Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities
HHOMOTOPY INVARIANCE OF THE SPACE OF METRICS WITH POSITIVESCALAR CURVATURE ON MANIFOLDS WITH SINGULARITIES
BORIS BOTVINNIK AND MARK G. WALSH
Abstract.
In this paper we study manifolds M Σ with fibered singularities, more specifically, arelevant space R psc ( X Σ ) of Riemannian metrics with positive scalar curvature. Our main goal isto prove that the space R psc ( X Σ ) is homotopy invariant under certain surgeries on M Σ . Introduction
Existence of a psc-metric.
A classical result in this subject concerns the existence of metricsof positive scalar curvature (psc-metrics) on a simply-connected smooth closed manifold X . Thereare two cases here: either X is a spin manifold or it is not. Recall that in the case when X is spin,there is an index α ( X ) ∈ KO − n of the Dirac operator valued in real K -theory. Here is the result: Theorem 1.1. (Gromov-Lawson [11], Stolz [14])
Let X be a smooth closed simply connected man-ifold of dimension n ≥ . (i) If X is spin, then X admits a psc-metric if and only if the index α ( X ) ∈ KO − n of theDirac operator on X vanishes. (ii) If X is not spin, then X always admits a psc-metric. We denote by R psc ( X ) the space of psc-metrics on X . Recall that one of the major tools usedto prove Theorem 1.1 is the surgery technique due to Gromov and Lawson (proved independentlyby Schoen and Yau). In particular, Gromov-Lawson observed that a psc-metric survives surgeriesof codimension at least three (such surgeries are called admissible ). It turns out that the homotopytype of the space R psc ( X ) is invariant under such surgeries, see [7, 8, 17].1.2. Existence of a psc-metric on a manifold with Baas-Sullivan singularities.
We startwith the simplest case, where the geometrical picture is transparent. Let (
L, g L ) be a closedRiemannian manifold, in which the metric g L is assumed to have zero scalar curvature. Let Y bea closed smooth manifold, such that the product Y × L is a boundary of a smooth manifold X : ∂X = Y × L . Here is a natural geometrical question: Question.
Does there exist a psc-metric g Y on Y , such that the product metric g Y + g L on ∂X = Y × L can be extended (being a product near ∂X ) to a psc-metric g X on X ? Date : May 8, 2020.2000
Mathematics Subject Classification.
Key words and phrases.
Positive scalar curvature metrics, manifolds with singularities, surgery. a r X i v : . [ m a t h . A T ] M a y t is convenient to denote βX := Y . Then we obtain a manifold with singularities of the type L (or just L -singularities) X Σ = X ∪ ∂X − βX × C ( L ), where C ( L ) is a cone over L . The metric g L on L easily extends to a scalar-flat metric g C ( L ) on the cone C ( L ) which is a product-metricnear its base L ⊂ C ( L ). We say that a metric g on X Σ is a well-addapted Riemannain metric on X Σ if(i) the restriction g | X is a regular Riemannian metric which is a product-metric near ∂X ;(ii) the restriction g | βX × C ( L ) splits as a product-metric g | βX × C ( L ) = g βX + g C ( L ) .We denote by R ( X Σ ) the space of all well-adapted Riemannian metrics on X Σ , and by R psc ( X Σ )its subspace of psc-metrics. Then the above geometrical question is asking whether the space R psc ( X Σ ) is non-empty. This existence question was addressed and even affirmatively resolved forsome particular examples of the singularity types L (provided that all manifolds involved are spinand both X and βX are simply-connected, see [2]).There is a particularly interesting example here. Let us consider spin manifolds, and choose L = S with a non-trivial spin structure, so that L represents the generator η ∈ Ω spin1 = Z . Wedenote by Ω spin ,η ∗ ( − ) the bordism theory of spin manifolds with η -singularities, and by MSpin η thecorresponding representing spectrum. It turns out, there exists a Dirac operator on spin-manifoldswith η -singularities. Furthermore, there is a natural transformation α η : Ω spin ,η ∗ → KO η ∗ whichevaluates the index of that Dirac operator, where the “ K -theory with η -singularities” KO η ∗ ( − )coincides with usual complex K -theory. Here is the result from [2]: Theorem 1.2. (Botvinnik, [2])
Let X be a simply connected spin manifold with nonempty η -singularity of dimension n ≥ . Assume βX (cid:54) = ∅ . Then X admits a metric of positive scalarcurvature if and only if α η ([ X ]) = 0 in the group KO ηn ∼ = KU n . Existence of a psc-metric on a manifold with fibered singularities.
There are moregeneral objects, “manifolds with fibered singularities” (or pseudomanifolds with singularities ofdepth one). Here again, we start with a manifold X with boundary ∂X (cid:54) = ∅ , which is a totalspace of the fiber bundle ∂X → βX with the fiber L . To get geometrically interesting objects, weassume that L is given a metric g L of non-negative constant scalar curvature and that the bundle ∂X → βX has a structure group G which is a subgroup of the isometry group Isom( g L ) of themetric g L . Then the bundle ∂X → βX is induced by a structure map f : βX → BG . Let C ( L )be a cone over L with a “cone metric” g C ( L ) which restricts to g L on the base and is scalar-flat.Furthemore, we assume the isometry action of the G extends to the one of the cone metric g C ( L ) .Then there is a fiber bundle N ( βX ) → βX , which is given by “inserting” the cone C ( L ) as afiber with the same structure group G . The actual manifold with fibered singularities is given as X Σ := X ∪ ∂X N ( βX ). Then a well-adapted Riemannain metric g on X Σ is a regular Riemannianmetric restricted to X (which is also a product near the boundary), and g | N ( βX ) is determinedby a requirement that the projection N ( βX ) → βX is a Riemannian submersion (which has astructure group G ⊂ Isom( g L )) and with the cone metric g C ( L ) on the fiber (we give a detailed efinition in Section 2). We denote by R ( X Σ ) the space of well-adapted Riemannain metrics on X Σ , and by R psc ( X Σ ) its subspace of psc-metrics. Below we describe two interesting cases.1.3.1. We assume that all manifolds are spin, and L = S representing η ∈ Ω spin1 , and G = S . Weobtain a corresponding bordism group Ω spin ,η - fb ∗ of such manifolds. Then there exists an appropriateDirac operator on X Σ , and index map α η - fb : Ω spin ,η - fb ∗ → KO η - fb ∗ evaluating the index of thatDirac operator. Here is the existence result for psc-metrics in that setting: Theorem 1.3. (Botvinnik-Rosenberg, [4])
Let X Σ = X ∪ N ( βX ) be a simply connected spinmanifold with fibered η -singularity (i.e. X and βX are simply-connected and spin) of dimension n ≥ . Then R psc ( X Σ ) (cid:54) = ∅ if and only if α η - fb ([ X Σ ]) = 0 in the group KO η - fb n . L, G )-manifolds are spin: here L is equipped with a metric g L with constant scalar curvature s L = (cid:96) ( (cid:96) − L = (cid:96) , and G is a subgroup of the isometrygroup of the metric g L . We assume L = ∂ ¯ L and the G -action on L extends to a G -action on¯ L . In this setting, an ( L, G )-manifold X Σ could be given as a triple ( X, βX, f ), where X is amanifold with boundary ∂X , which is a total space of a fiber bundle ∂X → βX (with a fiber L and a structure group G ) given by a map f : βX → BG . In this setting and with n = dim X , wehave indices α ( βX ) ∈ KO n − (cid:96) − and α cyl ( X ) ∈ KO n . Here are the existence results: Theorem 1.4. (Botvinnik-Piazza-Rosenberg [5])
Let ( X, ∂X, f ) define a closed ( L, G ) -singularspin manifold X Σ . Assume that X , βX , and G are all simply connected, that n − (cid:96) ≥ , andsuppose that L is a spin boundary, say L = ∂ ¯ L , with the standard metric g L on L extending to apsc-metric on ¯ L , and with the G -action on L extending to a G -action on ¯ L . Assume that the twoobstructions α ( βX ) ∈ KO n − (cid:96) − and α cyl ( X ) ∈ KO n both vanish. Then X Σ admits a well-adaptedpsc-metric. Theorem 1.5. (Botvinnik-Piazza-Rosenberg [5])
Let ( X, βX, f ) define a closed ( L, G ) -singularspin manifold X Σ , with L = HP k and G = Sp (2 k + 1) , n ≥ . Assume that ∂X = βX × L , i.e.,the L -bundle over βX is trivial, or in other words that the singularities are of Baas-Sullivan type.Then if X and βX are both simply connected and n − k ≥ , X Σ has an adapted psc-metric ifand only if the α -invariants α ( βX ) ∈ KO n − k − and α cyl ( X ) ∈ KO n both vanish. There are several other interesting cases and also much more general results when the pseudomanifold X Σ has non-trivial fundamental group; see [6]. Now we are ready to address our mainresult concerning homotopy invariance of corresponding spaces of psc-metrics on X Σ .1.4. Main result.
The homotopy-invariance of certain spaces of psc-metrics is a crucial propertywhich has allowed detection of their non-trivial homotopy groups. Let M be a closed spin manifold.An important consequence of the results due to Chernysh [7], Walsh [17, 18, 19] (see also recentwork by Ebert and Frenk [8]) is that the homotopy type of the space R psc ( M ) is an invariant ofthe bordism class [ M ] ∈ Ω spin n (provided M is simply-connected and n ≥ There is also a similar result for non-simply connected manifolds. otice that if X Σ = X ∪ ∂X N ( βX ) is a pseudomanifold with ( L, G )-singularities equippedwith structure map f : βX → BG , then there are two types of surgery possible on X :(i) a surgery on its “resolution”, i.e. the interior X ⊂ X Σ away from the boundary ∂X ;(ii) a surgery on the structure map f : βX → BG .In case (i) all constructions are the same as in the case of closed manifolds, however, in case (ii),we have to be a bit more careful. Indeed, let ¯ B : βX (cid:32) βX be the trace of a surgery on the map f : βX → BG , with ∂ ¯ B = βX (cid:116) − βX . Then the map f extends to a map ¯ f : ¯ B → BG whichgives a fiber bundle ¯ p : Z → B with the fiber L . This gives us a new manifold X = X ∪ ∂X Z with boundary ∂X , the total space over a new Bockstein βX with the same fiber L . Also weobtain a new conical part N ( βX ) as above. All of this results in a new pseudomanifold(1) X Σ , = X ∪ ∂X N ( βX ) , X = X ∪ ∂X Z, with structure map f = ¯ f | βX : βX → BG . Here is our main technical result: Theorem A.
Let X Σ = X ∪ ∂X N ( βX ) be a pseudomanifold with ( L, G ) -singularities, with dim X = n , dim L = (cid:96) . (i) Let i : S p ⊂ X be a sphere with trivial normal bundle, and X Σ , be the result of surgeryon X Σ along S p . Then if n − p ≥ , the spaces R psc ( X Σ ) and R psc ( X Σ , ) are weaklyhomotopy equivalent. (ii) Let i : S p ⊂ βX be a sphere with trivial normal bundle, and with f ◦ ι : S p → BG homotopic to zero. Let ¯ B be a trace of the surgery along S p ⊂ βX with ∂ ¯ B = βX (cid:116) − βX and a structure map ¯ f : ¯ B → BG . Then if n − (cid:96) − p ≥ , the spaces R psc ( X Σ ) and R psc ( X Σ , ) are homotopy equivalent, where X Σ , is given by (1) . Theorem A could be applied to a variety of interesting examples. Among these are:(1) Let L = (cid:104) k (cid:105) be the set of k points, and let G = Z k be its “isometry group”. Then a( (cid:104) k (cid:105) , Z k )-manifold X Σ = X ∪ ∂X N ( βX ) is assembled out of a manifold X with boundary ∂X equipped with free Z k -action, and a structure map βX → B Z k classifies the corre-sponding k -folded covering ∂X → βX = ∂X/ Z k . Here N ( βX ) is given by inserting thecone C (cid:104) k (cid:105) instead of (cid:104) k (cid:105) in the fiber bundle ∂X → βX . Assuming that all manifoldsare spin, we obtain corresponding bordism groups Ω spin , ( (cid:104) k (cid:105) , Z k ) - fb ∗ and the correspondingtransformation α ( (cid:104) k (cid:105) , Z k ) - fb : Ω spin , ( (cid:104) k (cid:105) , Z k ) - fb ∗ → KO ∗ ( B Z k ) which evaluates the index of thecorresponding Dirac operator.(2) Let η ∈ Ω spin1 be as above, i.e., [ L ] = η , and G = S . Then, similarly, we arrive at thebordism groups Ω spin ,η - fb ∗ and the index map α η - fb : Ω spin ,η - fb ∗ → KO η - fb ∗ , as in Theorem5.1 as above.The above examples lead to the following two corollaries of Theorem A: orollary B. Let X Σ be a spin ( (cid:104) k (cid:105) - fb) -manifold. Assume dim X ≥ and that X and βX are simply-connected. Then the homotopy type of the space R psc ( X Σ ) is a bordism invariant anddepends only on the bordism class [ X Σ ] ∈ Ω spin , (cid:104) k (cid:105) - fb n . Corollary C.
Let X Σ be a spin ( η - fb) -manifold. Assume dim X ≥ and that X and βX are simply-connected. Then the homotopy type of the space R psc ( X Σ ) is a bordism invariant anddepends only on the bordism class [ X Σ ] ∈ Ω spin ,η - fb n . The cases addressed in Theorems 1.4 and 1.5 give interesting implications.(3) Let L and G be as in Theorem 1.4, i.e. G is a simply connected Lie group, L is a spinboundary, say L = ∂ ¯ L , with the standard metric g L on L extending to a psc-metric on ¯ L ,and with the G -action on L extending to a G -action on ¯ L . Then an ( L, G )-singular spinmanifold X Σ determines an element in the relevant bordism group Ω spin , ( L,G ) − - fb n whichfits to an exact triangle (see [5]):Ω spin ∗ Ω spin , ( L,G ) - fb ∗ Ω spin ∗ ( BG ) . (cid:45) i (cid:8)(cid:8)(cid:8)(cid:8)(cid:25) β (cid:72)(cid:72)(cid:72)(cid:72)(cid:89) T Here the indices α ( βX ) ∈ KO n − (cid:96) − and α cyl ( X ) ∈ KO n can be thought of as homomor-phisms from Ω spin , ( L,G ) - fb ∗ to a relevant K -theory.(4) Let L = HP k and G = Sp (2 k + 1), n ≥
1. Assume that ∂X = βX × L , i.e., the L -bundleover βX is trivial, or in other words that the singularities are of Baas-Sullivan type. Thena closed ( L, G )-singular spin manifold X Σ determines an element in the correspondingbordism group Ω spin , ( L,G ) - fb n , and, as above, the index homomorphism from Ω spin , ( L,G ) - fb ∗ to a relevant K -theory.These examples lead to following corollary Corollary D.
In both of the cases described in (3) and (4), the homotopy type of the space R psc ( X Σ ) is a bordism invariant and depends on the bordism class [ X Σ ] ∈ Ω spin , ( L,G ) - fb n , provided n − (cid:96) ≥ (where (cid:96) = 2 k in the case (4)). In the last section we show that the cases (3) and (4) above lead to an interesting resultconcerning homotopy groups of R psc ( X Σ ).2. Preliminarities
Positive scalar survature on manifolds with boundary.
Here we recall the main con-structions and results from [18]. The set-up is as follows. Given a smooth compact n -dimensionalmanifold X (possibly with boundary ∂X (cid:54) = ∅ ), we denote by R ( X ), the space of all Riemannianmetrics on X . The space R ( X ) is equipped with the standard C ∞ -topology, giving it the structureof a Fr´echet manifold; see [15, Chapter 1] for details. For each metric g ∈ R ( X ), we denote by g : X → R the scalar curvature on X of the metric g and by R + ( X ) ⊂ R ( X ) the subspace ofpsc-metrics on X .In the case when ∂X (cid:54) = ∅ , it is necessary to consider certain subspaces of R + ( X ) wheremetrics satisfy particular boundary constraints. With this in mind, we specify a collar embedding c : ∂X × [0 , (cid:44) → X around ∂X and define the space R + ( X, ∂X ) as: R + ( X, ∂X ) := { h ∈ R + ( X ) : c ∗ h | ∂X × I = h | ∂X + dt } , where I := [0 , ⊂ [0 , g ∈ R + ( ∂X ), we define the subspace R + ( X, ∂X ) g ⊂ R + ( X, ∂X ) of all psc-metrics h ∈ R + ( X, ∂X ) where ( c ∗ h ) | ∂X ×{ } = g .Let Z : Y (cid:32) Y be a bordism between ( n − Y and Y given togetherwith collars c i : Y i × [0 , (cid:44) → Z , i = 0 , ∂Z = Y (cid:116) Y . Then R + ( Z, ∂Z )denotes the space of psc-metrics on Z which restrict as a product structure on the neighbourhood c i ( Y i × I ) ⊂ Z , i = 0 ,
1; i.e. c ∗ i ¯ g = g i + dt on Y i × I for some pair of metrics g i ∈ R + ( Y i ), i = 0 ,
1. Now we fix a pair of psc-metrics g ∈ R + ( Y ) and g ∈ R + ( Y ) and consider the followingsubspace of R + ( Z, ∂Z ): R + ( Z, ∂Z ) g ,g := { ¯ g ∈ R + ( Z, ∂Z ) | c ∗ i ¯ g = g i + dt on Y i × [0 , , i = 0 , } . We note that each metric ¯ g ∈ R + ( