Positive Scalar Curvature due to the Cokernel of the Classifying Map
PPositive scalar curvature due to the cokernel ofthe classifying map
Thomas Schick and Vito Felice Zenobi
Abstract
This paper contributes to the classification of positive scalar curvature metricsup to bordism and up to concordance.Let M be a closed spin manifold of dimension ≥ M up to bordism in terms of the corank of the canonical map KO ∗ ( M ) → KO ∗ ( B π ( M )) , provided the rational analytic Novikov conjecture istrue for π ( M ) . The study of metrics with positive scalar curvature is nowadays the focus of a veryactive area of research. The starting point typically will be a closed spin manifold M ,and one would like to get suitable information about the possible Riemannian metricson M .Stephan Stolz introduced a long exact sequence for the systematic bordism classi-fication of metrics of positive scalar curvature. For this, one has to fix an additionalreference space X . Then this sequence is given by (ending with n = · · · (cid:47) (cid:47) R spinn + ( Γ ) ∂ (cid:47) (cid:47) Pos spinn ( X ) (cid:47) (cid:47) Ω spinn ( X ) (cid:47) (cid:47) R spinn ( Γ ) ∂ (cid:47) (cid:47) · · · (1.1)Here Ω spinn ( X ) is the usual spin cobordism group, consisting of cycles f : M → X , with M a closed n -dimensional spin manifold; Pos spinn ( X ) is the group of bordism classesof metrics of positive scalar curvature on n -dimensional closed spin manifolds withreference map to X ; finally, R spinn ( Γ ) : = R spinn ( X ) is a relative group discussed in moredetail below, known to depend only on Γ : = π ( X ) . The group structure in each of thethree cases is given by disjoint union.Because the starting point typically is a fixed manifold M , one has to make a suit-able choice of X . The standard choice here is X = B Γ with Γ = π ( M ) . Note that with X = B Γ the Stolz sequence then contains in Pos spinn ( B Γ ) information for all spin mani-folds with fundamental group Γ at once. This is the situation discussed in the majorityof all the previous work. 1 a r X i v : . [ m a t h . K T ] J un n the current article we change the paragidm a bit. We argue that, starting with M , the choice of X = M is even more canonical, and we study Pos spinn ( M ) . The usualapplications to concordance classes of metrics of positive scalar curvature on M canstill be made, and the theory is richer and more specific.A very fruitful way to get information about the Stolz sequence (for arbitrary X )uses the index theory of the spin Dirac operator. A systematic approach was givenin [11], where the authors construct a mapping of (1.1) to the analytic surgery exactsequence of Higson and Roe (where Γ = π ( X ) ) · · · (cid:47) (cid:47) R spinn + ( Γ ) ∂ (cid:47) (cid:47) Ind Γ (cid:15) (cid:15) Pos spinn ( X ) (cid:47) (cid:47) (cid:36) (cid:15) (cid:15) Ω spinn ( X ) (cid:47) (cid:47) β (cid:15) (cid:15) R spinn ( Γ ) ∂ (cid:47) (cid:47) Ind Γ (cid:15) (cid:15) · · ·· · · µ Γ X (cid:47) (cid:47) K n + ( C ∗ Γ ) (cid:47) (cid:47) S Γ n ( (cid:101) X ) (cid:47) (cid:47) K n ( X ) µ Γ X (cid:47) (cid:47) K n ( C ∗ Γ ) (cid:47) (cid:47) · · · . (1.2)A successful strategy for detecting non-trivial elements in Pos spinn ( X ) goes as fol-lows: if one can construct a cycle ξ for R spinn + ( Γ ) such that Ind Γ ( ξ ) is in the cokernel of µ Γ X , then ∂ ( ξ ) is non-zero in Pos spinn ( X ) because its image through (cid:36) does not vanish.Indeed, this is the line followed by Weinberger and Yu in [19], where the authorsdefine the so-called finite part of the K-theory of the maximal group C*-algebra, whichis proven to lie in the cokernel of the assembly map. Along with this they give theconcrete construction of elements in R spinn + ( Γ ) whose higher index belongs to this finitepart. Xie, Yu and Zeidler in [21] have systematized those constructions and correctedsome mistakes, giving a more exhaustive description of the images of the vertical ar-rows in (1.2). These are complemented by a long line of results which instead makeuse of higher numerical invariants, such as [1, 8], where higher η -invariants are used,or [12], whereCheeger-Gromov l - (cid:36) -invariants play an important role.Another example of how K-theory methods could improve those which use numer-ical invariants can be appreciated by comparing [14] with [9], where η -invariants onend-periodic ends are used in order to study positive scalar curvature metrics on evendimensional