The action of the mapping class group on metrics of positive scalar curvature

We present a rigidity theorem for the action of the mapping class group $\pi_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the $2$-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected $7$-dimensional $\mathrm{Spin}$-manifolds.

from the mapping class group of M to the group of homotopy classes of homotopy selfequivalences of R + (M ). Our main result is that the image of this map is often very small. To state this precisely without too many technicalities, we confine ourselves to the special case where M is simply connected and Spin in this introduction, but remark that we prove results for all manifolds of dimension at least 6.
Let be a Spin-structure on M and recall that a Spin-diffeomorphism of (M, ) is a pair (f,f ) consisting of an orientation preserving diffeomorphism f : M → M and an isomorphismf : f * → of Spin-structures. We denote by Diff Spin    Theorem B. Let M be a simply connected, stably parallelizable manifold of dimension d − 1 ≥ 6, equipped with a Spin-structure. Let (f,f ) be a Spin-diffeomorphism.
Then the map f * : R + (M ) −→ R + (M ) is homotopic to the identity unless d ≡ 1, 2 (mod 8). In the latter case, (f 2 ) * is homotopic to the identity.
Remark. Any orientation-preserving diffeomorphism f of M can be lifted to a Spin-diffeomorphism if M is simply connected. Therefore f * : R + (M ) → R + (M ) is homotopic to the identity for each orientation preserving diffeomorphism of M if d ≡ 1, 2 (mod 8). This conclusion does not hold for orientation-reversing diffeomorphisms (for example it is false if f : S 7 → S 7 is an orthogonal matrix of determinant −1).
For more examples we refer to [Fre19,Chapter 4.1]. Using Theorem A one can also use computational results on π 0 (R + (M )) and π 0 (Diff x0 (M )) (for example [BERW17] and [GRW16]) to find elements in π 0 and π 1 of the observer moduli space of psc-metrics for certain manifolds. In the situation of Theorem A, assume that f * g is homotopic to g for one g ∈ R + (M ). Then the mapping torus admits a psc metric and hence α(T f ) = 0. This has an interesting consequence for manifolds of dimension 7. Recall that the map Ω Spin used the Atiyah-Singer index theorem to show that inddiff(g, f * g) = α(T f ). Hence, the α-invariant of the mapping torus of f is an obstruction to f acting trivially on π 0 (R + (M )). For S d−1 with d ≥ 9 and d ≡ 1, 2 ( mod 8) there exist diffeomorphisms f with α(T f ) = 0 which implies that R + (S d−1 ) is not connected in these dimensions. Theorem A shows that these are the only dimensions where simply connected, stably parallelizable manifolds admit such a diffeomorphism.
Remark. In [BERW17] a factorisation result similar to Theorem A is proven. It is shown that for certain manifolds the image of π 0 (Diff ∂ (M 2n )) → π 0 (hAutR + (M ))) is abelian, where Diff ∂ denotes those diffeomorphisms that fix a neighbourhood of the boundary point-wise. Using an obstruction theoretic argument they conclude that this map factors through π 1 (M T Spin(2n)). This has been upgraded in [ERW19a] and [ERW19b] to hold for a bigger class of manifolds. Theorem A directly implies abelianess of the image and improves the named results since the map π 1 (M T Spin(d − 1)) → Ω Spin d has nontrivial kernel.
1.2. Outline of the proof. Theorem A follows from a more general, cobordism theoretic result which we will develop in this outline. The main geometric ingredient is a parametrised version of the famous Gromov-Lawson-Schoen-Yau surgery theorem due to Chernysh. Let ϕ : S k−1 × D d−k → M be an embedding and let R + (M, ϕ) := {g ∈ R + (M )|ϕ * g = g • + g tor } be the space of those metrics that have a fixed standard form on the image of ϕ. where the first map is the homotopy inverse to the inclusion and the second one is given by cutting out ϕ * (g • + g tor ) and pasting in ϕ op * (g tor + g • ). Next we want to define the map S for general cobordisms. In this paper, a cobordism between (d − 1)-dimensional manifolds M 0 and M 1 is a triple (W, ψ 0 , ψ 1 ) consisting of a d-dimensional manifold W whose boundary has a decomposition ∂W = ∂ 0 W ∂ 1 W and diffeomorphisms ψ i : ∂ i W → M i for i = 0, 1. We will only consider Spinstructures on cobordisms in the final step of the proof. An admissible handle decomposition H of (W, ψ 0 , ψ 1 ) is a collection of manifolds N 1 , . . . , N n , embeddings ϕ i : S ki−1 × D d−ki → N i with d − k i ≥ 3 for i = 1, . . . , n and diffeomorphisms and (W, ψ 0 , ψ 1 ) is called an admissible cobordism if it admits an admissible handle decomposition. By the theory of handle cancellation developed by Smale [Sma62] (see also [Ker65] and [Wal71]), a cobordism is admissible if the inclusion ψ −1 1 : M 1 → W is 2-connected. For a cobordism W with an admissible handle decomposition H we define the surgery map S W,H : Lemma E. Let d ≥ 7. Then the homotopy class of S W,H is independent of the choice of admissible handle decomposition H. We will write S W := S W,H . If the inclusion ψ −1 0 : M 0 → W is 2-connected as well, S W is a weak homotopy equivalence.
