Spaces of positive intermediate curvature metrics

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.


Introduction
Given a compact manifold M with possibly nonempty boundary, the classification of Riemannian metrics on M satisfying a given curvature condition is a central problem in Riemannian geometry. In the present article we will study the uniqueness question. Of course, open conditions like positive scalar, Ricci or sectional curvature are preserved under small perturbations of a metric and so there cannot be a unique metric satisfying them. Therefore it is more reasonable to study uniqueness "up to continuous deformation", which translates into the following question: Is the space of Riemannian metrics on M satisfying a given curvature condition contractible?
In recent years, a lot of effort has gone into understanding the homotopy type of the space R scal>0 (M ) h of metrics of positive scalar curvature which restrict to h + dt 2 in a collar neighborhood of the boundary. For example, Botvinnik-Ebert-Randal-Williams in [BERW17] have studied this space for d-dimensional Spin-manifolds using the secondary index-invariant inddiff which is a well-defined homotopy class of a map inddiff : R scal>0 (M ) h × R scal>0 (M ) h → Ω ∞+d+1 KO first defined by Hitchin in [Hit74]. Fixing a base-point g ∈ R scal>0 (M ) h one obtains a homotopy class of a map inddiff g : R scal>0 (M ) h → Ω ∞+d+1 KO and they showed that this induces a nontrivial map on homotopy groups, provided that d ≥ 6, M admits a Spin-structure and the target is nontrivial. This shows that the space R scal>0 (M ) h is at least as complicated as the infinite loop space of the real K-theory spectrum. For more results on this space see [Wal14;CSS18;ERW19a]. (2) In [KKRW20], Krannich-Kupers-Randal-Williams showed that the image of the orbit map π 3 (Diff(HP 2 )) → π 3 (R sec>0 (HP 2 )) ֒→ π 3 (R p-curv>0 (HP 2 )) contains an element of infinite order for every p ≥ 0. Furthermore, the rational homotopy type of Diff(M )-invariant subspaces of R scal>0 (M ) has been studied by Reinhold and the first named author in [FR20]. Here it is shown that this space has non-vanishing higher rational cohomology, provided that M is a high-dimensional Spin-manifold and given by N #(S p × S q ) for p, q in a range. This is a generalization of the main result from [BEW20]. To the best of our knowledge, those are the only other known result about non-triviality of the higher rational homotopy type of spaces of positive pcurvature metrics (resp. k-positive Ricci curvature metrics) for p ≥ 1 (resp. k ≤ d − 1). (3) Concerning k-positive Ricci curvature, there is one other result besides [KKRW20] and [FR20] we would like to mention. Namely, Walsh-Wraith have shown in [WW20] that for d ≥ 3 and k ≥ 2 the space R k-Ric>0 (S d ) is an H-space and the component of the round metric is in fact a d-fold loop space.
The present article grew out of an attempt to extract the necessary geometric ingredients from [BERW17]. The main one is a parametrized version of the famous Gromov-Lawson-Schoen-Yau surgery theorem [GL80b;SY79] which is due to Chernysh [Che04] and has been first published by Walsh [Wal13], see also [EF20]. It states that the homotopy type of R scal>0 (M ) is invariant under surgeries with certain dimension and codimension restrictions. It turns out, that the above theorems follow from a more general result about so-called surgery-stable Diff(M )-invariant subsets F (M ) ⊂ R scal>0 (M ) with a few extra properties. We will give the general statement of our main result Theorem 2.21 in the course of Section 2, after we introduced the relevant notions.
Outline of the argument. Let F (M ) h ⊂ R scal>0 (M ) h be a Diff ∂ (M )-invariant subset. The strategy for proving Theorem A and Theorem B for manifolds of dimension 2n is to construct maps ρ : Ω ∞+1 MTθ c−1 (2n) → F (M ) h from the infinite loop space of the Madsen-Tillmann-Weiss spectrum MTθ c−1 (2n) associated to θ the tangential (c − 1)-type of M (cf. Section 2.7 for the definition). Afterwards one has to show that the composition with the maps from those theorems is weakly homotopic to the loop map ofÂ : Ω ∞ MTθ c−1 (2n) → Ω ∞+2n KO(pt) which is accomplished by index theoretic arguments from [BERW17]. Computations then show that ΩÂ induces a surjection on rational homotopy groups, whenever the target is nontrivial. The construction is first done for M a certain θ-nullcobordism of S 2n−1 which itself is θ-cobordant to the disk relative to the boundary. By gluing in k copies of K := ([0, 1]× S 2n−1 )#(S n × S n ) along the boundary, we obtain the manifold M k := M ∪ k · K. We will show that there is a metric g st ∈ F (K) h•,h• for h • the round metric on S 2n−1 with the property, that the map F (W ) hN ,h• → F (W ∪ K) hN ,h• gluing in g st is a homotopy equivalence for any cobordism W : N → S 2n−1 and any metric h N ∈ R(N ). Therefore Since there are stabilization maps F (M k ) h• → F (M k+1 ) h• and Diff ∂ (M k ) → Diff ∂ (M k+1 ) we get stabilization maps for the associated Borel constructions and after passing to the (homotopy) colimit, this yields the following fibration: (2) p ∞ : hocolim The space hocolim k→∞ BDiff ∂ (M k ) admits an acyclic map to by the work of Galatius-Randal-Williams [GRW14]. By an obstruction argument the fibration from (2) extends to a fibration p + ∞ : T + → Ω ∞ MTθ c−1 (2n), meaning that the associated diagram of fibrations is homotopy-cartesian, i. e. a homotopy pullback diagram. The main input for solving this obstruction problem is the fact that the pullback action Diff ∂ (M k ) F (M k ) h• factors up to homotopy through an abelian group for all k, which follows from surgery-stability combined with an argument in the style of Eckmann-Hilton. The desired map ρ is then given by the fiber transport map associated to the fibration p + ∞ composed with the homotopy-inverse of the stabilization map from (1). Using the additivity theorem for the index, this result is the propagated from M to any Spin-manifold of the same dimension. Jumping to the next dimension requires the spectral flow index theorem and the additional assumption that the map F (M ) → F (∂M ) restricting a metric to the boundary to be a fibration.