manifolds.All the work described so far uses fundamentally that the group Γ contains non-trivial torsion. In particular, as it is explicitly explained in [21], the ultimate sourceof those constructions is the difference between E Γ , the classifying space for properactions, and E Γ , the classifying space for proper and free actions.Now, if Γ is torsion-free, then E Γ and E Γ coincide. Therefore one has to find analternative source for non-trivial elements in R spinn + ( Γ ) whose higher index lies into thecokernel of the assembly map.The main method of this paper is to use the homological difference between M and B Γ for this purpose. More generally, given X (which could be M ) with a classifyingmap u : X → B Γ such that the homological difference between them is rich, we canconstruct non-trivial elements in R spinn + ( Γ ) . Although our main motivation was to ob-tain results for torsion-free fundamental groups, our constructions work for arbitrary Γ . In particular, we prove the following result.2 heorem 1.3. Let X be a finite connected CW-complex and n ≥ . Let us assume that therational Novikov Conjecture holds for Γ : = π ( X ) , i.e. the assembly map K ∗ ( B Γ ) ⊗ Q → K ∗ ( C ∗ Γ ) ⊗ Q is injective. Setk : = dim coker (cid:77) j ≥ H n + − j ( X ; Q ) u ∗ −→ (cid:77) j ≥ H n + − j ( B Γ ; Q ) , then rk Pos spinn ( X ) ≥ k .By standard surgery techniques we can refine the previous result to the followingone for a specifc manifold M . Theorem 1.4.
Let X, n, and Γ be as in Theorem 1.3. Assume that there exists a cycle in Pos spin n ( X ) , given by ( f : M → X , g ) such that f is 2-connected (i.e. inducing an isomorphismfor π and π and a surjection for π ).Then there are metrics with positive scalar curvature g , g , . . . , g k on M, together with thefixed map f : M → X, which(1) span an affine lattice of rank k in the abelian group
Pos spin n ( X ) and hence an affince spaceof dimension k in Pos spin n ( X ) ⊗ Q ;(2) in particular, they span an affine lattice of rank k in Pos spin n ( M ) (with reference map theidentity);(3) in particular, they span an affine lattice of rank k of concordance classes of positive scalarcurvature metrics on M. Perhaps the first result which uses index methods to classify metrics of positivescalar curvature is obtained by Carr [2], where infinitely many concordance classesof metrics with positive scalar curvature are constructed even on simply connectedmanifolds M like the sphere (of the right dimension). This is different in spirit to ourresult: we prove in Remark 3.10 that the classes of Carr are all equal in Pos spin n ( M ) ,i.e. although they are not concordant, they are all bordant.Recently, Ebert and Randal-Williams in [4] developed a very sophisticated bordismcategory approach to study R + ( M ) , the space of the metrics with positive scalar curva-ture on M . [4, Theorem C] implies that, if M has even dimension 2 n , the fundamentalgroup Γ verifies rationally the Baum-Connes conjecture and its homological dimensionis less or equal to 2 n +
1, then the so-called index difference map is a rational surjectionof π ( R + ( M )) onto KO n + ( C ∗ Γ ) .In our results, we only assume the rational injectivity instead of bijectivity of theBaum-Connes assembly map for Γ and, as remarked in comparison with Carr, we ob-tain metrics which are not only non-isotopic, but also non-bordant. On the other hand,in [4] the authors are mainly is interested in higher homotopy groups.Finally, we provide a detailed and pedestrian proof how to pass from a bordism W F −→ X between M f −→ X and M f −→ X to a bordism W (cid:48) F (cid:48) −→ X with same ends3hich we call Gromov-Lawson admissible , meaning that it is built from M by attachinghandles of codimension ≥
3, provided that f is 2-connected. This is certainly a wellknown and heavily used result, but doesn’t seem treated well in a pedestrian way withall details, which we try to provide here.The paper is organized as follows:• In Section 2 we prove Theorem 1.3, which gives a lower bound for the rank ofPos spinn ( X ) in term of the difference between X and B Γ .• In Section 3 we prove Theorem 1.4, which refine Theorem 1.3 to a result aboutconcordance classes. In particular we give details how bordisms can be madeGromov-Lawson admissible in the sense mentioned above. Acknowledgements
The authors thank the German Science Foundation and its prior-ity program “Geometry at Infinity” for partial support.
