The Gromov-Lawson-Chernysh surgery theorem

In this article, we give a complete and self--contained account of Chernysh's strengthening of the Gromov--Lawson surgery theorem for metrics of positive scalar curvature. No claim of originality is made.


Introduction
A famous result by Gromov-Lawson [8] and Schoen-Yau [21] states that if M d is a closed manifold with a metric of positive scalar curvature and ϕ : S d−k × R k → M a surgery datum of codimension k ≥ 3, then the surgered manifold M ϕ := M \ (S d−k × D k ) ∪ S d−k ×S k−1 D d−k+1 × S k−1 does have a metric of positive scalar curvature as well. This has been the basis for virtually all existence results for psc metrics on high-dimensional manifolds, the most prominent of which is [22].
A strengthening of the surgery theorem has been proven by Chernysh [3], based on Gromov-Lawson's proof. His result implies that the two spaces R + (M ) and R + (M ϕ ) of psc metrics have the same homotopy type if in addition to k ≥ 3 the condition d − k + 1 ≥ 3 is also satisfied.
To state Chernysh's theorems in full generality, some preliminaries are needed. In order to keep the length of this introduction at bay, we state the results somewhat informally and refer to the main body of the paper for precise definitions. We consider Riemannian metrics on compact manifolds with boundary (it is always assumed that the boundaries are equipped with collars). Let R(M ) be the space of all Riemannian metrics h on M such that h = g + dt 2 near ∂M , for some metric g on ∂M , and with respect to the given collar. Let R + (M ) ⊂ R(M ) be the subspace of metrics of positive scalar curvature. If h ∈ R + (M ) is of the form g + dt 2 near ∂M , then g has positive scalar curvature as well, and hence mapping h to g defines a continuous restriction map res : R + (M ) → R + (∂M ).
We define R + (M ) g := res −1 (g), the space of all Riemannian metrics of positive scalar curvature on M which near ∂M are equal to g + dt 2 . Theorem 1.1 (Chernysh [4]). The restriction map res : R + (M ) → R + (∂M ) is a Serre fibration.
In fact, this is a slight improvement of the main result of [4], where it is only shown that res is a quasifibration. The proof of Theorem 1.1 is given in §5.1 and follows largely the idea of [4]. Now let N be a compact manifold with collared boundary and let ϕ : N × R k → M be an open embedding such that ϕ −1 (∂M ) = ∂N × R k and such that ϕ is compatible with the chosen collars of M and N . Let g N ∈ R(N ) be a Riemannian metric on N , not necessarily of positive scalar curvature.
Let g k tor be a torpedo metric on R k such that scal(g k tor ) + scal(g N ) = scal(g N + g k tor ) > 0. The precise definition of a torpedo metric will be given in (2.9) below, and for the time being, let us only list the most important features. Firstly, g k tor is an O(k)-invariant metric on R k . Secondly, let ψ : (0, ∞) × S k−1 → R k be the polar coordinate map and dξ 2 be the round metric on S k−1 . We require that ψ * g k tor = dt 2 + δdξ 2 on [R, ∞) × S k−1 for some R > 0 and δ > 0. Thirdly, scal(g k tor ) ≥ 1 δ 2 (k − 1)(k − 2). We define the subspace R + (M, ϕ) := {h ∈ R + (M )|ϕ * h| N ×B k R = (g N + g k tor )| N ×B k R } ⊂ R + (M ). Theorem 1.2 (Chernysh [3], see also Walsh [28]). Let ϕ : N × R k → M be an open embedding as before with k ≥ 3. Let g N ∈ R(M ) be a Riemannian metric on N , not necessarily of positive scalar curvature. Let g k tor be a torpedo metric on R k so that the product metric g N + g k tor on N × R k has positive scalar curvature. Then the inclusion map is a weak homotopy equivalence.
The main bulk of this paper is devoted to a detailed discussion of the proof of Theorem 1.2. Remark 1. 3. What Gromov and Lawson proved is that under the hypotheses of Theorem 1.2 and for closed N , R + (M, ϕ) = ∅, provided that R + (M ) = ∅. Later, Gajer [6] improved their result and proved that the inclusion map R + (M, ϕ) → R + (M ) is 0-connected. Remark 1.4. If N = S d−k and g N is the round metric, one obtains a zig-zag where ϕ : S k−1 × R d−k+1 → M ϕ is the opposite surgery datum. It follows that R + (M ϕ ) = ∅ and R + (M ) R + (M ϕ ) if 3 ≤ k ≤ n − 2.
More generally, Theorem 1.2 implies the following cobordism invariance result.  The best-known special case is B = BSpin(d). In that case, the hypothesis that M i → BSpin(d) is 2-connected just means that M i is simply connected. For such manifolds, Theorem 1.5 follows in a straightforward manner from Theorem 1.2 and the proof of the h-cobordism theorem (see e.g. [14,Theorem VIII.4.1]), as explained in [28, §4]. The general case requires techniques from surgery and handlebody theory which are not so well-known, which is why we include the proof in §6.