Outline of the paper. In Section 2 we develop the basic notions needed in this paper, starting with the definition of Riemannian functors in Section 2.1. These will be contravariant functors on the category of manifolds with codimension 0 embeddings to the category of spaces, assigning to a manifold a subspace of Riemannian metrics. The main examples are given by curvature conditions, which is reviewed in Section 2.2 where we also give precise definitions of the intermediate curvature conditions. Afterwards we introduce the notions of surgery-stability and fibrancy for Riemannian functors in Section 2.3 and Section 2.4. We give a list of Riemannian functors satisfying these two conditions after proving a criterion for fibrancy. In Section 2.5 we are finally able to state the general version of our main result. The computations of the image of the mapÂ ⊗ Q mentioned above is then carried out in Section 2.7, where Madsen-Tillmann-Weiss spectra are introduced. The final Section 2.8 of the preliminaries is a recollection of the index-theoretic arguments from [BERW17] involved in the proof of our main result, which we included to give some context.
In Section 3 we carry out the proof of our main theorem. In Section 3.1 we show that the pullback action factors through an abelian group which builds the basis for the obstruction argument used in Section 3.2 to construct the map ρ mentioned above. Afterwards we deduce the propagation result in Section 3.3 which enables to extend the result from a particular manifold to all of them. For convenience we show how the proof of our main result assembles in Section 3.4.
We close this paper by giving an overview of other recent results about the homotopy type of R scal>0 (M ) in Section 4. The proofs of those also depend mainly on the parametrized surgery theorem from a geometrical point of view. We believe that many of them can also be generalized to hold for positive p-curvature and k-positive Ricci curvature, too.

Contents
Let Mfd denote the category which has compact manifolds with (possibly empty) boundary as objects and morphisms are given by smooth codim 0-embeddings.
(1) Since diffeomorphisms are codim 0-embeddings, (2) The pull-back of Riemannian metrics along a smooth embedding is a continuous map with respect to the C ∞ -topology on the spaces of Riemannian metrics.
Definition 2.3. We say that a Riemannian functor F implies positive scalar curvature, if F (M ) ⊂ R scal>0 (M ) for every manifold M .
(1) One of the most studied examples for a Riemannian functor arises from positive scalar curvature metrics, i.e. by the assignment where scal(g) : M → R denotes the scalar curvature function of the metric g. It is immediately clear that for g ∈ R scal>0 (N ) and a codim 0-embedding f : M → N the pull-back f * g is a metric of positive scalar curvature on M .
(2) Clearly, this example can be extended to more general (open) curvature conditions, which we will recall in the subsequent section. Note, however, that for the most common conditions "positive Ricci curvature" and "positive sectional curvature" on a manifold with non-empty boundary M , the space F (M ) is empty. This is implied by our assumption on boundary collars, since the cylindrical metric g + dt 2 on ∂M × R has neither positive Ricci, nor positive sectional curvature. subsets C ⊂ C B (E d ) invariant under this action are referred to as curvature conditions. We say that a Riemannian metric g on a smooth manifold M satisfies a curvature condition C ⊂ C B (E d ), if for every point p ∈ M the description of R p in terms of an orthonormal basis in T p M is contained in C.
Let d C ≥ 0 and let C = {C d } d≥dC with C d ⊂ C B (E d ) be a sequence of curvature condition. We define a Riemannian functor Our convention will be that R C : M d → ∅ for all M d ∈ Mfd with 0 ≤ d < d C . As can be seen from the following examples, d C can be thought of as the lowest dimension in which it makes sense to consider the curvature condition C.
(1) There exist corresponding subsets to all classical curvature bounds, e.g. bounds on the sectional, Ricci or scalar curvature. For example, we can express (globally point-wise) positive sectional, Ricci and scalar curvature as conditions Here we write sec(R)(X, an orthonormal basis of E d and tr(R) denotes the trace of the algebraic curvature operator, which coincides with its scalar curvature up to a factor of 1 2 . In these cases we have d sec>0 = d Ric>0 = d scal>0 = 2.
(2) The notion of p-curvature, where p is an integer, was proposed by Gromov (cf. [Lab97b,p.301]) and is a natural generalization of scalar and sectional curvature which provides an interpolation between both. Let (M, g) be a Riemannian manifold of dimension d ≥ p + 2 and let G p (TM ) denote the p-Graßmannian bundle over M and U (P ⊥ x ) be a neighborhood around 0 in the plane perpendicular to a p-plane P x ⊂ T x M . The map ) is referred to as p-curvature function. If s p is positive on all of G p (TM ), the metric g is said to have positive p-curvature.
The term p-curvature coincides with scalar curvature for p = 0 and with (the double of) sectional curvature for p = d − 2. Without much effort, one can show that positive p-curvature implies positive (p − 1)-curvature and thus ultimately positive scalar curvature.
, where for convenience we set sec(E i , E i ) := 0. It is easy to see in this description that s 1 (span(v)) = s 0 − 2Ric(v) for any element v ∈ S(T x M ), which is precisely double the value of the Einstein tensor E(v, v) = 1 2 scalg(v, v) − Ric(v). Positive p-curvature can be described as a curvature condition given by an open convex cone Hence for every fixed p ≥ 0, we obtain a sequence C d := (p-curv > 0) d of curvature conditions that yield a Riemannian functor as above with d p-curv>0 = p + 2.