In [11, Section 5] Piazza and Schick construct a map from the Stolz exact sequence tothe Higson-Roe exact sequence (see also [20, 23] for different approaches). Instead ofworking with complex C ∗ -algebras as in [11], one can without extra effort adapt thisconstruction to the setting of real C ∗ -algebras (compare [22]). All of the constructionsare natural. As a result, for a connected CW-complex X with Γ = π ( X ) and classi-fying map u : X → B Γ for its universal covering we obtain the following commutingdiagram of Stolz exact sequences (cid:47) (cid:47) Pos spinn + ( X ) (cid:47) (cid:47) u ∗ (cid:15) (cid:15) Ω spinn + ( X ) j X (cid:47) (cid:47) u ∗ (cid:15) (cid:15) R spinn + ( X ) ∂ X (cid:47) (cid:47) ∼ = (cid:15) (cid:15) Pos spinn ( X ) (cid:47) (cid:47) u ∗ (cid:15) (cid:15) (cid:47) (cid:47) Pos spinn + ( B Γ ) (cid:47) (cid:47) Ω spinn + ( B Γ ) j (cid:47) (cid:47) R spinn + ( B Γ ) ∂ (cid:47) (cid:47) Pos spinn ( B Γ ) (cid:47) (cid:47) (2.1)which is mapped to the corresponding diagram of Higson-Roe sequences (cid:47) (cid:47) S O Γ n + ( (cid:101) X ) (cid:47) (cid:47) u ∗ (cid:15) (cid:15) KO n + ( X ) Ind Γ X (cid:47) (cid:47) u ∗ (cid:15) (cid:15) KO n + ( C ∗ Γ ) ι X (cid:47) (cid:47) = (cid:15) (cid:15) S O Γ n ( (cid:101) X ) (cid:47) (cid:47) u ∗ (cid:15) (cid:15) (cid:47) (cid:47) S O Γ n + (cid:47) (cid:47) KO n + ( B Γ ) Ind Γ (cid:47) (cid:47) KO n + ( C ∗ Γ ) ι (cid:47) (cid:47) S O Γ n (cid:47) (cid:47) (2.2)Here, we set S O Γ n ( (cid:101) X ) : = KO n ( D ∗ R ( (cid:101) X ) Γ ) , the K-theory of the Roe’s D ∗ -algebra. More-over, the universal analytic structure group S O Γ n is the limit of S O Γ n ( Z ) over all Γ -compact subspaces Z of E Γ .Recall that the Pontrjagin character Ph : KO ∗ ( X ) → (cid:76) j ∈ Z H ∗ + j ( X ; Q ) is definedas the composition of the complexification map in K-homology KO ∗ ( X ) ⊗ C −−→ K ∗ ( X ) Ch : K ∗ ( X ) → (cid:76) k ∈ Z H ∗ + k ( X ; Q ) . It so happens that Ph takesvalues only in the subgroup (cid:76) j ∈ Z H ∗ + j ( X ; Q ) and is a rational isomorphism. Lemma 2.3.
Let X be a space and n ≥ . Then the composition Ω spinn + ( B Γ ) ⊗ Q β −→ KO n + ( B Γ ) ⊗ Q Ph −→ (cid:77) j ≥ H n + − j ( B Γ ; Q ) , which assigns to a cobordism class [ M f −→ X ] the class f ∗ ([ M ] ∩ ˆ A ( M )) , is surjective.Proof. If x ∈ H n − j ( X ; Q ) then by [21, Proposition 3.1] there exists a spin manifold M of dimension n − j and a map f : M → X such that a non-zero multiple of x is thePontrjagin character of β ([ f : M → X ]) = f ∗ [ / D M ] ∈ KO n − j ( X ) ⊗ Q . Finally, recallthe Kummer surface V , a spin manifold whose index generates KO ( ∗ ) ⊗ Q . Observethat the cartesian product of f : M → X with V j → ∗ is n -dimensional with a map to X such that the push-forward of its Pontrjagin character is still a non-zero multiple x ,thanks to the mulitplicativity of the Pontrjagin character.We are now able to prove the first main result of this paper. Proof of Theorem 1.3.