Chernysh also proved a version of Theorem 1.2 for a fixed boundary condition, which is used in an essential way in [5]. To state it, let ∂ϕ : ∂N × R k → ∂M be the induced embedding and let g ∈ R + (∂M, ∂ϕ) be a fixed boundary condition. We let R + (M, ϕ) g := R + (M, ϕ) ∩ R + (M ) g . Theorem 1.6 (Chernysh [4]). Under the hypotheses of Theorem 1.2, the inclusion map R + (M, ϕ) g → R + (M ) g is a weak homotopy equivalence.
The proof of Theorem 1.6 is only sketched in [4]. We give a detailed proof, somewhat different from the proof envisioned in [4], in §5.2. Besides Theorems 1.2 and 1.1, the proof uses the (elementary) corner smoothing technique which was developped in [5, §2].
When [3] appeared, his result was apparently perceived as a curiosity and drew little attention. This has changed in recent years: Theorem 1.2 is an irreplacable ingredient in the papers [2] and [5]. Important parts of [3] are written in a fairly obscure way, and the paper has never been published. Later Walsh published a paper [28] containing a proof of Theorem 1.2, but many relevant details are not addressed in [28]. Because of the importance of the result for [2] and [5], the first named autor wanted to make sure that the result is correct and that he understands the proof properly. He suggested checking [3] and [28] as a project for the second author's Master's thesis. The present paper is the result of this checking process. Let us summarize our findings.
(1) One half of the proof of Theorem 1.2 is virtually identical to the proof of the original Gromov-Lawson result. We found one small computational error, which is reproduced in various expositions of the result ( [20], [27]). This error looks harmless at first sight, but enforces an alternative argument at one key juncture of the proof. (2) All other arguments in Chernysh's paper are essentially correct and complete, albeit some parts of his paper are very intransparent and hard to decipher. (3) [28] leaves many questions open. In particular, it remains unclear to us how to fill in the details of the proof of Lemma 3.3 loc.cit., without using the quite technical computations of [3, §3] or computations of a similar delicacy. We only consider Riemannian metrics on M which have a simple structure near ∂M . More precisely, for c ∈ (0, 1), we denote by R(M ) c the space of all Riemannian metrics h on M such that h = g + dt 2 on ∂M × [0, c] for some metric g on ∂M .
We topologize R(M ) c as a subspace of the Fréchet space of smooth symmetric (2, 0)-tensor fields on M , with the usual C ∞ -topology. Now let R + (M ) c ⊂ R(M ) c be the subspace of all Riemannian metrics with positive scalar curvature (this is an open subspace). It follows from [17,Theorem 13] and [9,Proposition A.11] that R + (M ) c has the homotopy type of a CW complex.
If h ∈ R + (M ) c and h = g + dt 2 on ∂M × [0, c], then scal(g + dt 2 ) = scal(g) and so the metric g on ∂M necessarily has positive scalar curvature. This defines a restriction map There is no a priori reason why (2.1) should be a homeomorphism. However: Proof. The inclusion maps R + (M ) b → R + (M ) c and R + (M ) b g → R + (M ) c g are closed embeddings. Hence the Lemma then follows from the next one, which is a general fact.
. be a sequence of closed embeddings of Hausdorff spaces and let f n : X n → Y be a compatible sequence of maps. Then the continuous bijection ψ : colim n (f −1 n (y)) → (colim n f n ) −1 (y) is a weak homotopy equivalence.
Proof. It is enough to prove that if K is compact Hausdorff and h : K → colim n (f −1 n (y)) is a map (of sets), then h is continuous if and only if ψ • h is continuous. If g := ψ • h is continuous, then we can consider g as a map to colim n X n . By [23,Lemma 3.6], there is n and k : K → X n (continuous), so that g := i n • k (i n : X n → colim n X n is the natural map). Now k maps into f −1 n (y), and so h can be written as the composition K k → f −1 n (y) → colim n (f −1 n (y)) of continuous maps.

2.2.
The trace construction. For the proof of both, Theorem 1.2 and Theorem 1.1, we need a tedious but straightforward calculation using the standard formulas of Riemannian geometry. We include the proof because we do not know an explicit reference. Lemma 2.4. Let g : R → R(M ) be a smooth path. Let h := dt 2 + g(t) be the induced metric on R × M . Then the scalar curvature of h is given by the formula scal(h) = scal(g(t)) + 3 4 g ik g jl g ij,0 g kl,0 − g kl g kl,00 − 1 4 g ik g jl g ik,0 g jl,0 .
Here we use a local coordinate system in M and the Einstein summation convention. Moreover, g ij are the components of the metric tensor of g, g ij the components of its inverse. A symbol as g ij,k denotes the derivative of g ij with respect to the kth coordinate and similarly for higher derivatives. The 0th direction is the R-direction.