For technical reasons it is more convenient for us to replace k by (d − k) (cf. Remark 2.13). For fixed k ≥ 0 we define a sequence of curvature conditions There are further examples for curvature conditions such as positive s-curvature, point-wise almost non-negative curvature (cf. [Hoe16]) or positive Γ 2 -curvature (cf. [BL14]). This space can be thought of as the subspace of those metrics which have a fixed (standard) form on N . If M has boundary ∂M there is a restriction map res: F (M ) → R(∂M ) and for h ∂ ∈ R(∂M ) we write Since the boundary of N is collared, there is a collar of ι(N ) ⊂ M . If additionally ι(N ) lies in the interior of M , then by prolonging the collar yields a homotopy equivalence F (M, ι; h) ≃ F (M \ ι(N \ ∂N )) ι * (h| ∂N ) . We denote by g k • ∈ R(S k ) the round metric on the k-dimensional sphere.
A Riemannian functor F is called parametrized codimension c surgery-stable on d-dimensional manifolds if additionally the map is a homotopy equivalence. We will abbreviate F (M, ϕ; g k ) = F (M, ϕ). Usually there will be no chance of confusion and we will omit "on d-dimensional manifolds".
Let us first give an explanation for the wording "surgery-stability". For this, let c − 1 ≤ k ≤ d − c and let ϕ : S k × D d−k ֒→ M be an embedding. We denote by M ϕ the manifold obtained by performing surgery on M along ϕ. Let ϕ op : D k+1 × S d−k−1 ֒→ M ϕ be the obvious reversed surgery embedding. We now have the following observation: If F is codim c-surgery stable, then If F is parametrized codim c-surgery stable, all of these spaces are homotopy equivalent: Corollary 2.9. Let F be a parametrized codimension c surgery stable Riemannian functor and let ϕ : Then we get a zigzag of weak homotopy equivalences Remark 2.10.
(1) Note that with our definition codim c-surgery-stability obviously implies codim c ′ -surgery-stability for every c ′ ≥ c.
(2) We do not explicitly assume the existence of a metricg on the opposite surgery embedding in our definition of surgery stability, the reason being that it is not required in the proof of our main result. However, such a metric exists in all of the examples we know for surgery stability or if there is the symmetric lower bound on the index k of the surgery embedding.
(3) Note that for all c ≤ d we have that codim c-surgery-stability of F implies For some of the constructions later on, we will need that fixing a metric on only one disk instead of S 0 × D d also gives a homotopy equivalence. This is guaranteed by the following proposition if F is cellular.
Proof. Without loss of generality we may assume that F (M ) = ∅. Let ϕ : S 0 × D d ֒→ M be an embedding that extends ι and consider the composition which is a homotopy equivalence by parametrized surgery stability. Hence the second inclusion is surjective on all homotopy groups. For injectivity on homotopy groups let ϕ : S 0 × D d ֒→ M ∐ M denote the disjoint union of ι with itself and consider the following diagram: The horizontal maps are inclusions into the product and hence injective on homotopy groups and it follows that the inclusion F (M, ι) ֒→ F (M ) is injective on homotopy groups.
(1) It is well-known by the work of [GL80b] and [SY79] that positive scalar curvature is codimension 3 surgery-stable on d-manifolds in all dimensions d ≥ 3. Chernysh showed in [Che04] that it is in fact parametrized codimension 3 surgery-stable.
(2) A similar result is true for other open curvature conditions, which satisfy a condition specified by Hoelzel in [Hoe16]. This includes curvature conditions such as positive p-curvature and k-positive Ricci curvature, which are codimension p+3 (resp. max{3, d−k+2}) surgery-stable on d-manifolds for d ≥ 3. By work of the second named author [Kor20] these conditions are in fact parametrized surgery-stable with the same codimension restriction. (3) The condition sec < 0 gives rise to a Riemannian functor, which is codimension 2 surgery stable on 2-manifolds. (4) The Riemannian functor, which assigns to a manifolds its metrics that are simultaneously conformally flat and have scal ≥ 0 is codimension d surgery stable on d-manifolds (cf. [Hoe16, Theorem 6.3]).
Remark 2.13. Since we want the codimension restriction arising from surgerystability to be independent of the dimension, we choose to replace k-positive Ricci curvature by (d − k)-positive Ricci curvature, which is parametrized codimension max{3, k + 2}-surgery stable.
2.4. Fibrancy. In order to compare spaces of metrics on manifolds with different dimensions, we need the restriction map res : F (M ) → R(∂M ) to satisfy the properties from the following definition.
Definition 2.14. A Riemannian functor F is called fibrant if (1) res(F (M )) ⊂ F (∂M ) for all M ∈ Mfd with ∂M = ∅ and (2) the restriction map res : The Riemannian functor given by positive scalar curvature is fibrant. This was shown utilizing the method we generalize here in [EF20].
Proposition 2.15. A Riemannian functor F satisfies (1) in the above definition if and only if for every closed N ∈ Mfd and every g ∈ R(N ) with . Now let M ∈ Mfd with ∂M = ∅ and let g ∈ F (M ). Since we assumed M to be collared and the metric to be cylindrical in a neighborhood of the boundary, there is a codim 0-embedding c : [0, 1] × ∂M ֒→ M such that c * g = res(g) + dt 2 ∈ F (∂M × [0, 1]). By assumption, this implies that res(g) ∈ F (∂M ) Let F be a Riemannian functor. For every closed manifold N ∈ Mfd we have a continuous stabilization map The following is a criterion for curvature conditions for which R C is fibrant.
Before diving into the proof, let us give the consequences most important to us. Proof. It remains to show that g has positive p-curvature (resp. (d − k)-positive Ricci curvature) if and only if g + dt 2 has positive p-curvature (resp. (d + 1 − k)positive Ricci curvature).