Remark. In [Wal14], Walsh constructed a psc metric g H on (W, H) that restricts to a given metric g 0 on ∂ 0 W . He shows that the homotopy class of g H is independent of H. Using boundary identifications ψ i this gives a well defined map S W : π 0 (R + (M 0 )) → π 0 (R + (M 1 )). We adapt the proof from [Wal14] so that we obtain a well-defined homotopy class of a map of spaces inducing Walsh's map on π 0 .
To prove this one uses Cerf theory to show that different handle decompositions are related by a finite sequence of elementary moves. The parametrized handle exchange theorem of Igusa [Igu88] ensures that these moves keep the handle decomposition admissible. Igusa's theorem is the point where d ≥ 7 is used. Next we show surgery invariance of S W .
Lemma F. Let d ≥ 7, let M 0 , M 1 be two (d − 1)-manifolds, let W be an admissible cobordism and let Φ : Now we are able to derive the general cobordism theoretic result. LetΩ Spin d denote the following category: objects are given by simply connected, (d − 1)-dimensional Spin-manifolds M and morphisms from M 0 to M 1 are given by cobordism classes of d-dimensional Spin-cobordisms (W, ψ 0 , ψ 1 ). Note that every such cobordism class contains an admissible cobordism and two admissible cobordisms in the same class are related by a sequence of surgeries satisfying the index constraints from the previous Lemma. (1) On objects, S is given by is represented by (tr (ϕ), id, id) for tr (ϕ) the trace of a surgery datum ϕ with codimension at least 3, then S(α) = S ϕ . Furthermore, S is uniquely determined by these properties, up to natural isomorphism.

This immediately implies Theorem A: For a closed Spin-manifold
Acknowledgements. This paper is a streamlined version of the author's PhDthesis [Fre19] at WWU Münster. It is my great pleasure to thank Johannes Ebert for his guidance, lots of comments and many enlightening discussions. I also would like to thank Lukas Buggisch, Oliver Sommer and Rudolf Zeidler for many fruitful discussions.
2. Handle decompositions and the surgery map 2.1. Spaces of Riemannian metrics. For a closed manifold M we denote by R(M ) the contractible space of all Riemannian metrics on M equipped with the (weak) Whitney C ∞ -topology. The subspace of metrics whose scalar curvature is strictly positive will be denoted by R + (M ).
Definition 2.1. Let M and N be compact manifolds of dimension d − 1 ≥ 0 and let ϕ : N → M be an embedding. For a metric g on N , we define • denotes the round metric and g d−ki tor a torpedo metric 1 . If there is no chance of confusion, we will omit the dimension of these metrics.
There is the following generalization of the famous Gromov-Lawson-Schoen-Yau surgery theorem (cf. [GL80] and [SY79]) which is originally due to Chernysh [Che04] and has been first published by Walsh [Wal13]. A detailed exposition of Chernysh's proof can be found in [EF18]. Let M be a (d − 1)-manifold and for i = 1, . . . , n let N i be closed manifolds of dimension (k i − 1). Let d − k i ≥ 3 for all i and let g Ni be metrics on N i such that scal(g Ni + g tor ) > 0. Let N : is a weak homotopy equivalence. In particular, if M 1 is obtained from M 0 by performing surgery along ϕ : If furthermore k ≥ 3, the rightmost map in this composition is also a weak equivalence and we obtain a zig-zag of weak equivalences from R + (M 0 ) to R + (M 1 ).
Remark 2.3. The space R + (M ) is homotopy equivalent to a CW -complex (see [Pal66,Theorem 13]). By Whitehead's theorem, a weak homotopy equivalence of CW -complexes is an actual homotopy equivalence. Therefore we may assume that weak homotopy equivalences of R + (M ) have actual homotopy-inverses.
2.2. Handle decompositions of cobordisms. In this section we discuss handle decompositions of a cobordism W . First, we give a model for attaching a handle. We adapt the one given in [Per17,Construction 8.1] which is convenient.
Construction 2.4 (Standard trace). Let ε ∈ (0, 1 4 ) be fixed and let k ∈ {0, . . . , d}. We fix once and for all an with the following properties (see Figure 1  restricts to the round metric on the boundary. For precise definitions see [Che04], [Wal11] or [EF18]. (3) We have the following equalities for intersections (4) The boundary of T k is given by We call T k the standard trace of a k-surgery.
There is a Morse function h ϕ : tr (ϕ) → [0, 1] with precisely one critical point with value 1 2 and index k. We define For a surgery datum ϕ in M there is an obvious reversed surgery datum ϕ op : S d−k−1 × D k → M ϕ and there is a canonical diffeomorphism (M ϕ ) ϕ op ∼ = M . We define the attaching sphere of ϕ to be ϕ(S k−1 × {0}) ⊂ M and the belt sphere of ϕ as Definition 2.6.