Proof. Fix a local coordinate system (x 1 , . . . , x d ) on M and define a coordinate system on R × M by taking x 0 = t, the R-variable. We now let g ij be the components of g in these coordinates and h ij those of h. Let g ij and h ij be the components of the inverses of the metric tensors. Note that We write Γ k ij , R i jkl and S for the Christoffel symbols, the components of the curvature tensor and the scalar curvature of h and use the symbols γ k ij , r i jkl and s for those associated with g. Without any further comment, we use the Einstein summation convention. With these notations in place, we have, essentially by definition, ]. The symmetry property Γ k ij = Γ k ji , is obvious. The same formulas of course hold for g and its associated objects. Using these formulas, one computes The relevant components of the curvature tensor are R k 0k0 = + 1 4 g ki g jl g ij,0 g kl,0 − 1 2 g kl g kl,00 , R 0 j0l = − 1 2 g jl,00 + 1 4 g mi g lm,0 g ij,0 and (for j, k, l = 0) R k jkl = r k jkl + 1 4 g ki g il,0 g kj,0 − 1 4 g ki g ik,0 g lj,0 .
Lemma 2.5. [6] Let M be a compact manifold, P a compact space and let G : Proof. Lemma 2.4 shows that there is C > 0 so that which immediately implies the claim.

Rotationally invariant metrics. Let
→ tv be the polar coordinate map. We denote by dξ 2 the round metric on S k−1 . Furthermore, S k−1 r ⊂ R k denotes the sphere of radius r.
(2) a(t) ≡ 1 holds if and only if the rays t → tv are unit speed geodesics for all v ∈ S k−1 . In this case we call g a normalized rotationally symmetric metric. (3) Under the hypothesis of (2), f is the restriction of an odd smooth function f : R → R with f (0) = 1. We call f the warping function of g. (4) In that situation, the scalar curvature of g is given by Proof. For part (1), one uses that for each v ∈ S k−1 , there is an A ∈ O(k) such that Av = v and A| v ⊥ = − id. It follows that at each point 0 = x ∈ B k R , the spaces span{x} and T x (S k−1 x ) are orthogonal with respect to g. Since dξ 2 is, up to a constant multiple, the only O(k)-invariant metric on S k−1 , the claim follows. Part (2) is clear. Part (3) can be found in [18, §3.4], and the computation for part (4) in [18, p. 69].
We denote the scalar curvature of the metric dt 2 + f (t) 2 dξ 2 by The function f (t) = sin(t) on [0, π) gives a metric which is isometric to the usual round metric on S k . It has σ(f ) = k(k − 1). Let us now give the precise definition of the torpedo metrics. Definition 2.9. A torpedo function of radius δ > 0 is a function f : [0, ∞) → R which is the restriction of a smooth odd function with f (0) = 1, such that The metric dt 2 + f (t) 2 dξ 2 on R k is called a torpedo metric of radius δ.
Let us give a concrete construction of a torpedo function. Let > 0 be small and let u : [0, ∞) → R be a function satisfying • u ≤ 0 (together with the previous conditions, this implies 0 ≤ u ≤ 1).
We define h 1 (t) := sin(u(t)). By (2.7) we have , so that h 1 is indeed a torpedo function of radius 1 (with R ≥ π 2 + ). For δ > 0, the function is a torpedo function of radius δ.For the rest of this paper, we fix a torpedo function h 1 of radius 1, and define h δ (t) := δh 1 ( t δ ).

The parametrized Gromov-Lawson construction
In this and the following section, we prove Theorem 1.2, and we begin with the precise statement. Let N and M be compact manifolds with collared boundary and let ϕ : N × R k → M be an open embedding with k ≥ 3. We assume that ϕ −1 (∂M ) = (∂N ) × R k and let ∂ϕ : ∂N × R k → ∂M be the induced embedding. Furthermore, we assume that φ is compatible with the chosen collars, From now on, we usually identify N × R k with an open subset of M via ϕ.
Let g N be a Riemannian metric on N which is of the form g ∂N + dt 2 on ∂N × [0, 1). It is not required that scal(g N ) > 0. Let A := inf(scal(g N )) ∈ R and pick δ > 0 so that 1 tor be a torpedo metric on R k of radius δ, and let R > 0 be as in Definition 2.9. For c > 0, define are weak homotopy equivalences.
The proof that we give will apply simultaneously to both cases, and for notational simplicity, we deal only with R + (M, ϕ) → R + (M ). The proof is in two steps. We introduce an intermediate In this section, we show: The proof of Proposition 3.2 is essentially the same as the original argument by Gromov and Lawson [8] (but note that Rosenberg-Stolz [20] corrected mistakes in [8]).
3.1. Adapting tubular neighborhoods. In the proof of Theorem 1.2, we shall use several devices to change a Riemannian metric. One such device (which plays a minor, more technical role) is by suitable isotopies.
, is a unit speed geodesic.