If g has positive p-curvature, then the p-curvature of g + dt 2 is positive by the computation in Lemma A.1. Now let g + dt 2 have positive p-curvature and let P ⊂ T x M be a p-dimensional subspace. Then there is an orthonormal basis and we can compute Concerning (d − k)-positive Ricci we note that the eigenvalue of the Ricci curvature corresponding to ∂ t equals 0, the first sum of the first (d + 1 − k) eigenvalues of Ric(g + dt 2 ) is positive if and only if the sum of the first (d − k)-eigenvalues of Ric(g) is positive.
We have the following observation: (1) For every smooth path of Riemannian metrics for every function f : R → [0, 1] that is constant near 0 and 1 and satisfies |f ′ |, |f ′′ | ≤ Λ (2) Additionally, Λ can be chosen depending continuously on the family {g r }.
Proof. We obtain (1) immediately from a computation similar to [Gaj87, p.184] (cf. Lemma A.2), which yields the following correspondence between curvature tensors: where E 1 , E 2 , E 3 only depend on the path {g r } r∈[0,1] and its derivatives in rdirection. Since C is an open subset in C B (E n ), we find Λ accordingly. This also reveals that Λ can be chosen continuously and thus implies (2).
(1) The proof of Theorem 2.16 indeed shows the following: If Let us now turn to the proof of Theorem 2.16. The following lemma and its proof are adaptations from [EF20, Lemma 5.1] to a more general setting. It constructs a family of paths from a path of metrics, which stops at any particular point.
Lemma 2.20. Let F be an open Riemannian functor. Let N d−1 be a closed manifold, P be a compact topological space and let G : P × [0, 1] → F (N ) be a continuous map. Then there exists a continuous map with the properties If, additionally, F = R C satisfies the assumptions of Theorem 2.16, then there exists 0 < Λ ≤ 1 such that for every function f : Proof This converges uniformly to G in the sense that lim n→∞ C n (p, s, t) = G(p, s · t). Again using that F (N ) is open, we conclude there exists a sufficiently large n such that Im(C n ) ⊂ F (N ). We then let C := C n . If F satisfies the assumptions of Theorem 2.16, then stab( . Finally by (2) in Lemma 2.18, Λ (p,s) depends on (p, s) continuously and thus we choose Λ := min{Λ (p,s) }.
Proof of Theorem 2.16. To prove the statement, it suffices to find a solution to the following lifting problem: is of product form on the collar of length 2δ. Since F is open, G is homotopic relative to G| D×{0,1} to a mapG with {G(p, t)} t∈[0,1] a smooth path of metrics for every p ∈ P . We replace G byG. Now apply Lemma 2.20 to G to obtain a map C : D × [0, 1] 2 → F (∂M ) and 0 < Λ ≤ 1 accordingly. Choose a smooth function f : Thus a candidate for a lift is given bŷ This is well-defined, since along the gluing, we have (cf. Moreover, by construction of C (cf. (3) in Lemma 2.20) we have for p ∈ D: Theorem 2.21. Let n ≥ c ≥ 3 and let F be a cellular, parametrized codimension c-surgery stable Riemannian functor that implies positive scalar curvature. Let W be a Spin-manifold of dimension d = 2n. Let h ∈ R + (∂W ) and g ∈ F (W ) h . Then for all k ≥ 0 such that d + k + 1 ≡ 0(4) the composition is nontrivial. If additionally F is fibrant, this holds for all manifolds of dimension d ≥ 2c.
Theorem A and Theorem B now follow from the above theorem by Example 2.12 and Proposition 2.17. Note that the long list of adjectives in front of "Riemannian functor" does not imply lack of examples but rather is due to the fact that there are many examples and the aim to extract necessary assumptions out of these.
2.6. Stable metrics. The following Lemma states the existence of stable metrics (cf. [ERW19a]) in a special case. Let c ≥ 3 and let F be a parametrized codimension c surgery stable Riemannian functor.
relative to the boundary. Then there exists a metric g ∈ F (V ) g•,g• with the following property: If W : S d−1 ❀ S d−1 is cobordism and h ∈ R(S d−1 ) is a boundary condition then the two gluing maps are homotopy equivalences.
Definition 2.23. A metric g as in this Lemma is called an F -stable metric.
Proof of Lemma 2.22. By assumption, there exists a relative BO(d) c−1 -cobordism X : V ❀ S d−1 × [0, 1] and by performing surgery on the interior of X we may assume X has no handles of indices 0, . . . , c − 1, d + 1 − c + 1, . . ., d + 1. So S d−1 × [0, 1] is obtained from V by a sequences of surgeries in the interior with these indices.
where the first map is induced by the inclusion. By the Pontryagin-Thom construction the group π k (MTθ n (d)) is isomorphic to the cobordism group of triples . Note that because of n < d we have π n (BSO(d)) is either Z (for n ≡ 0(4)), Z/2 (for n ≡ 1, 2(8)) or 0 and hence it suffices to consider the case that n = 4m because in the other cases the map B n (d) → B n−1 (d) induces an isomorphism in rational cohomology. The Serre spectral sequence has the form −1 (d)). The cohomology of K(Z, n) is given by H * (K(Z, n)) ∼ = Q[α] for α in degree n. Furthermore, H p (B n (d)) = 0 unless p ≡ 0(4). Hence all differentials vanish, the spectral sequence collapses on the E 2 -page and we have Since H n (B n (d)) = 0, the preimage of p m is the class α generating H * (K(Z, n)) and therefore H * (B n (d), Q) ∼ = Q[p m+1 , . . . , p ⌊d/2⌋ ].
Corollary 2.26. The bordism group π k (MTθ n (d)) ⊗ Q consists of the classes in Ω θn d+k ⊗ Q which do not have nontrivial Pontryagin classes of degree greater than ⌊d/2⌋.