First recall that the natural map KO ∗ ( B Γ ) ⊗ Q → KO Γ ∗ ( E Γ ) ⊗ Q isinjective. Secondly, rationally the real and the complex Analytic Novikov conjectureare equivalent, compare e.g. [17]. Therefore, if the Strong Novikov Conjecture holdsfor Γ , it follows that Ind Γ : KO ∗ ( B Γ ) ⊗ Q → KO ∗ ( C ∗ Γ ) ⊗ Q is injective.By using Lemma 2.3 and the fact that k = dim coker (cid:77) j ≥ H n + − j ( X ; Q ) → (cid:77) j ≥ H n + − j ( B Γ ; Q ) ,one can pick x , . . . , x k ∈ Ω spin ( B Γ ) such that their images in (cid:76) j ≥ H n + − j ( B Γ ; Q ) span a k -dimensional subspace modulo the image of (cid:76) j ≥ H n + − j ( X ; Q ) .Consider the subspace of Ω spinn + ( B Γ ) ⊗ Q generated by x , . . . , x k . We want to provethat the following composition Ω spinn + ( B Γ ) ⊗ Q j (cid:47) (cid:47) R spinn + ( B Γ ) ⊗ Q R spinn + ( X ) ⊗ Q ∼ = (cid:111) (cid:111) ∂ X (cid:47) (cid:47) Pos spinn ( X ) ⊗ Q is injective when we restrict it to the subspace W generated by x , . . . , x k . The injectivityof the first arrow j on W is given by the commutativity of the following square Ω spinn + ( B Γ ) ⊗ Q j (cid:47) (cid:47) β (cid:15) (cid:15) R spinn + ( B Γ ) ⊗ Q Ind Γ (cid:15) (cid:15) KO n + ( B Γ ) ⊗ Q µ Γ B Γ (cid:47) (cid:47) KO n + ( C ∗ R Γ ) ⊗ Q and by the assumption that Γ verifies the Strong Novikov Conjecture, so that µ Γ B Γ isrationally injective. 5oncerning the third arrow ∂ X , we know that the subspace W generated by x , . . . , x k has trivial intersection with the image of u ∗ : Ω spinn + ( X ) ⊗ Q → Ω spinn + ( B Γ ) ⊗ Q , becausethe images of the x j in the homology of B Γ are linearly independent modulo the imageof the homology of X . By the commutativity of (2.1), this implies that the image of W in R spinn + ( X ) ⊗ Q has trivial intersection with the image of j X , hence, by exactness of(1.1), the restriction of ∂ X to W is injective. The basis of most constructions of positive scalar curvature metrics is the surgery the-orem of Gromov and Lawson, see [6] or [3] for full details. It says that, given a bordism W from M to M such that W is obtained from M by surgeries of codimension ≥ M can be extended to a metric of pos-itive scalar curvature on W with product structure near the boundary. In particular,one obtains a “transported” positive scalar curvature metric on M . We call bordismssatisfying the codimension condition Gromov-Lawson admissible .In the following, we discuss the details how Gromov-Lawson admissible bordism W can be obtained, focusing on the not quite so obvious question why finitely manysurgery steps suffice. The result appears also e.g. as [15, Theorem 2.2] where the finite-ness questions are not discussed or in a much more general setup in [7, Appendix 2]. Proposition 3.1.
Let F : W → X be a spin bordism with reference map between f : M → Xand f : M → X. Assume M is connected, f is -connected and n : = dim ( M ) ≥ . Thenwe can change W in the interior to F (cid:48) : W (cid:48) → X such that W (cid:48) is a Gromov-Lawson admissiblebordism from M to M .Proof. By standard results from surgery theory (compare [15, Proof of Theorem 2.2]),the desired bordism W (cid:48) is Gromov-Lawson admissible if the inclusion M (cid:44) → W (cid:48) is2-connected. We perform surgeries in the interior of W to achieve this. Connectedness . As M is connected, we have to modify W so that it becomes con-nected. This is achieved by (interior) connected sum of the finitely many componentsof W (cid:48) . Because also X is path-connected, the map F : W → X can be extended over theconnected sum of its components. Isomorphism on π . The composition π ( M ) → π ( W ) → π ( X ) is an isomor-phism, therefore the map π ( W ) → π ( X ) is surjective. We want to modify W withfurther surgeries which eliminate its kernel, then π ( W (cid:48) ) → π ( X ) and consequentlyalso π ( M ) → π ( W (cid:48) ) is an isomorphism. As π ( X ) is finitely presented and π ( W ) is finitely generated, this kernel is finitely generated as a normal subgroup, see Lemma3.2 below. So we have to do a finite number of surgeries along embedded circles (in theinterior of W ). Because W is oriented, these have automatically trivial normal bundle,so surgery is possible. The fact that we kill the kernel of π ( F ) means precisely that F can be extended over the disks and thus over the new bordism, which we continue todenote W by small abuse of notation. Epimorphism on π . We finally have to perform surgeries so that ι ∗ : π ( M ) → π ( W ) becomes surjective, where ι : M (cid:44) → W is the inclusion. We follow the proof of618, Lemma 5.6] adapted to our situation. Since M and W are compact manifolds, therelative 2-skeleton ( W , M ) ( ) of W is obtained by attaching a finite number of 2-cellsto M .Since after the previous step ι induces an isomorphism of the fundamental groups,these 2-cells are glued to M along contractible loops. Therefore the relative 2-skeleton ( W , M ) ( ) is homotopy equivalent to M ∨ ( (cid:87) j ∈ J S ) and the cokernel of ι i ∗ : π ( M ) → π ( W i ) is finitely generated by these spheres x j , j ∈ J which we can assume embeddedbecause n ≥