In order to compare different handle decompositions of a manifold, we need to describe a model for handle cancellation. Let W : M 0 ; M 1 be a cobordism which has a handle decomposition with two handles 2 : Let ϕ : S k−1 × D d−k → M 0 and ϕ : S k × D d−k−1 → (M 0 ) ϕ be two surgery data such that the belt sphere of ϕ and the attaching sphere of ϕ intersect transversely in a single point. By [Wal16,Theorem 5.4.3] there exists an embedding of a disk D d−1 ∼ = D ⊂ M 0 such that im ϕ ⊂ D and im ϕ ⊂ D ϕ . Therefore it suffices to have a closer look at handle cancellation on the sphere.
Since the belt sphere of ϕ and the attaching sphere of ϕ intersect transversally here, there is a disc S k Because of transversality we may isotopy ϕ such that Lemma 5.4.2.]) and also A : and hence we can change coordinates on S d−1 by changing the embedding D d−1 → M such that ϕ is the embedding of the first factor of the solid torus decomposition . We get an induced map , 2 For ease of notation we assume that all boundary identifications are given by the identity.
where we identify ( . Because of transversality we may isotope ϕ so that (a k ϕ ) • ϕ is equal to the inclusion of the first factor in This is a solid torus decomposition of (S d−1 ϕ ) ϕ . We get a diffeomorphism H k : and the lower boundary point-wise. We may also assume that H k restricts on the upper boundary to a diffeomorphism η k : For every k ∈ {0, . . . , d} we fix the diffeomorphisms H k (and hence η k ) once and for all. The following proposition is well known and can be proven by analyzing paths of generalized Morse functions using Cerf theory (see [GWW12,Theorem 3.4] or [Fre19, Proposition 1.5.7]).
Proposition 2.8. Let d ≥ 7. Then any two handle decompositions of W only differ by a finite sequence of the following moves: (1) An identifying diffeomorphism is replaced by an isotopic one.
(2) A surgery datum is replaced by an isotopic one. ( (4) The order of two surgery data with disjoint images is changed.
(5) Let ϕ and ϕ be k-and (k + 1)-surgery data such that the belt sphere of ϕ and the attaching sphere of ϕ intersect transversally in a single point. Then the two handles are replaced by the identifying diffeomorphism id # η k .

2.3.
Hatcher-Igusa's 2-index theorem. Since Theorem 2.2 has restrictions on the indices of surgery data, we need to consider handle decompositions with index constraints. Let (W d , ψ 0 , ψ 1 ) : M 0 ; M 1 be a cobordism.
Remark 2.10. It follows from the proof of the h-cobordism theorem due to Smale [Sma62] (see also [Ker65] and [Wal71]) that every admissible cobordism admits an admissible handle decomposition.
Next we want to analyze different admissible handle decompositions. Recall that a birth-death-singularity of a function f : In this case we call (λ − 1) the index of f at p. A function f : W → R that has only non-degenerate and birth-death-singularities is called a generalized Morse function. Theorem 2.12. Let d ≥ 7 and let M 1 → W be 2-connected. Then the space This follows from the parametrized handle exchange theorem. It was first proven by Hatcher [Hat75] "in a short and elegant paper which ignores most technical details" [Igu88, p. 5]. A complete and rigorous proof has been given by Igusa in [Igu88]. Note that there is an index shift: Igusa considers n + 1-dimensional cobordisms, whereas our cobordisms are d-dimensional.
Parametrized Handle Exchange Theorem ([Igu88, p. 211, Theorem 1.1]). Let i, j, k ∈ N and assume that There is a dual version of this: Assume that Proof of Theorem 2.12. Consider the chain of maps If M 1 → W is 2-connected and d ≥ 7, the last three maps are 1-connected. If M 0 → W is 2-connected, the first three maps are 1-connected as well. The theorem follows as H(W ) is connected.
Remark 2.13. There is a small mistake in [Wal14, Proof of Theorem 3.1], where he only requires d ≥ 6. But the map H 0,d−2 (W ) → H(W ) is only 0-connected, i. e. π 0 -surjective but not necessarily π 0 -injective under this assumption. Therefore it does not follow, that H 0,d−2 (W ) is path-connected as claimed in loc. cit.. However, if d ≥ 7 the map is not only π 0 -injective but also 1-connected which is more than needed.
The following result can again be proven by analyzing paths of generalized Morse functions with index constraints: Any two admissible handle decompositions arise from a Morse function having only critical points of index ≤ d − 3. By Theorem 2.12 there exists a path of generalized Morse functions also having only critical points of index ≤ d − 3 and birth-death-points of index ≤ d − 4. The rest of the proof is analogous to the one of Proposition 2.8 (again, see [GWW12, Theorem 3.4] or [Fre19, Proposition 1.5.7 and Proposition 1.6.4]).
Proposition 2.14 ([Fre19, 1.6.4]). Let W : M 0 ; M 1 be an admissible cobordism of dimension d ≥ 7. Then any two admissible handle decompositions of W only differ by a finite sequence of the 5 moves from Proposition 2.8 with the following difference: 5'. Let k ≤ d − 4 and let ϕ and ϕ be k-and (k + 1)-surgery data such that the belt sphere of ϕ and the attaching sphere of ϕ intersect transversally in a single point. Then the two handles are replaced by the identifying diffeomorphism id # η k .