Proof. Since the sphere spectrum is rationally an HQ-spectrum by Serre's finiteness theorem, the rational Hurewicz-homomorphism of spectra π k (MTθ n (d)) ⊗ Q → H k (MTθ n (d); Q) is an isomorphism. Composing with the Thom-isomorphism we get an isomorphism π k (MTθ n (d)) ⊗ Q → H k+d (B n (d), Q). The claim follows from = q · (⌊n/4⌋ + 1) + r and hence M := (M ⌊n/4⌋+1 ) q × M r has only Pontryagin classes of degree smaller than ⌊d/2⌋ and hence is an element of π k (MTθ n (d)) ⊗ Q by Corollary 2.26 with non-vanishinĝ A-genus which proves the theorem.
2.8. Index theoretic ingredients. This is mainly a recollection of index-theoretic arguments involved in the proof of our main result. Even though this is just a recollection from [BERW17], we decided to keep it in here to give some context. There is no claim of originality for this entire section.
2.8.1. KO-theory. Let us start by recalling the model for KO-theory that was used in [BERW17, Chapter 3], for a more detailed discussion see loc.cit.. Let X be a space H → X be a Hilbert bundle with separable fibers. An operator family is a fiber preserving and fiber-wise linear continuous map H 0 → H 1 of Hilbert bundles H 0 and H 1 . It is determined by a family (F x ) x∈X of bounded operators x∈X is an operator family and we denote the algebra of adjointable operators by Lin X (H). The * -ideal of compact operators on X is denoted by Kom X (H). We call an adjointable operator family F a Fredholm family if there exists a K ∈ Kom X (H) such that F + K is invertible.
Definition 2.27. Let V → X be a Riemannian vector bundle and let τ : V → V be a self-adjoint involution. A Cl(V τ )-Hilbert bundle is a triple (H, ι, c) where H → X is a Hilbert bundle, ι : H → H is a self-adjoint involution and c = (c x ) x∈X is a collection of maps c is a continuous section, then c x (s(x)) is an operator family We will omit x in c x (v) when there is no chance of confusion.
The opposite Cl(V τ )-Hilbert bundle is given by (H, −ι, −c). A Cl(V τ )-Hilbert bundle with V = V + ⊕ V − and τ (v 1 , v 2 ) = (v 1 , −v 2 ). It will also be called a Cl(V + ⊕ V − )-Hilbert bundle and if V + = R p and V − = R q we will abbreviate this by Cl p,q . A Cl(V + ⊕ V − )-module is a finite-dimensional Cl(V + ⊕ V − )-Hilbert bundle and a Cl p,q -Fredholm family is a Fredholm family on a Cl p,q -Hilbert bundle that is Cl p,q -linear and anti-commutes with the grading, i. e. F c(v) = c(v)F and F ι = −ιF .
Definition 2.28. Let (X, Y ) be a space pair. A (p, q)-cycle on X is a tuple (H, ι, c, F ) where (H, ι, c) is a Cl p,q -Hilbert bundle and F is a Cl p,q -Fredholm family. A relative (p, q)-cycle is a (p, q)-cycle on X such that F is invertible over Y . A concordance between (H 0 , ι 0 , c 0 , F 0 ) and (H 1 , ι 1 , c 1 We will sometimes abbreviate (H, ι, c, F ) by (H, F ) or x → (H x , F x ).
Definition 2.29. For a pair (X, Y ) of a paracompact space X and a closed subspace Y we define There is an isomorphism of abelian groups A Cl p,q -Hilbert space is called ample if it contains every finite dimensional irreducible Cl p,q -Hilbert space with infinite multiplicity. We fix an ample Cl p,q -Hilbert space U and we define Fred p,q to be the space of all Cl p,q -Fredholm operators on U with the norm topology and G p,q the (contractible) subspace of invertible ones. We have the following theorem.
Theorem 2.30 ([BERW17, Theorem 3.3 and below]). Let (X, Y ) be a CW-pair. Then the following holds (2) Every class b ∈ F p,q (X, Y ) corresponds to a unique homotopy class of a map (X, Y ) → (Ω ∞+p−q KO, * ) which we call the homotopy-theoretic realization of b D is a linear, formally self-adjoint, elliptic differential operator of order 1 and anti-commutes with the grading and the Clifford multiplication of R 0,d . Hence, after changing the Cl 0,d -multiplication to a Cl d,0 -multiplication via replacing c(v) by ιc(v), the Dirac operator D becomes Cl d,0 -linear. The relevance of the Dirac operator to positive scalar curvature geometry originates from the Schrödinger-Lichnerowicz formula: which forces the Dirac operator to be invertible if the scalar curvature is positive. Now, let X be a paracompact Hausdorff space and π : E → X a fiber bundle with possibly non-compact E x := π −1 ({x}) of dimension d such that the vertical tangent bundle T (v) E admits a Spin-structure. A fiber-wise Riemannian metric g x gives rise to a Spinor-bundle S E , a Cl(T (v) E + ⊕ R 0,d )-module that restricts to the Spinorbundle S x → E x with Dirac operator D x in each fiber. If the fibers are compact with boundary diffeomorphic to N and the boundary bundle is trivial as a Spinbundle, i. e. ∂E = X × ∂N , we can consider the elongation of (E, g). This is defined to be the bundleÊ := E ∪ ∂E (X × [0, ∞) × N ) with the metric (dt 2 + g x ) on the added cylinders.
Definition 2.31 ([BERW17, Definition 3.4]). Let t : E → R be fiber-wise smooth such that (π, t) : and E (a0,a1) := (π, t) −1 (X × (a 0 , a 1 )). We say the bundle E is cylindrical over and (a + , ∞) for some functions a − , a + : X → R. If E has cylindrical ends and there is a fiber-wise Riemannian metric g = (g x ) x∈X that is cylindrical over the ends, we say that (g x ) has positive scalar curvature at infinity if there exists a function ε : X → (0, ∞) such that on the ends of E x the metric g x has scalar curvature ≥ ε(x).
