ∗ ( W, g W )) + a k − l σ ( W ) + a k − l η ( B ( M, g M )) . By [KS; Lemma 2.7] we have Z W ( ˆ A + a k − l L )( p ∗ ( W ; g W )) = Z M d − ( ˆ A + a k − l L )( p ∗ ( M ; g M ))+ D ( ˆ A + a k − l L )( j − p ∗ ( W )) , [ W, M ] E . Combining the above two statements with Corollary 2.6 yields the result. ⊓⊔ If we assume that both g M and g M + g N have positive scalar curvature, (we can alwaysachieve this by scaling g M if necessary), then the term h ( M × N ; g M + g N ) in the statementof Proposition 2.7 is zero. Collecting together the boundary terms as in [KS] then leads to12 efinition 2.8(a). Given M and N as above (so in particular the real Pontrjagin classesof M all vanish and ˆ A ( N ) = 0 ), together with a positive scalar curvature metric g M on M and a metric g N on N such that the product metric g M + g N on M × N has positivescalar curvature, we set ˜ s ( M × N, g M + g N ) := ˆ A ( N ) hZ M d − ( ˆ A + a k − l L )( p ∗ ( M ; g M )) − a k − l η ( B ( M, g M )) i − η ( D ( M × N ; g M + g N ) ) . We now can writeind D + ( W × N ; g ) = ˜ s ( M × N, g M + g N ) − ˆ A ( N ) t ( W ) . ( † )Recalling the definition of the s -invariant (Definition 1.3) allows us to re-express ˜ s as˜ s ( M × N, g M + g N ) = ˆ A ( N ) s ( M, g M ) + 12 ˆ A ( N ) η ( M, g M ) − η ( M × N ; g M + g N ) . If the metric g on W × N has positive scalar curvature then the index term vanishesand we are left with Lemma 2.9.
With all manifolds and metrics as above, if g is a positive scalar curvaturemetric on W × N (which as always is a product dt + g M + g N near the boundary) then ˜ s ( M × N, g M + g N ) = ˆ A ( N ) t ( W ) . The right-hand side of this expression depends only on the topology of W × N , and isindependent of the choice of metrics. We now point out some key properties of ˜ s . The arguments needed here are essentiallythe same as those required to establish the equivalent properties for s . Although thesearguments are for the most part suppressed in [KS], they are explained in depth in Chapter5 of [TW], and we therefore omit the details here. (In relation to Lemma 2.11 below, weremark that the oriented manifold − ( M × N ) can be viewed as either ( − M ) × N or M × ( − N ) , with the same conclusion obtained in either case following the arguments onpages 45-46 of [TW].) Lemma 2.10. ˜ s is additive over disjoint unions in the following sense: ˜ s (( M × N ) ⊔ ( M × N ) , g M + g N ⊔ g M + g N ) = ˜ s ( M × N, g M + g N ) + ˜ s ( M × N, g M + g N ) . Lemma 2.11. ˜ s is sensitive to the orientation of M in the sense that ˜ s ( M × N, g M + g N ) = − ˜ s ( − ( M × N ) , g M + g N ) . Lemma 2.12. ˜ s is additive over connected sums in the following sense: ˜ s (( M ♯M ) × N ) , ( g M ♯g M ) + g N ) = ˜ s ( M × N, g M + g N ) + ˜ s ( M × N, g M + g N ) , here g M ♯g M is the (canonical) Gromov-Lawson positive scalar curvature metric on theconnected sum. Before embarking on the proof of Theorem 0.3, we would like to extend the scope ofthe invariant ˜ s in the following way. Consider a Riemannian spin manifold ( X, g ) witha Kreck-Stolz product structure (Definition 0.2), that is, suppose that (
X, g ) is isometricto (
M, g M ) × ( N, g N ) (with ( M, g M ), ( N, g N ) as before), via a spin structure preservingdiffeomorphism X → M × N. We would like to define ˜ s for the manifold ( X, g ) by declaring˜ s ( X, g ) := ˜ s ( M × N, g M + g N ) . However, we need to argue that such an extension to thedefinition of ˜ s is well-defined, and in order to this we recall: Lemma 2.13. ([EH; page 3075]) If ( X, g ) is a closed Riemannian manifold, then X de-composes as a Riemannian product of indecomposable factors, and this decomposition isunique in the sense that the corresponding foliations of X are uniquely determined. In our case, we want to suppose that (
X, g ) is isometric to the Riemannian product( M × N, g M + g N ) . In order for our claimed ˜ s extension to be well-defined, we need toshow that if ( M × N, g M + g N ) ∼ = ( M ′ × N ′ , g M ′ + g N ′ ) then ( M, g M ) ∼ = ( M ′ , g M ′ ) and( N, g N ) ∼ = ( N ′ , g N ′ ) . Now it could be that one or both of (
M, g M ) and ( N, g N ) are them-selves decomposable as a Riemannian product, so we cannot make the desired conclusiondirectly from Lemma 2.13. However, we will now demonstrate that the topological con-ditions imposed on M and N in fact provide enough extra structure for us to make thisclaim. Lemma 2.14.