2.4. The surgery datum category. We recall the following method to construct a category. For details see [Mac71,pp. 48].
where O and A are sets called object set and arrow set and ∂ 0 , ∂ 1 are maps A ⇒ O. We say that two arrows We define the category C(G) to have elements of O as objects and morphisms of C(G) are (possibly empty) strings of composable morphisms of A. We call C(G) the free category generated by G.
Proposition 2.17 ([Mac71, p. 51, Proposition 1]). Let C be a small category and let R be a binary relation, i. e. a map that assigns to each pair (a, b) of objects a subset of mor C (a, b) 2 . Then, there exists a category C/R with object set obj C and a functor Q : C → C/R (which is the identity on objects) such that Let Bord d denote the category with objects (d − 1)-manifolds and morphisms given by diffeomorphism classes of cobordisms (W, ψ 0 , ψ 1 ). The main goal of this chapter is to give a presentation of Bord d , i.e. a graph G, a relation R and an equivalence of Let us first construct the graph G. Objects in O are the objects of Bord d and arrows will be given by diffeomorphisms and elementary cobordisms: (1) For a diffeomorphism f : M 0 → M 1 we get an arrow I f ∈ A from M 0 to M 1 .
(2) For a surgery datum ϕ in M we get an arrow S ϕ ∈ A from M to M ϕ . Next, we need to construct the relation R on C(G). Recall that for a diffeomorphism f : M → M and a surgery datum ϕ in M there exists a canonical induced diffeomorphism f ϕ : M ϕ → M f •ϕ . Also, if ϕ and ϕ are two surgery embeddings into M with disjoint images, there are obvious induced surgery data ϕ ϕ and ϕ ϕ on M ϕ and (M ϕ ) ϕ ϕ = (M ϕ ) ϕ ϕ . We define R to be the relation on morphism sets of C(G) generated by the following: (1) I id = id. ( Let ϕ be a k-surgery datum in M and ϕ a (k + 1)-surgery datum in M ϕ such that the belt sphere of ϕ and the attaching sphere of ϕ intersect transversely in a single point. Then S ϕ • S ϕ = I id # η k , where η k is the diffeomorphism described Section 2.2, below Remark 2.7.
Remark 2.18. For isotopic surgery embeddings ϕ and ϕ we get a diffeotopy H of M such that H 0 = id and H 1 • ϕ = ϕ by the isotopy extension theorem. Then Definition 2.19. We define the surgery datum category X d to be C(G)/R and Q : C(G) → X d shall denote the projection functor.
2.5. A presentation of the cobordism category. In this section we prove that the surgery datum gives a presentation of the category Bord d . This is the main result of this chapter.
Theorem 2.20. Let P : C(G) → Bord d denote the functor which is the identity on objects and is given on morphisms by (2) For a surgery datum ϕ in M , S ϕ is mapped to (tr (ϕ), id, id). Then P descends to a functor P : X d → Bord d which is an equivalence of categories.
Proof. First we check well-definedness. By Proposition 2.17 it suffices to show that P respects the relations of X d . ( , id, f • g) and the diffeomorphism is given by the identity on M 0 × [0, 1] and by the map (p, t) → (g −1 (p), t + 1) for (p, t) ∈ M 1 × [0, 1].
(3) Let ϕ be a surgery embedding into M 0 and let f : We will show that both of these are diffeomorphic to X : which is the identity on a collar of the boundary. (7) This is precisely the situation discussed below Remark 2.7. Therefore there is an essentially surjective functor P : X d → Bord d . Every cobordism admits a handle decomposition and hence this functor is full. It is faithful by Proposition 2.8: Any two preimages of a cobordism W under P only differ by a finite sequence of the seven relations of X d .
(2) G a,b to be the graph with the same object set as G and morphisms as follows: connecting M 0 and M 1 and for every surgery embedding ϕ : Proof. The proof goes along the same lines as the proof of Theorem 2.20. For fullness we note that if the inclusions ψ −1 1 : M 1 → W is 2-connected respectively, there exists a Morse function with all indices ≤ d − 3 by Theorem 2.12. Faithfulness follows from Proposition 2.14.
2.6. Definition of the surgery map. Let hTop denote the homotopy category of spaces, i. e. the category with spaces as objects and homotopy classes of maps as morphisms.
(2) For a diffeomorphism f : M 0 where the first map in this chain is the homotopy inverse to the inclusion (cf. Theorem 2.2) and the second one works as follows: For a metricg on M \im ϕ, the metricg ∪ ϕ * (g k−1 ). We will abbreviate S f := S(I f ) and S ϕ := S(S ϕ ).
Lemma 2.25. S induces a well-defined functor X −1,2 Proof. For d ≤ 2 the statement and the proof of this theorem is trivial since mor X −1,2 d is generated by diffeomorphisms and it suffices to note that isotopic diffeomorphisms induce homotopic maps. Therefore we may assume d ≥ 3 throughout this proof. Throughout this proof we will use dashed arrows for maps that contain inverses of weak homotopy equivalences (cf. Remark 2.3).