Let L 2 (E, S E ) x denote the Hilbert space of L 2 -sections of the spinor bundle S x → E x . These assemble to a Cl d,0 -Hilbert bundle over X. The Dirac operator is densely defined symmetric unbounded operator on L 2 (E, S E ) x and its closure is self-adjoint. Applying the functional calculus for f (x) = x √ 1+x 2 we get the bounded transform If g has positive scalar curvature at infinity this is a bounded Cl d,0 -Fredholm operator. The collection (F x ) is a Cl d,0 -Fredholm family over X. We define Dir(E, g) to be the (d, 0)-cycle given by x → (L 2 (E, S E ) x , F x ) and if we assume that D y is invertible for all y ∈ Y we obtain a class We have the following Lemma.
Lemma 2.32 ([BERW17, Lemma 3.7]). Let π : E → X be a Spin-manifold bundle with cylindrical ends and let g 0 , g 1 be fiber-wise metrics with psc at infinity and agree on the ends. Let Y ⊂ X and assume that g 0 and g 1 agree and have invertible Dirac operators over Y . Then In particular, if g is a fiber-wise metric only defined over the ends we still get a well defined class Ind(E, g) and if E is closed, then Ind(E, g) does not depend on g. Also note that for the bundle E op with the opposite Spin-structure we have Ind(E op , g) = −Ind(E, g).
From now on let F be a Riemannian functor that implies positive scalar curvature.
2.8.3. The two definitions of inddiff. Let again W d be a Spin-manifold and let h ∈ R(∂W ) such that F (W ) h = ∅. Since F implies positive scalar curvature we deduce that h ∈ R scal>0 (∂W ). Let us consider the trivial W -bundle over I × F (W ) h × F (W ) h . A fiber-wise Riemannian metric G is given by G (t,g0,g1) := 1−t 2 g 0 + 1+t 2 g 1 in the fiber over (t, g 0 , g 1 ). The elongation has psc at infinity and invertible Dirac operators for t = ±1. We therefore get an element where ∆ denotes the diagonal. This is Hitchin's definition of the index-difference, cf. [Hit74]. Fixing a base-point g ∈ F (W ) h we obtain an element inddiff g ∈ ΩKO −d (F (W ) h , g) and a homotopy theoretic realization Remark 2.33. Note that the definition of Ind only depends on the Dirac-operator associated to the metric and hence we get the following homotopy commutative triangle The second definition of the index-difference goes back to Gromov-Lawson [GL80a]. Let W d be a closed Spin-manifold and consider the trivial R × W -bundle over F (W ) × F (W ). Choose a smooth function ψ : R → [0, 1] that is constantly equal to 1 on [1, ∞) and equal to 0 on (−∞, 0]. We get a fiber-wise metric G := dt 2 + (1 − ψ(t))g 0 + ψ(t)g 1 in the fiber over (g 0 , g 1 ). This has positive scalar curvature at infinity and we hence get an element Again, after fixing a base point g ∈ F (W ) we get inddiff GL g ∈ KO −d−1 (F (W ), g) and a homotopy theoretic realization inddiff GL,F g : (F (W ) h , g) → (Ω ∞+d+1 KO, * ). Remark 2.35. Recall that two maps f 0 , f 1 : X → Y are called weakly homotopic if for every map α : K → X from a finite CW -complex K we have that f 0 • α is homotopic to f 1 • α. Weakly homotopic maps induce equal maps on homotopy groups.
2.8.4. The additivity theorem for the index-difference. One of the main tools for computing the index-difference is the additivity theorem. In order to state it we need some notation. Let X be a paracompact Hausdorff space and let E → X, E ′ → X be two Riemannian Spin-manifold bundles of fiber dimension d. Let g, g ′ be metrics with cylindrical ends such that E and E ′ have psc at infinity. Assume that there exist functions a 0 < a 1 : X → R such that E and E ′ are cylindrical over X × (a 0 , a 1 ) and agree there E (a0,a1) = E ′ (a0,a1) . Assume that the Dirac operators are invertible over E (a0,a1) . Let and for (i, j) ∈ {(0, 1), (2, 3), (0, 3), (1, 2)} let Theorem 2.36 ([BERW17, Theorem 3.12]).
If E 01 and E 23 have invertible Dirac operators over a closed subspace Y ⊂ X, then this equation holds in KO −d (X, Y ).
There is the following restatement in terms of the index-difference.
. Then the following diagram commutes up to homotopy: The homotopy fiber of f at y is defined as and we have the canonical map ε y0 : f −1 (y 0 ) → hofib y0 (f ), x → (x, const y0 ) which is a pointed map by considering * := (x 0 , const y0 ) as the base-point of hofib y0 (f ). There is a natural map if t ≥ 0.
Note that η y0 • (id I × ε y0 ) is homotopic to ι y0 : (t, x) → (x, 1 2 (t + 1)) as a map of pairs. The fiber transport map is defined as For a class α ∈ KO −p (f ) there is an associated base class bas(α) := i * α ∈ KO −p (Y ) and a transgression trg(α) := η * y0 α ∈ ΩKO −p (hofib y0 (f )). The loop map is defined by l : I × Ω y0 Y → Y, (t, c) → c( 1 2 (t + 1)) and we write Ω := l * : Lemma 2.38 ([BERW17, Lemma 3.19]). We have This lemma can be illustrated by the following homotopy-commutative diagram An instructive way to think about these class proposed in [BERW17] is the following: j * α for j : X × [0, 1] ֒→ Cyl(f ) is a concordance in the sense of Definition 2.28 and j * α| X×{0} is acyclic. Hence the class f * bas(α) = j * α| X×{1} = 0 ∈ KO −p (X, x 0 ) and hence we get the following homotopy-commutative diagram where the columns are homotopy fiber sequences: bas(α) f 2.8.6. Increasing the dimension. As a consequence of the abstract setting described in the previous section we can now derive the following propagation result allowing us to increase the dimension. For this we further assume that F is fibrant.
