Positive scalar curvature due to the cokernel of the classifying map

This paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold of dimension $\ge 5$ which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics on $M$ up to bordism in terms of the corank of the canonical map $KO_*(M)\to KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for $\pi_1(M)$.


Introduction
The study of metrics with positive scalar curvature is nowadays the focus of a very active 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 metrics on M.
Stephan Stolz introduced a long exact sequence for the systematic bordism classification of metrics of positive scalar curvature. For this, one has to fix an additional reference space X. Then this sequence is given by (ending with n = 5 at the right) · · · / / R spin n+1 (Γ) ∂ / / Pos spin n (X) / / Ω spin n (X) / / R spin n (Γ) ∂ / / · · · (1.1) Here Ω spin n (X) is the usual spin cobordism group, consisting of cycles f : M → X, with M a closed n-dimensional spin manifold; Pos spin n (X) is the group of bordism classes of metrics of positive scalar curvature on n-dimensional closed spin manifolds with reference map to X; finally, R spin n (Γ) := R spin n (X) is a relative group discussed in more detail below, known to depend only on Γ := π 1 (X). The group structure in each of the three cases is given by disjoint union.
Because the starting point typically is a fixed manifold M, one has to make a suitable choice of X. The standard choice here is X = BΓ with Γ = π 1 (M). Note that with X = BΓ the Stolz sequence then contains in Pos spin n (BΓ) information for all spin manifolds with fundamental group Γ at once. This is the situation discussed in the majority of all the previous work.
In 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 spin n (M). The usual applications to concordance classes of metrics of positive scalar curvature on M can still 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 given in [11], where the authors construct a mapping of (1.1) to the analytic surgery exact sequence of Higson and Roe (where Γ = π 1 (X)) · · · / / R spin n+1 (Γ) (1.2) A successful strategy for detecting non-trivial elements in Pos spin n (X) goes as follows: if one can construct a cycle ξ for R spin n+1 (Γ) such that Ind Γ (ξ) is in the cokernel of µ Γ X , then ∂(ξ) is non-zero in Pos spin n (X) because its image through does not vanish. Indeed, this is the line followed by Weinberger and Yu in [19], where the authors define the so-called finite part of the K-theory of the maximal group C*-algebra, which is proven to lie in the cokernel of the assembly map. Along with this they give the concrete construction of elements in R spin n+1 (Γ) whose higher index belongs to this finite part. Xie, Yu and Zeidler in [21] have systematized those constructions and corrected some mistakes, giving a more exhaustive description of the images of the vertical arrows in (1.2). These are complemented by a long line of results which instead make use of higher numerical invariants, such as [1,8], where higher η-invariants are used, or [12], whereCheeger-Gromov l 2 --invariants play an important role.
Another example of how K-theory methods could improve those which use numerical invariants can be appreciated by comparing [14] with [9], where η-invariants on end-periodic ends are used in order to study positive scalar curvature metrics on even dimensional manifolds.
All the work described so far uses fundamentally that the group Γ contains nontrivial torsion. In particular, as it is explicitly explained in [21], the ultimate source of those constructions is the difference between EΓ, the classifying space for proper actions, and EΓ, the classifying space for proper and free actions. Now, if Γ is torsion-free, then EΓ and EΓ coincide. Therefore one has to find an alternative source for non-trivial elements in R spin n+1 (Γ) whose higher index lies into the cokernel of the assembly map.
The main method of this paper is to use the homological difference between M and BΓ for this purpose. More generally, given X (which could be M) with a classifying map u : X → BΓ such that the homological difference between them is rich, we can construct non-trivial elements in R spin n+1 (Γ). Although our main motivation was to obtain results for torsion-free fundamental groups, our constructions work for arbitrary Γ. In particular, we prove the following result. Theorem 1.3. Let X be a finite connected CW-complex and n ≥ 5. Let us assume that the rational Novikov Conjecture holds for Γ := π 1 (X), i.e. the assembly map K * (BΓ) By standard surgery techniques we can refine the previous result to the following one 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 isomorphism for π 0 and π 1 and a surjection for π 2 ).
Then there are metrics with positive scalar curvature g, g 1 , . . . , g k on M, together with the fixed 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 space of 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 the identity); (3) in particular, they span an affine lattice of rank k of concordance classes of positive scalar curvature metrics on M.