We need to show that the relations R from Definition 2.19 do not change the homotopy class of S(α) for α ∈ mor X −1,2 d (M 0 , M 1 ). This is obvious for relations 1, 2 and 4. For relation 5 this is easy as well, because g Also, S f •ϕ • I f and I fϕ • S ϕ give homotopic maps because of the following homotopy-commutative diagram.
is homotopic to the inclusion ι: By the Theorem 2.2, the inclusion map ι is a weak homotopy equivalence since d ≥ 4 and hence S ϕ • S ϕ is homotopic to f * . Let g ∈ R + (D, ϕ) g• be a metric in the component of g tor ∈ R + (D) g• which exists by Theorem 2.2. Consider the following diagram: The composition of the top maps is given by gluing in g and the composition of the lower maps is given by gluing in g tor . These two metrics are homotopic relative to the boundary and hence this diagram commutes up to homotopy. The bottom map and the right-hand vertical map are weak equivalences by Theorem 2.2 because d ≥ 4 and k ≤ d − 4. Hence, the inclusion map R + (M, D; g) → R + (M, ϕ) is a weak equivalence as well. Let g ϕ be the metric obtained from g by cutting out ϕ * (g k−1 • + g d−k tor ) and gluing in ϕ op ). The following diagram where the horizontal maps are given by replacing g with g ϕ commutes on the nose with the non-dashed arrows and up to homotopy with the dashed arrow: It again follows that the right-hand vertical map and the right-hand diagonal map are weak equivalences. Note that the composition of the bottom horizontal maps is precisely the map S ϕ . Now letg ∈ R + (D ϕ , ϕ ) g• be a metric in the component of g ϕ ∈ R + (D ϕ ) g• . We get the following diagram which is homotopy-commutative asg and g ϕ are homotopic. The righthand vertical map is a weak equivalence because d − k − 1 ≥ 3 and we deduce that is a weak equivalence as well. Letg ϕ be the metric obtained fromg by cutting out ϕ * (g k • + g d−k−1 tor ) and gluing in ϕ op ). We get the analogous homotopy-commutative diagram: This accumulates to the following diagram where all arrows are weak equivalences: (1) f * Here, the map (1) is given by cutting out g ϕ and gluing ing. Since these are homotopic relative to the boundary, the inside triangle and hence the entire diagram commutes up to homotopy. Therefore, the composition f * • S ϕ • S ϕ • ι is homotopic to the inclusion if and only if the top row composition in this diagram is. In contrast to f * • S ϕ • S ϕ • ι this composition only consists of actual maps which are given as follows: We will denote the path component of a psc-metric g on M by [g] ∈ π 0 (R + (M )). By the above argument it suffices to show that [f * g ϕ ] = [g tor ] ∈ π 0 (R + (D) g• ). This is implied by Lemma 2.26 as follows: We can assume that D ⊂ S d−1 is a hemisphere and we have f * • S ϕ • S ϕ ([g tor ∪ g tor ]) ∼ [g tor ∪ f * g ϕ ] by the above argument for M = S d−1 and h = g tor . After possibly changing the coordinates of the disk D we may assume the following: If a k : is the solid torus decomposition then a k • ϕ is given by the inclusion of the first factor and a k ϕ • ϕ : is also given by the inclusion of the first factor (cf. Section 2.2). In this case we have f = η k . The metric [g tor ∪ g tor ] is homotopic to the round metric by [Wal11, Lemma 1.9] and we have Lemma 2.26 Also g 1 := g tor ∪ f * g ϕ and g 2 := g tor ∪ g tor are both in the image of the inclusion map R + (D) g• → R + (S d−1 ) which is a weak equivalence and since [g 1 ] = [g 2 ] it follows that [g tor ] = [f * g ϕ ] ∈ π 0 (R + (D) g• ).
Lemma 2.26. Let g • ∈ R + (S d−1 ) be the round metric and let a k : be surgery data such that a k • ϕ and a k ϕ • ϕ are both given by the inclusion of the respective first factor. Then Proof. Let g k mtor := (g k−1 ) denote the mixed torpedo metric on(S k−1 ×D d−k )∪(D k ×S d−k−1 ). By [Wal11, Lemma 1.9]) we have (a k ) * g k mtor ∼ g • and hence Now a k • ϕ is given by the inclusion and hence We can now compute But η k was chosen such that (a k ϕ ) ϕ • η k = a k+1 and therefore η * k (a k ϕ ) ϕ * g k+1 mtor = (a k+1 ) * g k+1 mtor ∼ g • .
We get the following Corollary which follows immediately from Lemma 2.25 and Theorem 2.22. Proof. First we note that for 3 ≤ k ≤ d − 3, W Φ is again an admissible cobordism: We have the following diagram: We choose γ, so that the boundaries of all of these are smooth. Then W 1 M 1 ∨S k−1 , Figure 3. Surgery on the cobordism W Note that W 1 and W 1 have the same boundary M 1 given by Next, we show that W 0 , W 1 , W 1 and W op 1 are again admissible. Because of Therefore W 0 → W is (d − k)-connected and we have the following diagram.
and hence M 1 → W 0 is 2-connected, too. So we get a decompositions into admissible cobordisms W = W 0 ∪ W 1 and W Φ = W 0 ∪ W 1 which implies S W = S W1 • S W0 and S WΦ = S W 1 • S W0 . In the homotopy category hTop we have  Figure 4) 5 For k = 3, the cobordism W 1 might not be admissible which is why this case is treated separately. and these diffeomorphisms are supported on a small neighbourhood of M 1 and hence relative to the boundary. This finishes the proof for the case k = 3.  For the case k = 3 we need a different argument, because W 1 might not be admissible in this case. Consider the map given by shrinking the interval and composing with the inclusion of the collar. We will use the following Lemma.