Theorem 2.39 ([BERW17, Theorem 3.22]). Let W be a Spin-manifold of dimension d, h ∈ F (∂W ) and g ∈ F (W ) h . Then the following diagram is weakly homotopy commutative where T denotes the fiber transport map after identifying F (W ) h with hofib h (res) via ε h0 .
2.8.7. Relating inddiff to Ind. Let W be a d-dimensional Spin-manifold with boundary M , such that (W, M ) is 1-connected. Let π : E → X be a smooth fiber bundle with fiber W over a paracompact base X and associated structure group Diff ∂ (W ). This as a Spin-structure on the vertical tangent bundle which is constant along the boundary sub-bundle. Let h 0 ∈ R(M ) be a fixed boundary condition such that F (W ) h0 = ∅. We get an associated fiber bundle Let x 0 ∈ X be a base point and let us identify π −1 (x 0 ) = W . Then p −1 (x 0 ) can be identified with F (W ) h0 and we choose a base point g 0 ∈ p −1 (x 0 ). We will now construct an element β ∈ KO −d (p), depending only on the bundle π and the metric g 0 . Let k be a fiber-wise Riemannian metric on π such that (1) the restriction of k to π −1 (x 0 ) = W is equal to g 0 , (2) near the boundary sub-bundle ∂E, the restriction of k is a cylinder on h 0 . Such a metric can be constructed using a partition of unity and k is not assumed to be in F (π −1 (x)) h0 for all x. LetẼ := pr * E for the natural map pr : Cyl(p) → X.Ẽ then inherits a fiber-wise metrick as follows: over x ∈ X ⊂ Cyl(p) we takek x := k x and over a point (x, g, t) ∈ Q × Diff ∂ (W ) F (W ) h0 × [0, 1] we letk x := (1 − t)g + tk x . Thenk also satisfies boundary condition h 0 and it has positive scalar curvature for t = 0 and (x, g) = (x 0 , g 0 ). SinceẼ also has a Spin-structure on the vertical tangent bundle there is a family Dirac operator and hence a well-defined class β ∈ KO −d (p). This has the following properties. (1) bas(β) = Ind(E, k) ∈ KO −d (X).
(3) β is natural with respect to fiber bundles.
(4) Let V : M → M ′ is a Spin-cobordism and m ∈ F (V ) h0,h1 . Let be the bundle obtained by gluing in V in each fiber. Then there is a commutative diagram

Proof of main results
For this entire section let F be a parametrized codimension c ≥ 3 surgery stable, cellular Riemannian functor that implies positive scalar curvature. • is an abelian group.
For the proof we will use the following Lemma of Eckmann-Hilton style.
Lemma 3.2 ([BERW17, Lemma 4.2]). Let C be a nonunital topological category with objects the integers and let G be a topological group which acts on C, i.e. G acts on all morphism spaces and the composition in C is G-equivariant. We will denote the composition of x and y by x · y. Suppose that (1) C(m, n) = ∅ for n ≤ m.
Furthermore, let h ∈ F (T ) be the metric obtained by cutting out the metric ι * g 0,1 on ι(D), where g 0,1 is the metric from Proposition 2.11. h restricts to the round metric on all three boundary components. We get the sequence of maps The composition is given by gluing in h which is homotopic to gluing in g • +dt 2 and therefore is a homotopy equivalence. The right-most map is a homotopy equivalence by Proposition 2.11 and so µ h also is a homotopy equivalence, too. Let V := M ∪ S d−1 ×{0} T and let us consider this as a cobordism We now apply Lemma 3.2 to the following scenario: Let G := Diff ∂ (M ) and let We abbreviate D k := Diff ∂ (W k ), B k := BD k and π k : There is a homomorphism D k → D k+1 given by extending by the identity and we get induced maps ι k : B k → B k+1 on classifying spaces. Furthermore we write F k := F (W k ) g• , T k := ED k × D k F k and we denote by p k : T k → B k the projection maps and by µ k := µ( , h k ) : F k → F k+1 the maps gluing in the stable metrics h k ∈ F (K| [k,k+1] ) g•,g• which exist by Lemma 2.22. The map µ k is D k -equivariant and so there is an induced map between the Borel constructions We introduce the following notation: The construction (2.8.7) gives classes β k ∈ KO −2n (p k ) that assemble to a class β ∞ ∈ KO −2n (p ∞ ) (cf. [BERW17, Proposition 4.9]).
Lemma 3.3. There exists a cobordism W : ∅ ❀ S 2n−1 such that there exists an acyclic map Ψ : B ∞ → MTθ c−1 (2n) and the maps are weakly homotopic. and since the Spin-structures on all bundles π k are compatible, we obtain a map The map α ∞ is acyclic for our choice of W which follows from [GRW17, Theorem 1.5] in the same way as demonstrated in the proof of [BERW17,Theorem 4.19]. It remains to show that the maps ι * k,∞ α * ∞ Ω ∞ λ −2n and Ind(E k , h • ) are weakly homotopic. For this note that a W -bundle E → X with a θ c−1 -structure on the vertical tangent bundle also admits a Spin-structure and we get the following diagram The maps in the bottom triangle are weakly homotopic by [BERW17,Proposition 4.16] which is a version of the Atiyah-Singer index theorem. The left-hand triangle commutes since the Spin-structure on T (v) E is precisely the one induced by the θ c−1 -structure and the right-hand triangle commutes by definition. Therefore the entire diagram commutes and hence we obtain that Ω ∞ λ −2n • α θ E and Ind(E, h • ) are weakly homotopy which specifies to our claim for X = B ∞ .
Theorem 3.5. The map Ω ∞+1 λ −2n and the composition induce the same map on homotopy groups.

3.3.