Perhaps the first result which uses index methods to classify metrics of positive scalar curvature is obtained by Carr [2], where infinitely many concordance classes of metrics with positive scalar curvature are constructed even on simply connected manifolds M like the sphere (of the right dimension). This is different in spirit to our result: 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 bordism category approach to study R + (M), the space of the metrics with positive scalar curvature on M. [4,Theorem C] implies that, if M has even dimension 2n, the fundamental group Γ verifies rationally the Baum-Connes conjecture and its homological dimension is less or equal to 2n + 1, then the so-called index difference map is a rational surjection of π 0 (R + (M)) onto KO 2n+1 (C * Γ).
In our results, we only assume the rational injectivity instead of bijectivity of the Baum-Connes assembly map for Γ and, as remarked in comparison with Carr, we obtain 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 which we call Gromov-Lawson admissible, meaning that it is built from M 0 by attaching handles of codimension ≥ 3, provided that f 1 is 2-connected. This is certainly a well known and heavily used result, but doesn't seem treated well in a pedestrian way with all 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 of Pos spin n (X) in term of the difference between X and BΓ.
• In Section 3 we prove Theorem 1.4, which refine Theorem 1.3 to a result about concordance classes. In particular we give details how bordisms can be made Gromov-Lawson admissible in the sense mentioned above.

Acknowledgements
The authors thank the German Science Foundation and its priority program "Geometry at Infinity" for partial support.

Mapping psc to analysis to detect bordism classes
In [11, Section 5] Piazza and Schick construct a map from the Stolz exact sequence to the Higson-Roe exact sequence (see also [20,23] for different approaches). Instead of working with complex C * -algebras as in [11], one can without extra effort adapt this construction to the setting of real C * -algebras (compare [22]). All of the constructions are natural. As a result, for a connected CW-complex X with Γ = π 1 (X) and classifying map u : X → BΓ for its universal covering we obtain the following commuting diagram of Stolz exact sequences which is mapped to the corresponding diagram of Higson-Roe sequences Here, we set SO Γ n ( X) := KO n (D * R ( X) Γ ), the K-theory of the Roe's D * -algebra. Moreover, the universal analytic structure group SO Γ n is the limit of SO Γ n (Z) over all Γcompact subspaces Z of EΓ.
Recall that the Pontrjagin character Ph : KO * (X) → j∈Z H * +4j (X; Q) is defined as the composition of the complexification map in K-homology KO * (X) ⊗C − − → K * (X) and the Chern character Ch : K * (X) → k∈Z H * +2k (X; Q). It so happens that Ph takes values only in the subgroup j∈Z H * +4j (X; Q) and is a rational isomorphism. Lemma 2.3. Let X be a space and n ≥ 0. Then the composition Finally, recall the Kummer surface V, a spin manifold whose index generates KO 4 ( * ) ⊗ Q. Observe that 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.
Secondly, rationally the real and the complex Analytic Novikov conjecture are equivalent, compare e.g. [17]. Therefore, if the Strong Novikov Conjecture holds for Γ, it follows that Ind Γ : By using Lemma 2.3 and the fact that one can pick x 1 , . . . , x k ∈ Ω spin (BΓ) such that their images in j≥0 H n+1−4j (BΓ; Q) span a k-dimensional subspace modulo the image of j≥0 H n+1−4j (X; Q).
Consider the subspace of Ω spin n+1 (BΓ) ⊗ Q generated by x 1 , . . . , x k . We want to prove that the following composition is injective when we restrict it to the subspace W generated by x 1 , . . . , x k . The injectivity of the first arrow j on W is given by the commutativity of the following square and by the assumption that Γ verifies the Strong Novikov Conjecture, so that µ Γ BΓ is rationally injective.
Concerning the third arrow ∂ X , we know that the subspace W generated by x 1 , . . . , x k has trivial intersection with the image of u * : Ω spin n+1 (X) ⊗ Q → Ω spin n+1 (BΓ) ⊗ Q, because the images of the x j in the homology of BΓ are linearly independent modulo the image of the homology of X. By the commutativity of (2.1), this implies that the image of W in R spin n+1 (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.

Concordance classes
The basis of most constructions of positive scalar curvature metrics is the surgery theorem of Gromov and Lawson, see [6] or [3] for full details. It says that, given a bordism W from M 0 to M 1 such that W is obtained from M 0 by surgeries of codimension ≥ 3, then a metric of positive scalar curvature on M 0 can be extended to a metric of positive scalar curvature on W with product structure near the boundary. In particular, one obtains a "transported" positive scalar curvature metric on M 1 . We call bordisms satisfying 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 many surgery steps suffice. The result appears also e.g. as [15,Theorem 2.2] where the finiteness questions are not discussed or in a much more general setup in [7, Appendix 2]. Proof. By standard results from surgery theory (compare [15, Proof of Theorem 2.2]), the desired bordism W is Gromov-Lawson admissible if the inclusion M 1 → W is 2-connected. We perform surgeries in the interior of W to achieve this.