On (M \ im ϕ) × I the diffeomorphism α shall be given by the identity. Next we take diffeomorphisms On the D 3 × D d−3 -parts it is given by the inclusion of the lower or upper hemisphere The entire diffeomorphism is visualized in Figure 5. Therefore we have S (M ×I)Φ ∼ S tr ϕ op • S tr ϕ ∼ id ∼ S M ×I and the proof is finished modulo Lemma 2.31. Proof of Lemma 2.31. We have the following diagram where Mon denotes the space of bundle monomorphisms. Note that the bottommost vertical maps are homeomorphisms because S 2 is stably parallelizable and the middle ones are homotopy equivalences by the Smale-Hirsch immersion theorem (cf. [Ada93, Section 3.9]). The map (1) is 0-connected because of the Whitney embedding (cf. [Hir76,pp. 26]) and the maps (5) and (6) are π 0 -bijections again by the Whitney-embedding theorem. It remains to show that (2) and (3) are 0-connected. Then the map (4) is 0-connected, too. For (2) consider the following diagram of fibrations.
Since d − 4 ≥ 3, the map (2) is 0-connected. The map (3) fits into a similar diagram: Since M 1 → W is 2-connected, the bottom-most map is 0-connected and hence so is the map (3).

Tangential structures and proof of main result
3.1. Tangential structures. In order to get rid of the connectivity assumptions of the category Bord −1,2 d , we need tangential structures. For d ≥ 0 let BO(d + 1) be the classifying space of the (d + 1)-dimensional orthogonal group and let U d+1 be the universal vector bundle over BO(d + 1). Let θ : B → BO(d + 1) be a fibration. We call θ a tangential structure.
Definition 3.1. A θ-structure on a real rank(d + 1)-vector bundle V → X is a bundle mapl : V → θ * U d+1 . A θ-structure on a manifold W d+1 is a θ-structure on T W and a θ-manifold is a pair (W,l) consisting of a manifold W and a θ- An important source of tangential structures are covers of BO(d + 1). For example we have BSO(d + 1) → BO(d + 1) or BSpin(d + 1) → BO(d + 1) or more generally BO(d + 1) k → BO(d + 1), where BO(d + 1) k denotes the k-connected cover of BO(d + 1). Other sources of tangential structures are Moore-Postnikov towers:  (1) The stabilized tangential 2-type of a connected Spin-manifold M of dimension at least 3 is BSpin(d + 1) × Bπ 1 (M ).
(2) The stabilized tangential 2-type of a simply connected, non-spinnable manifold M of dimension at least 3 is BSO(d + 1).
Recall the following lemma which is frequently used when working with surgery results concerning positive scalar curvature.

3.2.
Proof of the main result. We will now prove the general version of Theorem G which is the main result of this article.
Definition 3.5. We define Ω d,2 to be the category given by the following: Objects are given by tupels (M, B, θ,l) where -M is a closed (d − 1)-dimensional manifold.
-l is a stabilized θ-structure such that the underlying map l : M → B is 2-connected. Morphisms (M 0 , B 0 , θ 0 ,l 0 ) to (M 1 , B 1 , θ 1 ,l 1 ) are given by equivalence classes of tupels (W, ψ 0 , ψ 1 ,ˆ , h) where -h : B 0 → B 1 is a map over BO(d + 1). This gives an induced map where −l 1 denotes the bundle map given bŷ Figure 6. A representative of a morphism in Ω d,2 and there exists a (d + 1)dimensional θ 1 -manifold (X, X ) with corners such that there exists a partition of ∂X = i=0,3 ∂ i X together with diffeomorphisms such that θ-structures and diffeomorphisms fit together (see Figure 7). Composition is given by gluing cobordisms along the common boundary: Theorem 3.6. Let d ≥ 7. There is a functor S : Ω d,2 −→ hTop with the following properties: (1) On objects, S is given by S(M, B, θ,l) = R + (M ).
Proof. Let V := (V, ψ 0 , ψ 1 , V ) : (M 0 ,ĥ •l 0 ) ; (M 1 ,l 1 ) be a θ 1 -cobordism. By Lemma 3.4, there exists an admissible θ 1 -cobordism V : M 0 ; M 1 in the same cobordism class. We define S V := S V . By definition of S it is clear that this fulfils the desired properties and is compatible with composition. It remains to show that this is well-defined. Let X : V 0 ; V 1 be a θ 1 -cobordism relative to ∂V 0 = ∂V 1 and let X i : V i ; V i be relative θ 1 -cobordisms such that (V i , M 1 ) is admissible for i = 0, 1. We get a relative θ 1 -cobordism X := X op 0 ∪ X ∪ X 1 : V 0 ; V 0 ; V 1 ; V 1 . Again, by Lemma 3.4, we may assume that ( X, V i ) is 2-connected. So, V 1 is obtained from V 0 by a sequence of surgeries of index k ∈ {3, . . . , d − 2}. One can order these surgeries, so that one first performs the 3-surgeries, the 4-surgeries next and so on up to the d − 3-surgeries. By Lemma 2.30 all of these do not change the homotopy class of S and we may assume that V 1 is obtained from V 0 by a finite sequence of d − 2-surgeries. Reversing these surgeries we deduce that V 0 is obtained from V 1 by a finite sequence of 3-surgeries and by Lemma 2.30 the map S V 0 is homotopic to S V 1 . Hence S is well-defined.