For example, each g ∈ R + rot (M ) is, by definition, normalized on the R-tube around N . Proposition 3.4 (Adapting tubular neighborhoods). Let (K, L) be a finite CW-pair and let G : Then there exists r 0 ∈ (0, r] and a continuous map F : The first is the fixed isomorphism, the second is induced by the bundle metric g and the third is the Riemannian exponential map of g (and is only partially defined). The metric g is normalized on the r 0 -tube around N if and only if φ g is defined on N × B k r0 and agrees with φ there. Since N and K are compact, there is r 0 > 0 so that φ G(x) is defined on N × B k r0 , injective and has image in N × R k ⊂ M . There is an isotopy In the case K = * , this follows from the well-known result that tubular neighborhoods are unique up to isotopy [11,Theorem 4.5.3,4.6.5]. The proof given in loc.cit. carries over to the parametrized and relative case without change.
An instance of the parametrized isotopy extension theorem [26, Theorem 6.1.1] shows that there exists F : 3.2. Gromov-Lawson curves. One important step in the proof of Theorem 1.2 (well-explained in e.g. [20], [27]) is to obtain a deformation of a psc metric g on M by a deformation of M inside M × R and to take the metric induced by g + dt 2 .
(5) Γ λ (0) lies on the y-axis, and this is the only point where Γ λ meets the y-axis. Moreover, it does so at a right angle and follows the arc of a circle (of possibly infinite radius) in the region where r ≤ 1 2 r ∞ .
A typical Gomov-Lawson curve is shown in figure 3.2. The indicated points (y i , r i ) are important for the construction of these curves.
r-axis y-axis , and so we can extend E Γ (λ, ) as the identity over all of M . Note that E Γ (0, ) is just the inclusion x → (x, 0). Let g Γ λ be the Riemannian metric on M obtained by restricting the product metric on M × R to the image of E Γ (λ, ) and pulling back to M . The key argument for the proof of Proposition 3.2 is the following result.
Proposition 3.7. Let K ⊂ R(M ) be compact. Suppose that each g ∈ K is normalized on the r 0 -tube around N and that scal(g) ≥ B g for some B g . For every 0 > 0, η > 0, there exists a Gromov-Lawson curve Γ such that (1) the outer width of Γ is at most r 0 , (2) the inner width of Γ is at most 0 , (4) Moreover, for each > 0, we can arrange the length to be at least .

Construction of the Gromov-Lawson curve.
In this subsection, we prove Proposition 3.7. We need a formula for the scalar curvature of the metric g Γ λ .
Let I ⊂ R be an interval and let γ : I → R 2 + := {(y, r)|r ≥ 0} be a smooth embedded curve. We assume that whenever γ(t) lies on the y-axis, then near t, γ follows a circle of possibly infinite radius perpendicular to the y-axis. Consider the hypersurface (which is smooth because of the condition on γ near the y-axis). Let us recall some formulas from the geometry of plane curves. In the situation we consider, the derivative vector of γ will lie in the fourth quadrant. We let θ be the angle between γ and the negative r-axis and let κ be the signed curvature of γ. If γ is parametrized by arc-length, the curvature is given bÿ The angle is given by the formula If γ meets the y-axis in a circle of radius ρ < ∞, then κ = − 1 ρ , r = sin(θ)ρ near that point.
For a given Riemannian metric g on N × R k , we get the Riemannian metric g γ on Q γ , obtained by restricting the product metric g + dt 2 on N × R k+1 to Q γ . Lemma 3.10. Let K ⊂ R(M ) be compact. Assume that all g ∈ K are normalized on the r 0 -tube around N . Then there exists 0 < r 1 ≤ r 0 and C > 0 such that for all g ∈ K, and for all immersed curves in the region {(y, t) ∈ R 2 |0 < y ≤ r 1 }, we have Remark 3.11. This estimate originates from the curvature formula computed in [8], [20], [3] or [27]. These papers however contain a small computational error: There the formula has either κ sin(θ) k−1 r instead of κ sin(θ) 2(k−1) . We will point out in the proof of Lemma 3.10 where the error occurs and in Remark 3.22 below, we discuss what impact this has on the proof of Proposition 3.2.
Proof of 3.10. We will abbreviate this proof. A detailed computation can be found in the appendix of [27]. The curvature of g γ is given by where λ i are the principal curvatures of the hypersurface Q γ ⊂ N × R k × R. These are given by where the O-terms arise from the coefficients of the Taylor expansion of g and depend continuously on g. Therefore we get 1 : Rearranging all terms finishes the proof.
Assume that all g ∈ K are normalized on the r 0 -tube around N . Then for each B ∈ R, there exists 0 > 0 such that for ∈ (0, 0 ), the restriction of g to N × S k−1 has scalar curvature at least B.
Proof. Consider the curve γ(s) := (s, ). The restriction of the product metric g + dt 2 to Q γ is the product of g| N ×S k−1 and dt 2 . Hence scal(g γ ) = scal(g| N ×S k−1 ). In the case at hand, the curvature of γ is κ = 0, and θ = π 2 . Hence from Lemma 3.10, we get dominates all other terms.