5. Because W is spin, the normal bundle of these embedded spheres isautomatically trivial and surgery along them is possible.Since ( f ) ∗ : π ( M ) → π ( X ) is surjective, there exist elements { y j ∈ π ( M ) } j ∈ J such that ( f ) ∗ ( y j ) = F ∗ ( x j ) . It follows that the alternative generators of the cokernelgiven by ι ∗ ( y − j ) x j satisfy F ∗ ( ι i ∗ ( y − j ) x j ) = ( ι i ◦ F ) ∗ ( y − j ) F ∗ ( x j ) = ( f ) ∗ ( y − j ) F ∗ ( x j ) = ∀ j .Because of this, we can extend F over the surgeries along the alternative generators ι ∗ ( y − j ) x j and we obtain the desired cobordism F (cid:48) : W (cid:48) → X such that the inclusion of M into W (cid:48) is a 2-equivalence. Lemma 3.2.
Let α : Γ (cid:48) → Γ be a surjective group homomorphism between finitely generatedgroups. Assume in addition that Γ is finitely presented. Then the kernel of α is finitely generatedas a normal subgroup of Γ (cid:48) .Proof. Let us fix a finite presentation Γ = (cid:104) x , . . . , x h ; r , . . . , r k (cid:105) , where the relations r j are given by fixed words w j ( x ± , . . . , x ± h ) . Let us fix also a finite set of gener-ators { y , . . . , y n } for Γ (cid:48) . Pick a , . . . , a h ∈ Γ (cid:48) such that α ( a j ) = x j for all j and set w (cid:48) l ( x ± , . . . , x ± h ) : = α ( y l ) . Then it follows that { w ( a ± , . . . , a ± h ) , . . . , w k ( x ± , . . . , x ± h ) , y − w (cid:48) ( x ± , . . . , x ± h ) , . . . , y − n w (cid:48) n ( x ± , . . . , x ± h ) } is a finite set of generators as a normal subgroup for ker α .Now we are ready for the proof of the main result of this section. Proof of Theorem 1.4.
Let us consider again the situation of Theorem 1.3. Then we haveclasses x = [ M f −→ X , h ] , . . . , x k = [ M k f k −→ X , h k ] in Pos spin n ( X ) which span a sub-group of rank k , but are trivial when mapped to Ω spin n ( X ) (and a fortiori to Ω spin n ( B Γ ) ),in particular they are null-bordant. Let us pick such null-bordisms F i : Y i → X , so that M i is the boundary of Y i and f i is the restriction of F i to the boundary.For i ∈ {
1, . . . , k } , the disjoint union of M and M i is spin bordant to M , with bor-dism G i : W i → X given by the disjoint union of f × id : M × [
0, 1 ] → X and F i : Y i → X .By Proposition 3.1 we can modify these bordisms and then assume that W i is Gromov-Lawson admissible.Now we can use the Gromov-Lawson surgery theorem to “push” the given metrics g (cid:113) h i of positive scalar curvature from M (cid:113) M i through the new bordism to positivescalar curvature metrics g i on M . This finishes the proof.7 emark 3.3. Denote by P + ( M ) the set of concordance classes of metrics with positivescalar curvature on an n -dimensional closed spin manifold M . In the third point ofTheorem 1.4, we speak about a lattice of concordance classes, a notion which needsmore structure then being just a set on P + ( M ) to make sense. In the proof of [18,Theorem 5.2], in order to construct a free and transitive action of R spinn + ( B Γ ) on P + ( M ) ,Stolz defines a “difference” map i : P + ( M ) × P + ( M ) → R spinn + ( B Γ ) such that• i ( g , g ) = i ( g , g (cid:48) ) + i ( g (cid:48) , g (cid:48)(cid:48) ) = i ( g , g (cid:48)(cid:48) ) for all g , g (cid:48) , g (cid:48)(cid:48) ∈ P + ( M ) ;• the map i g : P + ( M ) → R spinn + ( B Γ ) , which sends g (cid:48) to i ( g , g (cid:48) ) is bijective for all g ∈ P + ( M ) .This induces on P + ( M ) the structure of an R spinn + ( B Γ ) -torsor, or the structure of affinespace modelled on R spinn + ( B Γ ) . After picking any point g of P + ( M ) as the identity, P + ( M ) acquires a group structure isomorphic to R spinn + ( B Γ ) . But this is non-canonicalas it depends on g . This group structure seems only useful if there is a preferred g (e.g. one which bounds a metric of positive scalar curvature, as the standard metric on S n ). This kind of structure is studied (and improved to an H-space structure on thespace of metrics of positive scalar curvature) in [5].Let us spell out the special case X = M of Theorem 1.4: Corollary 3.4.