Remark 3.7. Note that if M 0 and M 1 have the same tangential 2-type, there exists an admissible cobordism V in the same cobordism class as V such that (V ) op is admissible as well. Then S (V ) op is an inverse for S V Remark 3.8. As mentioned in Remark 2.29 (see also [Wal11]), Walsh constructed a psc-metric G on an admissible self-cobordism W : M ; M extending a given psc-metric g 0 on the incoming boundary using the same construction used here. He showed that the homotopy class of G restricted to the outgoing boundary does not depend on the handle presentation [Wal14, Theorem 1.3]. Therefore he obtained a map f W ∈ Aut(π 0 (R + (M ))) given by [g 0 ] → [G| M ×{1} ]. By separating the cobordism part of the picture(Section 2.2 to Section 2.5) from the scalar curvature part of the picture (Section 2.6 and Section 2.7) we upgraded this to give an actual homotopy class of a map S W ∈ π 0 (hAut(R + (M ))) inducing Walsh's map on π 0 (R + (M )). The second improvement lies in the cobordism-invariance of S which drastically enlarges its kernel and enables us to define S W for any θ-cobordism W .
Before we start deriving the general version of Theorem A, let us list two interesting facts about the surgery map. The first one is proven by an argument similar to the reduction step in the proof of Lemma 2.25 and uses the notion of left-/right-stable metrics (cf. [ERW19a]).
Let M 0 be a manifold and let M Proposition 3.9. Let M 0 be such that there exists a metric g = g rst ∪g lst ∈ R + (M 0 ) which is the union of a right-stable metric g rst ∈ R + (M Proof of Proposition 3.9. Since W and W are admissible, they consist of handles glued along surgery data with codimension at least 3. By transversality we may assume that all handles are attached in the interior of Q 0 . Hence we can decompose (2) 0 × [0, 1] and a relative cobordism V : Q 0 ; Q 1 (resp. V ). Let g V lst and g V lst represent the resulting path components of S V (g lst ) and S V (g lst ). Since g lst is left-stable µ( , g lst ) is a weak equivalence and S W = µ( , g lst ) −1 • µ( , g V lst ) and S W = µ( , g lst ) −1 • µ( , g V lst ). By assumption g rst ∪ g V lst is homotopic to g rst ∪ g V lst and because g rst is right-stable, g V lst is homotopic to g V lst . Therefore µ( , g V lst ) ∼ µ( , g V lst ) and hence S W ∼ S W . The second fact states that the surgery map induces a well defined map on concordance classes of psc-metrics which will be used in forthcoming work [Fre20]. Let us first recall the notion of concordance of psc-metrics.
Definition 3.11. Let g 0 , g 1 ∈ R + (M ). We say g 0 and g 1 are concordant if R + (M × [0, 1]) g0,g1 = ∅. This defines an equivalence relation and we denote the set of concordance classes of R + (M ) byπ 0 (R + (M )).  3.3. The Structured Mapping Class Group. In this section we will give the definitions and present two models for the structured mapping class group of a manifold. For the next two sections let θ : B → BO(d + 1) be a fixed tangential structure.    Remark 3.18. The mapping torus M γ has a θ-structure on the vertical tangent bundle. Since the tangent bundle of the circle is trivial, this gives a θ-structure on M γ .
Since the case of B = BSpin(d + 1) is of great interest in the present paper we will have a closer look at it. Let us recall the more traditional description of Spin-structures (cf. [Ebe06, Chapter 3]): A Spin-structure σ on a manifold M is a pair (P, α) consisting of a Spin(d + 1)-principal bundle P and an isomorphism α : An isomorphism of Spin-structures σ 0 = (P 0 , α 0 ) and σ 1 = (P 1 , α 1 ) is an isomorphism β : P 0  Proof. Since M is simply connected, the Spin-structure σ of an oriented manifold is unique up to isomorphism. So for every orientation preserving diffeomorphism If θ is an arbitrary tangential structure we also have a different model for Γ θ (M,l). Example 3.22. Since we usually will be interested in the case where θ is the (stabilized) tangential 2-type of a high-dimensional manifold M , let us consider at the case B = BSpin(d + 1) × BG. The map θ : BSpin(d + 1) × BG → BO(d + 1) factors through the 3-connected cover θ Spin : BSpin(d + 1) → BO(d + 1) and we get BG). So, a θ-structurel on M is given by a Spin-structure σ on M and a map M → BG. Let ψ := [f, L] ∈ B θ (M,l). Then f is an orientation preserving diffeomorphism of M and L is the homotopy class of a path connecting the bundle mapsl Spin ,l Spin • df : T M ⊕ R 2 → θ * Spin U d+1 together with the homotopy class of a path connecting the maps α and α • f : M → BG. If G = π 1 (M, x) for some base-point x ∈ M , this means that the induced map f * : π 1 (M, x) → π 1 (M, f (x)) is given by conjugation by a path γ : [0, 1] → M with γ(0) = x and γ(1) = f (x). We say that f acts on the fundamental group by an inner automorphism in this case. the θ-structure on W . Since W → W × [0, 1] is a homotopy equivalence there is a unique extension up to homotopy where the vertical map sends v ∈ R >0 to the inwards pointing vector. This gives a θ-structure on W × I and by restriction a θ-structure on W op . Now we can prove another useful tool.