Another auxiliary result is needed for the proof of Proposition 3.7.
Lemma 3.13. Let a > 0 and consider the ordinary differential equation (3.14) For any choice of initial values h(t 0 ) > 0 and h (t 0 ) < 0, there is T > t 0 and a solution h : be a maximal solution. We do not want to decide whether T 1 = ∞ or T 1 < ∞ and show that both cases lead to the desired conclusion. The quantity is constant as can be seen by differentiating. Also C(t) > 0 because of the initial conditions, and h is bounded from below by C(t 0 ) a .
If T 1 < ∞, we consider the trajectory of (h(t), h (t)) in the phase diagram. Since C(t) is constant, this trajectory lies on the level set C −1 (C(t 0 )). Because T 1 < ∞, this trajectory leaves every compact subset of R 2 . The shape of the level set is so that this implies lim t→T1 h(t) = +∞. Hence by Rolle's theorem, h (T ) = 0 for some (minimal) T > t 0 .
1 This is where the error [27] occurs: k−1 Remark 3.15. One can solve (3.14) explicitly, using that C is conserved. The above proof seems more efficient to us, though.
Proof of Proposition 3.7. We first construct a piecewise C 2 curve α and a homotopy α λ of such curves. By a smoothing procedure, we obtain a homotopy β λ of smooth curves which will yield Γ λ by a suitable reparametrization. We begin with the curve α = α 1 . Let us pick some constants first.
Let us explain the choice of r 2 .
To see this, estimate Together with Lemma 3.10, these two inequalities establish Claim 3.16.
Let us now construct the first part of α. One device to construct a (unit speed) curve is by prescribing its curvature function. More precisely, let J ⊂ R be an interval and s 0 ∈ J. If a function κ : J → R and initial values γ(s 0 ) andγ(s 0 ) (the latter of unit length) are given, then the solution to the differential equation is a unit speed curve with curvature function κ. If κ is piecewise continuous, then γ is piecewise C 2 . We write θ(s) for the angle of the curve γ(s).
By construction, the curves α λ satisfy the psc condition scal(g α λ ) ≥ B g − η 2 , α 0 is the straight line on the r-axis, and α 1 = α. The curves α λ are C 1 and piecewise C 2 , and we need to smoothen them.
For small enough u, the curve β λ,u satisfies the positive scalar curvature condition, namely scal(g β λ,u ) ≥ B g − η. This is no issue at point near which κ λ is continuous. Near the discontinuity points, the angle and height of β λ,u is close to that for α λ , while the curvature of β λ,u oscillates between the minimum and maximum value of κ λ . The decisive estimates (3.17), (3.19) and (3.21) all hold if κ lies between 0 and the allowed maximum value. Note, however, that we might loose a bit scalar curvature.
These conditions enforce that γ 0 lies on the r-axis. The construction of such curves is easy and left to the reader. By Claim 3.16, there is no problem with the psc condition.
It is extended to all of [0, ∞), so that above ρ, it is just the curve s → (0, s).
Remark 3.22. The above proof is almost the same as that of the corresponding result in [20] or [27]. The difference is that in loc.cit., the slightly incorrect version of the curvature formula (3.10) is used. This allows the choice a = 2 in the quoted papers. In that case, the differential equation (3.14) has a simple explicit solution. We can pick a = 2 if k > 4, but if k = 3, we need a > 2, and the argument in loc.cit. does not work as stated there.

3.4.
Completion of the proof of Proposition 3.2. We now give the proof of Proposition 3.2. We shall use the following well-known criterion for a map to be a weak equivalence. Proposition 3.23. Let j : X → Y be the inclusion of a subspace. Then the following are equivalent: (1) j is a weak homotopy equivalence, (2) for every n ≥ 0 and every map G 0 : D n → Y such that G(S n−1 ) ⊂ X, there exists a homotopy G λ starting with G 0 such that G 1 (D n ) ⊂ X and G λ (S n−1 ) ⊂ X for all λ ∈ [0, 1].  In short, we make the metrics G(0, x) normalized on some tube, but in addition, we also take a crude interpolation of G(0, x) to some rotationally invariant metric, without taking the psc condition into account.

So we let
Proof. Choose a Riemannian metric g on M such that g| N ×B k R = g N +g , where g is a a rotationally symmetric normalized metric on B k R . For example, we can take g to be the euclidean metric. Let Choose η > 0 so that ∀x ∈ D n : inf scal(G (0, x)) − 2η ≥ 0 and ∀x ∈ S n−1 : inf scal(G (0, x)| N ×B k R ) − 2η ≥ A. The second condition is implied by the first one if A ≤ 0. If A > 0, then for each point g ∈ R + rot (M ) which is of the form g N + g 0 on N × B k R , we have scal(g| N ×B k R ) > A, since otherwise g 0 won't be a psc metric. Therefore, if we can produce a homotopy G : [0, 1] × D n → R + (M ), so that (1) G(0, ) = G (0, ), (2) inf scal(G(λ, x)) ≥ inf scal(G (0, x)) − η and (3) G(λ, x)| N ×B k R is of the form g N + g 0 (λ, x) for (λ, x) ∈ ([0, 1] × S n−1 ) ∪ ({1} × D n ), with some rotationally invariant normalized metric g 0 (λ, x), then g 0 (λ, x) will have positive scalar curvature for all (λ, x) ∈ [0, 1] × S n−1 , and G is a relative homotopy, and so we have finished the proof of Proposition 3.2.