Let ( M , g ) be an n-dimensional connected spin manifold of positive scalarcurvature with π ( M ) = Γ , n ≥ . Let u : M → B Γ be an isomorphism on fundamentalgroups. Set k : = ∑ ≤ j ≤ n + j − n ≡ ( mod 4 ) dim (cid:0) coker ( u ∗ : H j ( M ; Q ) → H j ( B Γ ; Q )) (cid:1) . Then M admits metrics g , . . . , g k of positive scalar curvature such that the elements ( M , g ) , ( M , g ) , . . . , ( M , g k ) span a k-dimensional affine subspace of Pos spin n ( M ) ⊗ Q . In particular,these metrics form a k-dimensional lattice of non-concordant metrics of positive scalar curvatureon M. Example 3.5.
Let ( M , g ) be a connected n -dimensional spin manifold of positive scalarcurvature such that dim ( H n + ( B π ( M ) ; Q )) = k . Then the cokernel of the map in-duced by the inclusion in homology of degree n + H n + ( B π ( M )) , as H n + ( M ) = k metrics g , . . . , g k of positive scalar curvatureon M which span together with g an affine space of rank k in Pos spin n ( M ) and, in par-ticular, give rise to a lattice of rank k of concordance classes of positive scalar curvaturemetrics on M .For example, if π ( M ) ∼ = Z N then we have dim ( H n + ( Z N ; Q )) = ( Nn + ) .8 xample 3.6. Assume that n ≥ Γ = (cid:104) x . . . , x k ; r , . . . , r h (cid:105) is finitely presented.Then there exists a closed spin manifold M of dimension n with fundamental group Γ which admits a metric g of positive scalar curvature.Indeed, take the wedge of k circles and, for each relation r i , attach a two cell. Denoteby X this 2-dimensional CW-complex. Finally embed X into R n + and consider a tubu-lar neighbourhood N of X . Then M : = ∂ N is an n dimensional spin manifold withfundamental group Γ . Observe that N is a spin B Γ null-bordism for M or, after cuttingout a disk, a B Γ bordism to S n . By Proposition 3.1 we can assume that this bordismis Gromov-Lawson admissible. By the Gromov-Lawson surgery theorem therefore M admits a metric g of positive scalar curvature.In addition, observe that for the manifold we constructed we have a factorization u : M → N → B Γ and N is homotopy equivalent to a 2-dimensional CW-complex.Therefore im ( u ∗ : H k ( M ) → H k ( B Γ )) = { } ∀ k ≥ k = ∑ ≤ j ≤ n + j ≡ n + ( mod 4 ) dim ( H j ( B Γ ; Q )) we find metrics g , . . . , g k of positive scalar curvature on M such that the ( M , g i ) together with ( M , g ) span a k -dimensional affine subspace of Pos spin n ( M ) ⊗ Q . Remark 3.7.
In special situations, the different metrics constructed in Theorem 1.4,Corollary 3.4 and the examples remain different also in the moduli space of Rieman-nian metrics of positive scalar curvature on M , the quotient by the action of the diffeo-morphisms group. This is worked out in detail in [13]. As indicated in the introduction,this is based on the use of higher numeric rho invariants, whose behavior under theaction of the diffeomorphism group can be controlled.To make this work despite the fact that diffeomorphisms give rise to an outer ac-tion on the fundamental group, in [13] situations are considered where the cokernelof u ∗ : H j ( M ; Q ) → H j ( B Γ ; Q ) is one-dimensional. The following construction showsthat also these abound. Example 3.8.