Connectedness. As M 1 is connected, we have to modify W so that it becomes connected. This is achieved by (interior) connected sum of the finitely many components of W . Because also X is path-connected, the map F : W → X can be extended over the connected sum of its components.
Isomorphism on π 1 . The composition π 1 (M 1 ) → π 1 (W) → π 1 (X) is an isomorphism, therefore the map π 1 (W) → π 1 (X) is surjective. We want to modify W with further surgeries which eliminate its kernel, then π 1 (W ) → π 1 (X) and consequently also π 1 (M 1 ) → π 1 (W ) is an isomorphism. As π 1 (X) is finitely presented and π 1 (W) is finitely generated, this kernel is finitely generated as a normal subgroup, see Lemma 3.2 below. So we have to do a finite number of surgeries along embedded circles (in the interior 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 1 (F) means precisely that F can be extended over the disks and thus over the new bordism, which we continue to denote W by small abuse of notation.
Epimorphism on π 2 . We finally have to perform surgeries so that ι * : π 2 (M 1 ) → π 2 (W) becomes surjective, where ι : M 1 → W is the inclusion. We follow the proof of [18, Lemma 5.6] adapted to our situation. Since M 1 and W are compact manifolds, the relative 2-skeleton (W, M 1 ) (2) of W is obtained by attaching a finite number of 2-cells to 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 1 ) (2) is homotopy equivalent to M ∨ ( j∈J S 2 ) and the cokernel of ι i * : π 2 (M 1 ) → π 2 (W i ) is finitely generated by these spheres x j , j ∈ J which we can assume embedded because n ≥ 5. Because W is spin, the normal bundle of these embedded spheres is automatically trivial and surgery along them is possible.
Since ( f 1 ) * : π 2 (M 1 ) → π 2 (X) is surjective, there exist elements {y j ∈ π 2 (M 1 )} j∈J such that ( f 1 ) * (y j ) = F * (x j ). It follows that the alternative generators of the cokernel given by ι * (y −1 j )x j satisfy Because of this, we can extend F over the surgeries along the alternative generators ι * (y −1 j )x j and we obtain the desired cobordism F : W → X such that the inclusion of M 1 into W is a 2-equivalence.

Lemma 3.2.
Let α : Γ → Γ be a surjective group homomorphism between finitely generated groups. Assume in addition that Γ is finitely presented. Then the kernel of α is finitely generated as a normal subgroup of Γ .
Proof. Let us fix a finite presentation Γ = x 1 , . . . , x h ; r 1 , . . . , r k , where the relations r j are given by fixed words w j (x ±1 1 , . . . , x ±1 h ). Let us fix also a finite set of generators {y 1 , . . . , y n } for Γ . Pick a 1 , . . . , a h ∈ Γ such that α(a j ) = x j for all j and set w l (x ±1 1 , . . . , x ±1 h ) := α(y l ). Then it follows that in Pos spin n (X) which span a subgroup of rank k, but are trivial when mapped to Ω spin n (X) (and a fortiori to Ω spin n (BΓ)), 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 bordism 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 h i of positive scalar curvature from M M i through the new bordism to positive scalar curvature metrics g i on M. This finishes the proof. Remark 3.3. Denote by P + (M) the set of concordance classes of metrics with positive scalar curvature on an n-dimensional closed spin manifold M. In the third point of Theorem 1.4, we speak about a lattice of concordance classes, a notion which needs more 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 spin n+1 (BΓ) on P + (M), Stolz defines a "difference" map such that • i(g, g) = 0 and i(g, g ) + i(g , g ) = i(g, g ) for all g, g , g ∈ P + (M); • the map i g : P + (M) → R spin n+1 (BΓ), which sends g to i(g, g ) is bijective for all g ∈ P + (M).
This induces on P + (M) the structure of an R spin n+1 (BΓ)-torsor, or the structure of affine space modelled on R spin n+1 (BΓ). After picking any point g 0 of P + (M) as the identity, P + (M) acquires a group structure isomorphic to R spin n+1 (BΓ). But this is non-canonical as it depends on g 0 . This group structure seems only useful if there is a preferred g 0 (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 the space of metrics of positive scalar curvature) in [5].