Propagating the detection result. One consequence of the index additivity theorem is the following propagation result. Let F be a cellular, parametrized codimension c ≥ 3 surgery stable Riemannian functor and implies positive scalar curvature. Let d ≥ 2c and let W : ∅ ❀ S d−1 be a (c − 2)-connected BO c − 1manifold and X : W ❀ D d a BO c − 1 -cobordism relative to the boundary. By removing an embedded disk D ⊂ W we obtain a bordism W 0 : S d−1 ❀ S d−1 which is BO c − 1 -bordant to the cylinder relative to the boundary via X 0 . Therefore, there exists a stable metricg on W 0 by Lemma 2.22. Let g := g 0,1 ∪g ∈ F (W ) h• . By performing surgery on X 0 we may assume that X 0 is (c − 2)-connected and after choosing an appropriate handle decomposition of X 0 we get a surgery map S F ,X0,H (cf. Corollary 2.9 and the discussion below it). Thus, we obtain a (homotopy class of a) metricg := S F ,X0,H (g • + dt 2 ) and g := g 0,1 ∪g ∈ R + (W ) h• for g 0,1 the metric from Proposition 2.11. Proposition 3.6. Let W ′ : ∅ ❀ M ′ be an arbitrary compact Spin-cobordism with h ′ ∈ R(M ′ ) and g ′ ∈ F (W ′ ) h ′ . We get the following result. If there exists a CW complex X with a mapâ : X → Ω ∞+d+1 KO and a factorization ofâ up to homotopy, then there exists a factorization is again a factorization ofâ up to homotopy. By Proposition 2.11 there exists a metricg ′ on W ′ \ D such that g 0,1 ∪g ′ is homotopic to g ′ . Again, by Theorem 2.37 the composition given by is homotopic toâ.
Remark 3.7. Note that we do not require the manifold W ′ to admit a BO(d) c−1 or to be (c − 2)-connected. The proof reveals that the map ρ ′ from Proposition 3.6 actually factors through F (D d ) h• . The requirement of W ′ being Spin stems from requiring the existence of the map inddiff g ′ .
3.4. Proof of Theorem 2.21. The proof of the general statement of our main results now consists of assembling the parts. First we note that the result for the manifold W from Theorem 3.5 follows directly from Theorem 3.5 and Theorem 2.24. Since W is BO(d) c−1 -cobordant to D 2n relative to the boundary, Proposition 3.6 implies that it is true for every manifold W ′ of dimension 2n, in particular it is true for S 2n . If F is fibrant, then by Theorem 2.39 it holds for D 2n+1 and again by Proposition 3.6 it holds for all manifolds of dimension 2n + 1.

Epilog
[BERW17] is not the only paper about the space of positive scalar curvature which geometrically mostly depends on the parametrized surgery theorem. We therefore believe, that many other recent results can be proven for positive p-curvature und (d − k)-positive Ricci curvature. The following gives a (probably very incomplete) list of results that could possibly be generalized, with adapted dimension and connectivity assumptions, of course: [BHSW10] Here it is shown that there exist elements of infinite order in the some homotopy groups of the observer moduli space M scal>0 x0 (M ). For k ≥ 2 and k-positive Ricci curvature this follows from [BWW19] and the fact, that k-positive Ricci curvature is codimension 0 surgery stable. For positive p-curvature however this is not known. [Wal14] Here it is shown that the component of the round metric in R scal>0 (S d ) is a d-fold loop space for d ≥ 3. This has recently been upgraded in [WW20] to be true for k-positive Ricci curvature, k ≥ 2 and also d ≥ 3. Again, an analogue for positive p-curvature does not exist, but we suspect that it's true for d ≥ 3 and positive p-curvature for p ≤ d − 3. [ERW19a] Here, the existence of (left-)stable metrics (cf. Definition 2.23) is shown on cobordisms where the inclusion of the outgoing boundary is 2-connected. Also this contains an extension of the results from [BERW17] taking the fundamental group into account. [ERW19b] Here it is shown that the component of the round metric in R scal>0 (S d ) is an infinite loop space for d ≥ 6. However, the given proof does not work in low dimensions and we expect the dimension and connectivity assumptions to be worse worse compared to the ones from [Wal14]. [Fre19] For manifolds M, N of dimension d ≥ 6 it is shown that the surgery map (cf. Corollary 2.9 and the discussion below for the definition) is independent of the handle decomposition H and only depends on the θ-cobordism class of the cobordism X relative to M ∐ N , for θ the tangential 2-type of the outgoing boundary N . This is then utilized to show that for a simply connected Spin-manifold of dimension at least 6 the pullback action π 0 (Diff Spin (M )) π 0 (R scal>0 (M )) factors through the Spincobordism group. [Fre20] Here it is shown using the results from [Fre19] that R scal>0 (M ) is an Hspace for every manifold M of dimension at least 6 which is nullbordant in its own tangential 2-type. It is also shown that in dimensions at least 6 the underlying H-spaces structures from [ERW19b] and [Wal14] are equivalent.
Appendix A. Curvature Computations Lemma A.1. If g has positive p-curvature then g + dt 2 does as well.
Proof. Let (M, g) be a closed, d-dimensional Riemannian manifold. Let P ⊂ T x M be a p-dimensional subspace and choose an orthonormal base E 1 , . . . , E d−p of V ⊥ .
Since g has positive p-curvature we get: sec(E i , E j ) > 0.
Again, let W ⊂ T (x,t) (M × [0, 1]) be a p-dimensional subspace and suppose W ⊥ is not fully contained in T x M (otherwise positivity of s p,g+dt 2 (W ) follows directly from positivity of s p,g ). Then choose an orthonormal base E 1 , . . . , E d+1−p−1 for W ⊥ ∩ T x M and extend this to an orthonormal basis of W ⊥ by a vector E d+1−p = sin(φ)X + cos(φ)Y , where φ ∈ [0, π 2 ), X ∈ T x M and Y ∈ T t [0, 1] are vectors of unit length. This yields the following description.
By (11) we only need to determine the following terms: g(∇ Xj X k , ∇ Xi T ) = g(