Let Γ be a finitely presented group, n ≥ ≤ j < j < · · · < j r ≤ n − ≤ dim ( H j k ( B Γ ; Q )) < ∞ .Then we find a closed spin manifold N of dimension n of positive scalar curvaturewith fundamental group Γ and classifying map u : N → B Γ such that the cokernel of u ∗ : H j k ( N ; Q ) → H j k ( B Γ ; Q ) is 1-dimensional for all k =
1, . . . , r . Proof.
We start with the manifold M of Example 3.6. The orientation map Ω spin ∗ ( Y ) ⊗ Q → H ∗ ( Y ; Q ) is well known to be surjective for every CW-complex Y . When choosinga basis a , . . . , a d of H j k ( B Γ ; Q ) we can therefore assume without loss of generality thatthere are j k -dimensional spin manifolds and maps f j : M j → B Γ representing a j for j =
1, . . . , d .The connected sum of M with all but one M j × S n − j k will then be a manifold withpositive scalar curvature and map f j ◦ pr M j to B Γ such that the induced map on H j k hascodimension 1, while the image in homology is unchanged in all other degrees. Onestill has to adjust the fundamental group to obtain the desired manifold N by surgerieskilling the kernel of the map on π . This is possible by Proposition 3.1. This will cor-rect the fundamental group and change the homology only in degrees 1, 2, n −
2, and9 −
1. The manifold N obtained by doing this in all relevant degrees therefore satis-fies the required conditions, except potentially for the case j k = n −
2. In this criticalcase, for easy of presentation we assume that N is obtained from M by surgery alongan embedded S × D n − . Then the pair sequence implies that we inclusion inducesan isomorphism H n − ( M \ S × D n − ) ∼ = −→ H n − ( N ) . On the other hand, the pair se-quence also shows that H n − ( M \ S × D n − ) → H n − ( M ) is surjective. As all this iscompatible with the map to B Γ , we see that the image in H n − ( B Γ ) is unchanged if wepass from M to N and the claim follows. Remark 3.9.
Consider the map i from Remark 3.3. If we compose it with the map Ind Γ in (1.2), it is easy to see that we obtain the mapInd diff Γ : P + ( M ) × P + ( M ) → KO n + ( C ∗ Γ ) used in [4, Section 5.3]. More precisely, in [4] the map is defined on the space of isotopyclasses of metrics with positive scalar curvature, but it descends to P + ( M ) .It is straightforward to see that, rationally, the affine subspace generated by the lat-tice of P + ( M ) in Theorem 1.4, ( ) is mapped surjectively onto the image of the rationalassembly map KO n + ( B Γ ) ⊗ Q → KO n + ( C ∗ Γ ) ⊗ Q . Remark 3.10.
As a predecessor construction of concordance classes which does notmake use of non-trivial torsion, let us recall the construction of Carr.First, consider the sphere S n − . Carr takes a 2-connected 4 n dimensional spin man-ifold B with ˆ A ( B ) = W from S n − to S n − . Positive scalar curvature surgery produces a metric of positive scalar curvatureon W starting with the canonical metric on S n − and ending with a non-concordantnew metric of positive scalar curvature on S n − .However, these metrics are equal in Pos spin4 n − ( S n − ) . To see this, we have just to con-struct the reference map F : W → S n − which restricts to the identity on the boundarycomponents. For this, choose a path which is a clean embedding of the closed intervalinto W , joining two points in the two boundary spheres. Choose then a tubular neigh-bourhood of this one dimensional submanifold of W , which is necessarily trivial. Nowa trivialization of the tubular neighbourhood defines a collapse map from W to S n − ,whose restriction to the boundary components is homotopic to the identity. Puttingthese homotopies on collar neighbourhoods of the boundary components, we obtainthe desired map F .More generally, given an arbitrary closed spin manifold M of dimension 4 n − g , Carr makes a connected sum of M × [
0, 1 ] with W along a path parallel to the previously chosen one, to obtain a psc bordism V from ( M , g ) to ( M , g (cid:48) ) . These two metrics have non-zero index difference and therefore theyare not concordant. Nevertheless, they are equal in Pos spin4 n − ( M ) . We obtain the desiredreference map from V to M by connected sum of the previous map with the projectionfrom M × [