Next, we determine the parameters 0 and for the Gromov-Lawson curve.
• Let 1 > 0 be small enough, so that for all ∈ (0, 1 ) and for all (s, x), the restriction of G (s, x) to the sphere N × S k−1 has scalar curvature > max(0, A). This is possible by Corollary 3.12.
Lemma 3.26. For all B ∈ R, there exists 0 ∈ (0, 1 ] such that for all (s, x) ∈ [0, 1] × D n , the metric (G (s, x)) γ has scalar curvature at least B, where γ is a curve in the plane with the following properties.
Proof. This is an application of Lemma 3.10, but much easier than Proposition 3.7. Pick r 1 so that the curvature estimate of Lemma 3.10 holds for all G(s, x), with some constant C. At the points where γ is a circle, we have κ = − 1 and sin(θ) = r. There, Lemma 3.10 yields In the region √ 2 ≤ r ≤ , Lemma 3.10 yields • Now we choose 0 > 0 so that 0 < r 0 and that the conclusion of Lemma 3.26 holds with B = A + 2η. According to Proposition 3.7, there exists a Gromov-Lawson curve Γ with parameters η and 0 , which has an inner width r ∞ ≤ 0 .
Let g s,x := G (s, x)| N ×S k−1 r∞ . This metric on N × S k−1 r∞ has scalar curvature at least A + 2η by For each L > 0, we get an induced map The first two derivatives of t → λf ( t L ) are • We pick L so large that λ L b, λ L 2 b ≤ Λ, where Λ > 0 is the constant provided by Lemma 2.5. With these choices, we obtain scal(g λf ( t L ),x + dt 2 ) ≥ A + η. and for λ ∈ [ 1 2 , 1] by By construction, scal(G(λ, x)) > 0 for all x, λ, and if x ∈ S n−1 , then scal(G(λ, x)) > A. These metrics are not normalized, but the curve t → (p, tv) is a variable speed geodesic. This can be rectified by a reparametrization (pull back by an isotopy of N × R k which is the identity outside a compact set and which is of the form (p, v) → (p, h λ ( v )v) for a smooth odd function h λ ). After such a reparametrization, the metrics G(λ, x) are normalized on the r 0 -tube. If x ∈ S n−1 , they stay rotationally symmetric, and the geometric size of the region where they are does not decrease with λ. Hence after reparametrization, G(λ, x) is rotationally symmetric and normalized on the R-tube, for all x ∈ S n−1 . This completes the proof.

Rotationally symmetric metrics
In this section, we complete the proof of Theorem 3.1. Let us first recall some notation. Let A := inf scal(g N ) ∈ R. We choose δ > 0 so that 1 δ 2 (k − 1)(k − 2) + A > 0 and pick a torpedo metric g k tor on R k of radius δ, which is cylindrical outside the disc of radius R, for some R > 0. Recall that R + rot (M ) ⊂ R + (M ) is the space of all psc metrics g on M such that g| N ×B k R = g N + g 0 for some rotationally symmetric normalized psc metric g 0 on B k R . Furthermore, R + (M, ϕ) ⊂ R + rot (M ) is the subspace of those g such that g 0 = g k tor . The goal is to prove the following result, which together with Proposition 3.2 completes the proof of Theorem 3.1.

Proposition 4.1. The inclusion map
is a weak homotopy equivalence.

Preliminary remarks.
A rotationally symmetric normalized metric on B k R is of the form g = dt 2 + f (t) 2 dξ 2 , for some warping function f : [0, R] → R with the properties stated in Lemma 2.6. We also recall the curvature formula The torpedo metric g k tor is given by the warping function h δ as in (2.10). In order for the metric g N + dt 2 +f (t) 2 dξ 2 to have positive scalar curvature, we need to have To allow for more convenient notation when A ≤ 0, we introduce , and the condition on f becomes The most delicate step in the proof of Proposition 4.1 is the following.

Introducing collars.
In order to prove Proposition 4.3, we will change the warping function f by composition with another function h or a 1-parameter family thereof. The composition f • h will have a different domain of definition. In order to obtain a well-defined family of Riemannian metrics on M , we introduce the following construction.