With
M, N as before, decompose the manifolds as smooth products M ×· · · M p and N × · · · × N q , where the various factors cannot be further decomposed assmooth products. Then no factor of M is homeomorphic to any factor of N . Proof.
Let us consider real Pontrjagin classes, since all such classes for M vanish byassumption. Suppose that N splits as a smooth (but not necessarily Riemannian) product N = N × · · · N q , and that one of these factors, N say, is also a factor of M up tohomeomorphism, i.e. M ∼ = N × K, for some K . Note that we can work here withhomeomorphisms, since real Pontrjagin classes are homeomorphism invariants of smoothmanifolds. Now p i ( M ) = P j + k = i π ∗ p j ( N ) ∪ π ∗ p k ( K ) , where π , π indicate the projectionmaps onto the first, respectively second factors of N × K. Since ˆ A ( N ) = 0 by assumptionand the ˆ A -genus is multiplicative for products, it follows that ˆ A ( N ) = 0 also. In particularthis means that N has some non-vanishing real Pontrjagin classes. Suppose that p j ( N ) =0 for some j . We then have p j ( M ) = π ∗ p j ( N ) + X r + s = j,r With ( M, g M ) , ( N, g N ) as before, if ( M × N, g M + g N ) ∼ = ( M ′ × N ′ , g M ′ + g N ′ ) then ( M, g M ) ∼ = ( M ′ , g M ′ ) and ( N, g N ) ∼ = ( N ′ , g N ′ ) . Proof. Decompose ( M, g M ) and ( N, g N ) into Riemannian products with indecomposablefactors, and similarly decompose ( M ′ , g M ′ ) and ( N ′ , g N ′ ). The image of the isometry( M × N, g M + g N ) → ( M ′ × N ′ , g M ′ + g N ′ ) provides us with a second isometric splittingof ( M ′ × N ′ , g M ′ + g N ′ ) as a Riemannian product. By Lemma 2.13 these two splittingsmust coincide, so our isometry splits as a product of isometries from the factors of ( M × N, g M + g N ) to the factors of ( M ′ × N ′ , g M ′ + g N ′ ) . By Lemma 2.14, M and N respectively M ′ and N ′ have no common factors, hence we can assemble these factorwise maps intoisometries ( M, g M ) ∼ = ( M ′ , g M ′ ) and ( N, g N ) ∼ = ( N ′ , g N ′ ) as claimed. ⊓⊔ We also note that if the isometry ( M × N, g M + g N ) ∼ = ( M ′ × N ′ , g M ′ + g N ′ ) isorientation preserving, then the isometries ( M, g M ) ∼ = ( M ′ , g M ′ ) and ( N, g N ) ∼ = ( N ′ , g N ′ )are either both orientation preserving or both orientation reversing.From Corollary 2.15 we immediately deduce: Corollary 2.16. If ( X, g ) has a Kreck-Stolz product structure with respect to a product ( M × N, g M + g N ) , then the Kreck-Stolz product structure is unique up to isometries of theindividual product factors which are either both orientation preserving or both orientationreversing. Observe that ˜ s in Definition 2.8(a) is clearly invariant under orientation preservingisometries of the factors M and N , and using the arguments underpinning Lemma 2.11(see [TW] pages 45-46) also invariant under orientation reversing isometries of both factors.We can therefore now complete the definition of ˜ s : Definition 2.8(b). Given a closed Riemannian spin manifold ( X, g ) with a Kreck-Stolzproduct structure involving an oriented isometry to a product ( M × N, g M + g N ) as inDefinition 2.8(a), we set ˜ s ( X, g ) = ˜ s ( M × N, g M + g N ) , where the latter quantity is thatdefined in