0, 1 ] to M . 10 eferences [1] Boris Botvinnik and Peter B. Gilkey, The eta invariant and metrics of positive scalar curvature , Math.Ann. (1995), no. 3, 507–517, DOI 10.1007/BF01444505. MR1339924[2] Rodney Carr,
Construction of manifolds of positive scalar curvature , Trans. Amer. Math. Soc. (1988),no. 1, 63–74, DOI 10.2307/2000751. MR936805[3] Johannes Ebert and Georg Frenck,
The Gromov-Lawson-Chernysh surgery theorem , 2018.arXiv:1807.06311.[4] Johannes Ebert and Oscar Randal-Williams,
Infinite loop spaces and positive scalar curvature in the pres-ence of a fundamental group , Geom. Topol. (2019), no. 3, 1549–1610, DOI 10.2140/gt.2019.23.1549.MR3956897[5] Georg Frenck, H-Space structures on spaces of metrics of positive scalar curvature , 2019.arXiv:2004.01033.[6] Mikhael Gromov and H. Blaine Lawson Jr.,
The classification of simply connected manifolds of positivescalar curvature , Ann. of Math. (2) (1980), no. 3, 423–434, DOI 10.2307/1971103. MR577131[7] Fabian Hebestreit and Michael Joachim,
Twisted spin cobordism and positive scalar curvature , J. Topol. (2020), no. 1, 1–58, DOI 10.1112/topo.12122. MR3999671[8] Eric Leichtnam and Paolo Piazza, On higher eta-invariants and metrics of positive scalar curvature , K -Theory (2001), no. 4, 341–359, DOI 10.1023/A:1014079307698. MR1885126[9] Demetre Kazaras, Daniel Ruberman, and Nikolai Saveliev, On positive scalar curvature cobordismsand the conformal Laplacian on end-periodic manifolds , 2019. to appear in Communications in Analysisand Geometry.[10] Paolo Piazza and Thomas Schick,
Groups with torsion, bordism and rho invariants , Pacific J. Math. (2007), no. 2, 355–378, DOI 10.2140/pjm.2007.232.355. MR2366359[11] ,
Rho-classes, index theory and Stolz’ positive scalar curvature sequence , J. Topol. (2014), no. 4,965–1004, DOI 10.1112/jtopol/jtt048. MR3286895[12] , Groups with torsion, bordism and rho invariants , Pacific J. Math. (2007), no. 2, 355–378,DOI 10.2140/pjm.2007.232.355. MR2366359[13] Paolo Piazza, Thomas Schick, and Vito Felice Zenobi,
Higher rho numbers and the mapping of analyticsurgery to homology , 2019. arXiv:1905.11861.[14] ,
On positive scalar curvature bordism , 2019. To appear in Communications in Analysis andGeometry.[15] J. Rosenberg, C ∗ -algebras, positive scalar curvature and the Novikov conjecture. II , Geometric methodsin operator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech.,Harlow, 1986, pp. 341–374. MR866507[16] Jonathan Rosenberg and Stephan Stolz, Metrics of positive scalar curvature and connections withsurgery , Surveys on surgery theory, Vol. 2, Ann. of Math. Stud., vol. 149, Princeton Univ. Press,Princeton, NJ, 2001, pp. 353–386. MR1818778[17] Thomas Schick,
Real versus complex K-theory using Kasparov’s bivariant KK-theory , Algebr. Geom.Topol. (2004), 333–346, DOI 10.2140/agt.2004.4.333. MR2077669[18] Stephan Stolz, Concordance classes of positive scalar curvature metrics
Finite part of operator K-theory for groups finitely embeddable intoHilbert space and the degree of nonrigidity of manifolds , Geom. Topol. (2015), no. 5, 2767–2799, DOI10.2140/gt.2015.19.2767. MR3416114
20] Zhizhang Xie and Guoliang Yu,
Positive scalar curvature, higher rho invariants and localization algebras ,Adv. Math. (2014), 823–866, DOI 10.1016/j.aim.2014.06.001. MR3228443[21] Zhizhang Xie, Guoliang Yu, and Rudolf Zeidler,
On the range of the relative higher index and the higherrho-invariant for positive scalar curvature , 2017. arXiv:1712.03722v2.[22] Rudolf Zeidler,
Positive scalar curvature and product formulas for secondary index invariants , J. Topol. (2016), no. 3, 687–724, DOI 10.1112/jtopol/jtw005. MR3551834[23] Vito Felice Zenobi, Adiabatic groupoid and secondary invariants in K-theory , Adv. Math. (2019),940–1001, DOI 10.1016/j.aim.2019.03.003. MR3922452(2019),940–1001, DOI 10.1016/j.aim.2019.03.003. MR3922452