We fix, once and for all, diffeomorphisms ϕ a,b of (0, ∞) for each 0 < a ≤ b such that These are compactly supported and can be extended by the identity to M .
defines a smooth Riemannian metric on M in each of the following cases: ( Proof. We need to show that Let us record some further simple properties of this construction. We omit the easy proof.
(1) In the situation of Lemma 4.4, Λ(g, h, S) is rotationally symmetric and normalized on B k S ⊃ B k R .
(2) Let X be a space and let g : be continuous maps such that h(x) and S(x) satisfy the requirements of Lemma 4.4 and assume that for each x ∈ X, one of the two conditions from Lemma 4.4 is satisfied. Then From now on, we only change the warping function inside the R-disc. Note that the metric Λ(g, h, S), restricted to the complement of N × B k S , is isometric to the metric g. Hence we only need to control the scalar curvature of Λ(g, h, S) inside B k S , where it is determined by (4.2). In particular, our consideration will only involve the metrics on B k S , not on N . Let us make a few observations: If scal(dt 2 + f (t) 2 dξ 2 ) ≥ B > 0, then L'Hôpital's rule shows that and hence .
If h is a function as in Lemma 4.4, then the scalar curvature of dt 2 + (f • h) 2 dξ 2 is given by using the self-explanatory notation f (h) := f • h.
We call u r,s a sloping function with parameters a, b, p and q the resulting slope. The situation is depicted in the following figure. Proof. We construct u r,s by constructing its second derivative, and do this by first constructing a piecewise continuous approximation to the second derivative of u r,s . Choose q ≤ b 10 p. Define a piecewise continuous function w r,s : R → R by 19 20 a]    A straightforward, but lenghty integral computation reveals that w r,s has all the desired properties, except that it is only piecewise C 2 . For suffienctly small , consider the convoluted function v r,s := ξ * w r,s .
The point d r,s is the unique point with v r,s (d r,s ) = β.
But now f ≤ 1 in the relevant region, which implies f (t) ≤ t. Since 0 ≤ v ≤ C t , the last term is using that f ≤ T . But this is nonnegative, by our choice of T . (3) Now we turn to the region x ≥ 1 2 .
Here we have f x = h δ , the torpedo function of radius δ. In the region 1 2 ≤ x ≤ 2 3 and 0 ≤ λ 2 3 , we merely change the point where the warping function obtained by composition with v ... or u ... is glued to the original metric, until this gluing is done in the region where the warping function f x = h δ is constant. There is no problem in doing this, as all the functions u 1,s and v 1,s are linear with slope 1 beyond c 1,s and d 1,s . The concrete realization by formulas is the gluing now takes place in the region where f x = h δ is constant. Hence (compare Lemma 4.4), we are now free to change the functions u 1,3λ and v 1,3λ−1 by functions whose derivative at the relevant point is not equal to 1. We use this additional freedom to construct the homotopy in the region x ≥ 2 3 . (4) In the region 2 3 ≤ x ≤ 5 6 , we use the first parameter in the sloping and bending function and "dampen" those. To that end, let us pick η ∈ (0, 1) (we have to pick η small enough so that the next step goes through). We set, for 2 3 ≤ x ≤ 5 6 , wherec x ,3λ is the unique point with u 6(η−1) x +5−4η,3λ (c x ,3λ ) = R andd x ,3λ−1 is the unique point with v 6(η−1) x +5−4η,3λ−1 (d x ,3λ − 1) =c x ,1 . All the curvature estimates done in this proof so far apply as well when the sloping function u 1,3λ is replaced by u r,3λ and the bending function v 1,3λ−1 is replaced by v r,3λ−1 , for r > 0. Hence the above formula defines metrics of positive scalar curvature.
For x = 5 6 , we can write , and a λ ∈ (0, ∞) is the point with h λ (a λ ) = R. By Lemma 4.10, Lemma 4.11 and the chain rule, we have (4.12) (5) In the region 5 6 ≤ x ≤ 1, we change the function h λ and "pull it away from zero to the region around R". More precisely, we set In other words, the metric G(λ, x) coincides with the original metric on B k R (but it is changed outside this disc), so that we indeed get a relative homotopy. It remains to show that (4.13) defines a psc metric, and for this, we have to pick η sufficiently small. The functions h λ,s are just translated versions of h λ , and so their second derivatives still satisfies (4.12). But recall Lemma 4.8 and Remark 4.9: together, they show that (4.13) defines a psc metric, as long as we pick η small enough so that ,1]×D n of smooth functions and a λ,x ∈ R, depending continuously on λ and x so that (4.14) By Proposition 4.3, we may assume that G 0 (x) is given, on N × B k R , by g N + dt 2 + f x (t) 2 dξ 2 , where the warping function f x satisfies • for x ≥ 1 2 , f x is the δ-torpedo function h δ (in 4.3, this is only required for x = 1, but an obvious homotopy achieves this condition for x ≥ 1 2 ).