Definition 2.8(a). Remark: It is easily observed that Lemmas 2.10 and 2.11 can be immediately extended toincorporate the more general ˜ s definition in 2.8(b). Proof of Theorem 0.3. Consider a path of positive scalar curvature metrics g M × N ( t )on M × N for t ∈ [0 , 1] say, where g M × N (0) and g M × N (1) are both product metricswith respect to the smooth product structure on M × N . (Note that there is no needto assume that g M × N ( t ) is a product metric for any t = 0 , . ) We first establish that˜ s ( M × N, g M × N (0)) = ˜ s ( M × N, g M × N (1)) . It follows from a well-known observation about paths of positive scalar curvature met-rics (see for example [Wr1; Lemma 6.3]) that g ( t ) can be adjusted to give a metric g M × N × I on M × N × I for some interval I , which has positive scalar curvature globally, agrees withthe metrics g M × N (0) respectively g M × N (1) when restricted to the two boundary compo-nents, and moreover is a product with respect to the t parameter near these boundary15omponents. Thus taking W = M × I , we see that by Lemma 2.9 we have˜ s ( M × N ⊔ ( − M ) × N, g M × N (0) ⊔ g M × N (1)) = ˆ A ( N ) t ( M × I ) . By Lemmas 2.10 and 2.11 the left-hand side of this expression is equal to˜ s ( M × N, g M × N (0)) − ˜ s ( M × N, g M × N (1)) . We claim that t ( M × I ) = 0 . Now the p i ( M × I ) vanish as the Pontrjagin classes of I and (byassumption) the Pontrjagin classes of M both vanish. Thus the h ( ˆ A + a k − l L )( { j − p i ( M × I ) } ) , [ M × I, ∂ ( M × I )] i term in t ( M × I ) must also be zero. It remains to show thatthe signature σ ( M × I ) = 0, but this follows since M × I ≃ M and so H k ( M × I ) = H k ( M k − ) = 0 . Thus we have shown that ˜ s is an invariant of product metrics on M × N belonging to the same path-component of positive scalar curvature metrics.More generally suppose that both ( X, g ) and ( X, g ′ ) both have a Kreck-Stolz productstructure involving the same smooth product M × N and the same diffeomorphism φ. Let g ( t ) , t ∈ [0 , 1] be a path of positive scalar curvature metrics on X with g (0) = g and g (1) = g ′ . The push-forward metrics φ ∗ ( g ( t )) give a path of positive scalar curvaturemetrics on M × N beginning with a product metric g M + g N and ending with a productmetric g ′ M + g ′ N . According to Definition 2.8(b) we have ˜ s ( X, g ) = ˜ s ( M × N, φ ∗ g ) , and bythe above paragraph we have ˜ s ( M × N, φ ∗ g ) = ˜ s ( M × N, φ ∗ g ′ ) . By Definition 2.8(b) thislast term is equal to ˜ s ( X, g ′ ) , and so we deduce that ˜ s ( X, g ) = ˜ s ( X, g ′ ) . The assertion 0.3(i), that for g, g ′ as in the preceding paragraph (though not nec-essarily in the same path-component of positive scalar curvature metrics), the relativeindex is given by i ( g, g ′ ) = ˜ s ( X, g ) − ˜ s ( X, g ′ ) , now follows from equation ( † ) after 2.8(a) inconjunction with the above arguments. In detail, we have˜ s ( X, g ) − ˜ s ( X, g ′ ) = ˜ s ( M × N, φ ∗ ( g )) − ˜ s ( M × N, φ ∗ ( g ′ )) . If g ( t ) is any smooth path of metrics on X (i.e. with no condition on the scalar curvature)satisfying g (0) = g, g (1) = g ′ , then for the path φ ∗ ( g ( t )) on M × N we have i ( φ ∗ ( g ) , φ ∗ ( g ′ )) = ind