Proof. Since disjoint union is associative up to cobordism and disjoint union with the emptyset is the identity and this really defines a group action. If by Proposition 3.25. It remains to show thatΦ L (W ) L = (W ∪ L op ) L is cobordant to W . First we note that (W, ψ 0 , ψ 1 ) is diffeomorphic to (M 0 × I ∪ ψ0 W ∪ ψ −1 1 M 1 × I, id, id) and so it suffices to consider the case that all boundary identifications are given by the identity. We now decompose (W ∪ L op ) L as follows: By identifying ∂ + V 0 and ∂V 2 with ∂V 1 and ∂V 3 in different ways we obtain We will now construct the cobordism X : We construct this by taking V i × I for every i = 0, 1, 2, 3, introducing corners at the boundary (and at ∂ + V ) respectively) as shown in Figure 9. We then glue together these manifolds as follows: We identify . This is shown in Figure 10. The θ-structures are given byˆ Vi ⊕ id R (the arrows in Figure 10 indicate the incoming and outgoing boundary of X).
Remark 3.27. Proposition 3.26 can also be proven using structured cobordism categories. The presented proof however is much more direct. Proof. It is a group homomorphism because The rest follows from Proposition 3.26.
Remark 3.29. The inverse is given by mapping (W, ψ 0 , ψ 1 ) to the manifold obtained by gluing ∂ 1 W to ∂ 0 W along the diffeomorphism ψ −1 0 • ψ 1 .  3.5. The action of the mapping class group. We will now give the general statement of Theorem A. For a space X let hAut(X) denote the group-like H-space of weak homotopy equivalences of X.
Corollary 3.32. Let d ≥ 7, let M be a (d−1)-dimensional manifold and let θ : B → BO(d + 1) be the stabilized tangential 2-type of M wherel : T M ⊕ R 2 → θ * U d+1 is a θ-structure. Then there exists a group homomorphism SE : Ω θ d −→ π 0 (hAut(R + (M ))), such that the following diagram, where F is the forgetful map and T is the mapping torus map, commutes  Proof of Theorem B. Since M ist simply connected and stably parallelizable, the tangential 2-type of M is given by BSpin(d + 1). Proof of Proposition D. By Proposition 3.19 we may assume that f is a Spindiffeomorphism. Let W : S d−1 → S d−1 be an admissible cobordism Spin-cobordant to S d−1 × [0, 1] T f . Then f * ∼ SE T f ∼ S W and by Proposition 3.9 and Remark 3.10 this is homotopic to the identity if S W (g • ) is homotopic to g • .
(3) f * is homotopic to the identity.
Proof. The implications 3. ⇒ 4. and 4. ⇒ 5. are obvious and the implication 2. ⇒ 3 follows from Corollary 3.32. For 1. ⇒ 2. we note that where β denotes the Bott manifold withÂ(β) = 1 and sign(β) = 0. Furthermore, sign(HP 2 ) = 0 andÂ(HP 2 ) = 0. Since for T f both these invariants vanish, it has to be Spin-nullbordant. Finally 5. ⇒ 1. is proven as follows: Let g t be an isotopy between f * g and g. Since isotopy of psc-metrics implies concordance of psc-metrics, there exists a psc-metric G on M × [0, 1] restricting to f * g and g. Then G induces a psc-metric on T f as one can identify the metrics on the boundary along f * and henceÂ(T f ) = 0.
Remark 3.34. Since M is simply connected we have Diff Spin (M ) Diff + (M ). Hence the above Corollary classifies the action of Γ + (M ) on R + (M ) for every simply connected 7-dimensional Spin-manifold.
Proposition 3.35. Let M be a (d−1)-dimensional, simply connected Spin-manifold and let W d be a closed Spin-manifold with α(W ) = 0. Then SE W (g) ∼ g for every psc-metric g on M .
Proof. By Lemma 3.4 we can perform (Spin-)surgery on the interior of M ×[0, 1] W to get an admissible cobordism V : M ; M . If there exists a psc-metric g 0 ∈ R + (M ) such that SE W (g 0 ) ∼ g 0 , there exists a psc-metric G on V that restricts to g 0 on both boundaries by Remark 2.29. We obtain a psc-metric on the manifold V given by gluing the boundaries of V together along the identity. So, α(V ) = 0 by the Lichnerowicz-formula and since α is Spin-cobordism invariant we get This shows that vanishing of the α-invariant of W is necessary condition for SE(W ) to be homotopic to the identity. We close with the following question. If the answer to Question 3.36 were yes, we would get the following diagram.