The desired deformation of warping functions is given on the interval [0, R] by the formula (note and pick R ∞ > R large enough so that there exists an a as in 4.15. Finally, on the interval [R, 5. The fibration theorem and Theorem 1.  This has all the desired properties, except that the scalar curvature of C n (p, s, t) is not necessarily positive. A routine application of compactness proves that lim n→∞ C n (p, s, t) = G(p, st), uniformly in all variables (the target space has the Fréchet topology, as usual). Since R + (M ) ⊂ R(M ) is open, we find that for sufficiently large n, the scalar curvature of C n (p, s, t) is positive for all p, s, t.
Define C := C n for such an n.
The existence of the function Λ with the asserted property follows from the properties of C, from the compactness of P and from Lemma 2.4.
Proof of Theorem 1.1. Let P be a disc and consider a lifting problem Since P is compact, we find δ > 0 such that F (P × 0) ⊂ R + (W ) 2δ , by the observation made in the proof of Lemma 2.2. We will construct a continuous map K :

5.2.
Proof of Theorem 1.6. The proof uses an auxiliary construction. For r ≥ 0, we let W r := W ∪(M ×[0, r]) be the result of gluing an external collar of length r to W and define N r analogously. We extend the metric g N on N to one denoted g N,r on N r cylindrically. The embedding ϕ : N × R k → W gets extended in the obvious way to an embedding ϕ r : N r × R k → W r . We let R + (W r , ϕ r ) ⊂ R + (W r ) be the space of psc metrics which are of the form g N,r + g k tor on ϕ r (N r ×B k R ). Extending psc metrics cylindrically over M ×[0, r] defines maps R + (W r ) → R + (W s ) and R + (W r , ϕ r ) → R + (W s , ϕ s ) for s > r. Restriction to the boundary of W r defines restriction maps res : R + (W r ) → R + (M ) and R + (W r , ϕ r ) → R + (M, ∂ϕ) which are compatible. In the colimit, we obtain a commutative diagram colim r→∞ R + (W r , ϕ r ) colimr→∞ res / / colim r→∞ R + (W r )  Let us postpone the proof of Lemma 5.5 for the moment, and explain how to finish the proof of Theorem 1.6.
Proof of Theorem 1.6. The bottom horizontal map in (5.4) is a weak equivalence, by Theorem 1.2. The inclusion maps R + (W r ) → R + (W s ) are weak homotopy equivalences, and hence so is the inclusion map R + (W ) → colim r→∞ R + (W r ). This is proven in the same way as the elementary [2, Lemma 2.1 and Corollary 2.3].
Each individual map R + (W r , ϕ r ) → R + (W r ) is a weak equivalence, by Theorem 1.2 and hence so is the top horizontal map in (5.4). It follows that the inclusion R + (W, ϕ) → colim r→∞ R + (W r , ϕ r ) is a weak equivalence.
The right vertical map in (5.4) is a Serre fibration. This follows easily from Theorem 1.1 and a colimit argument.
Together with Lemma 5.5, it follows that the induced map on fibres is a weak equivalence. Over g ∈ R + (M, ∂ϕ), this is the bottom map of the diagram R + (W, ϕ) g / / R + (W ) g colim r→∞ R + (W r , ϕ r ) g / / colim r→∞ R + (W r ) g , and we want to know that the top map is a weak equivalence. Thus to conclude the proof of Theorem 1.6, it enough to prove that the two vertical maps in (5.6) are weak equivalences. For the right hand vertical map, this follows immediately from [2, Lemma 2.1], and for the left hand side map, we use [5, Corollary 2.5.4] (whose proof is elementary, but slightly lengthy).
Proof of Lemma 5.5. This is similar to, but easier than the proof of Theorem 1.1. Let P be a disc and consider a lifting problem By compactness of P , there is r ≥ 0 so that F (P × 0) ⊂ R + (W r , ϕ r ). Define L(p, s) by the formula (5.3) and let b as in loc.cit. Define H(p, s) ∈ R + (W r+b , ϕ r+b ) to be equal to L(P, s) on M ×[r, r +b] and equal to F (p, s) on W r . This has all the desired properties.

Cobordism invariance of the space of psc metrics
This section is devoted to the proof of Theorem 1.5.   shows that M 0 → W • is 2-connected, and so is M 0 → W Proof of Proposition 6.4. Part (1) follows quickly from handle trading [13], [25]. If d = 6, handle trading implies that W is of handle type [2,3]. But handle trading is not enough: we can only barter the 2-handles for 4-handles. In fact, it is not correct that a 6-dimensional cobordism W : M 0 ; M 1 with both inclusions 2-connected has handle type [3,3]: look at a 6-dimensional h-cobordism with nontrivial Whitehead torsion to see why. To deal with part (2), we invoke the following result, whose proof is contained in the proof of [7, Lemma 6.21] (and which has its origins in [15]). Let W : M 0 ; M 1 be a cobordism of dimension 2n ≥ 6 such that both inclusions M i → W are (n − 1)-connected. Then for sufficiently large r, the cobordism W r (S n × S n ) has handle type [n, n].