D + ( M × N × I, φ ∗ ( g ( t )) + dt )= ˜ s ( M × N, φ ∗ ( g )) − ˜ s ( M × N, φ ∗ ( g ′ )) , where the second equality follows from equation ( † ) together with Lemmas 2.10 and 2.11.We also have i ( g, g ′ ) = ind D + ( X × I, g ( t ) + dt ) . Now φ × id I : ( X × I, g ( t ) + dt ) → ( M × N × I, φ ∗ ( g ( t )) + dt ) is an orientation preservingisometry. As spin structures are uniquely determined in our circumstances by the orienta-tion, we see that φ × id I is also spin structure preserving. As the index is invariant underspin structure preserving isometries, we deduce that i ( g, g ′ ) = i ( φ ∗ ( g ) , φ ∗ ( g ′ )) , i ( g, g ′ ) = ˜ s ( X, g ) − ˜ s ( X, g ′ ) as claimed.To establish assertion 0.3(ii), that the space of positive scalar curvature metrics on X has infinitely many path-components distinguished by ˜ s , we begin by noting that theargument here is analogous to that of [KS; 2.15]. By [Ca] there is a positive scalar curvaturemetric g on S k − l ) − which is extendable to a positive scalar curvature metric on a certainparallelisable bounding manifold (constructed by plumbing disc bundles). This boundingmanifold has non-zero signature and vanishing Pontrjagin classes. It follows from Lemma2.9 that ˜ s ( S k − l ) − × N, g + g N ) is a (non-zero) multiple of the (non-zero) signature ofthe bounding manifold. Consider the manifold(( M ♯S k − l ) − ♯ · · · ♯S k − l ) − p ) × N, ( g M ♯g♯ · · · ♯g ) + g N ) . This is isometric to ( M × N ; g p + g N ) for some positive scalar curvature metric g p on M .Applying Lemma 2.9 to this latter manifold, or Lemma 2.12 to the former, we obtain adifferent ˜ s -value for each p ∈ N . Hence the result in this case.More generally, consider ( X, g ) orientation preserving isometric to ( M × N, g M + g N ) . For any p ∈ N we have a diffeomorphism M × N ∼ = ( M ♯S k − l ) − ♯ · · · ♯S k − l ) − p ) × N. Composing this isometry and diffeomorphism, then pulling-back the metric ( g M ♯g♯ · · · ♯g )+ g N to X via this composition, gives a Riemannian manifold ( X, h p ) with a Kreck-Stolzproduct structure. By Definition 2.8(b) we have that ˜ s ( X, h p ) = ˜ s ( M × N, g p + g N ) , whichcompletes the argument in the general case.Finally, we turn our attention to the moduli space of positive scalar curvature metricson X . First note that the group of diffeomorphisms of X acts on the set of metrics witha Kreck-Stolz product structure, so if g is such a metric, then every representative of itsmoduli space class [ g ] has a Kreck-Stolz product structure. Moreover this action is suchthat all Kreck-Stolz structures belonging to a given orbit involve the same Riemannianproduct ( M × N, g M + g N ) . If metrics g and h on X differ by an orientation preservingdiffeomorphism, it is an immediate consequence of Definition 2.8(b) that ˜ s ( X, g ) = ˜ s ( X, h ) . However a diffeomorphism X → X could reverse orientation, and therefore fail to preservespin structures. In this case, though, we know from Lemma 2.11 (and the remark afterDefinition 2.8(b)) that the sign of ˜ s changes. Thus we conclude that | ˜ s | is invariant underthe pull-back action of Diff( X ), establishing 0.3(iii). 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