Let us spell out the special case X = M of Theorem 1.4:  For example, if π 1 (M) ∼ = Z N then we have dim(H n+1 (Z N ; Q)) = ( N n+1 ). Example 3.6. Assume that n ≥ 5 and Γ = x 1 . . . , x k ; r 1 , . . . , r h 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. Denote by X this 2-dimensional CW-complex. Finally embed X into R n+1 and consider a tubular neighbourhood N of X. Then M := ∂N is an n dimensional spin manifold with fundamental group Γ. Observe that N is a spin BΓ null-bordism for M or, after cutting out a disk, a BΓ bordism to S n . By Proposition 3.1 we can assume that this bordism is 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 → BΓ and N is homotopy equivalent to a 2-dimensional CW-complex. Therefore im(u * : By Corollary 3.4, with k = ∑ 3≤j≤n+1, j≡n+1 (mod 4) dim(H j (BΓ; Q)) we find metrics g 1 , . . . , 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 Riemannian metrics of positive scalar curvature on M, the quotient by the action of the diffeomorphisms 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 the action of the diffeomorphism group can be controlled.
To make this work despite the fact that diffeomorphisms give rise to an outer action on the fundamental group, in [13] situations are considered where the cokernel of u * : H j (M; Q) → H j (BΓ; Q) is one-dimensional. The following construction shows that also these abound. Example 3.8. Let Γ be a finitely presented group, n ≥ 5 and assume for 3 ≤ j 1 < j 2 < · · · < j r ≤ n − 2 that 1 ≤ dim(H j k (BΓ; Q)) < ∞.
Then we find a closed spin manifold N of dimension n of positive scalar curvature with fundamental group Γ and classifying map u : N → BΓ such that the cokernel of u * : H j k (N; Q) → H j k (BΓ; 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 choosing a basis a 1 , . . . , a d of H j k (BΓ; Q) we can therefore assume without loss of generality that there are j k -dimensional spin manifolds and maps f j : M j → BΓ 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 with positive scalar curvature and map f j • pr M j to BΓ such that the induced map on H j k has codimension 1, while the image in homology is unchanged in all other degrees. One still has to adjust the fundamental group to obtain the desired manifold N by surgeries killing the kernel of the map on π 1 . This is possible by Proposition 3.1. This will correct the fundamental group and change the homology only in degrees 1, 2, n − 2, and n − 1. The manifold N obtained by doing this in all relevant degrees therefore satisfies the required conditions, except potentially for the case j k = n − 2. In this critical case, for easy of presentation we assume that N is obtained from M by surgery along an embedded S 1 × D n−1 . Then the pair sequence implies that we inclusion induces an isomorphism H n−2 (M \ S 1 × D n−1 ) ∼ = − → H n−2 (N). On the other hand, the pair sequence also shows that H n−2 (M \ S 1 × D n−1 ) → H n−2 (M) is surjective. As all this is compatible with the map to BΓ, we see that the image in H n−2 (BΓ) is unchanged if we pass from M to N and the claim follows. used in [4,Section 5.3]. More precisely, in [4] the map is defined on the space of isotopy classes 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 lattice of P + (M) in Theorem 1.4, (3) is mapped surjectively onto the image of the rational assembly map KO n+1 (BΓ) ⊗ Q → KO n+1 (C * Γ) ⊗ Q. Remark 3.10. As a predecessor construction of concordance classes which does not make use of non-trivial torsion, let us recall the construction of Carr.
First, consider the sphere S 4n−1 . Carr takes a 2-connected 4n dimensional spin manifold B withÂ(B) = 1 and removes two disks to obtain a bordism W from S 4n−1 to S 4n−1 . Positive scalar curvature surgery produces a metric of positive scalar curvature on W starting with the canonical metric on S 4n−1 and ending with a non-concordant new metric of positive scalar curvature on S 4n−1 .
However, these metrics are equal in Pos spin 4n−1 (S 4n−1 ). To see this, we have just to construct the reference map F : W → S 4n−1 which restricts to the identity on the boundary components. For this, choose a path which is a clean embedding of the closed interval into W, joining two points in the two boundary spheres. Choose then a tubular neighbourhood of this one dimensional submanifold of W, which is necessarily trivial. Now a trivialization of the tubular neighbourhood defines a collapse map from W to S 4n−1 , whose restriction to the boundary components is homotopic to the identity. Putting these homotopies on collar neighbourhoods of the boundary components, we obtain the desired map F.
More generally, given an arbitrary closed spin manifold M of dimension 4n − 1 with positive scalar curvature metric 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 ). These two metrics have non-zero index difference and therefore they are not concordant. Nevertheless, they are equal in Pos spin 4n−1 (M). We obtain the desired reference map from V to M by connected sum of the previous map with the projection from M × [0, 1] to M.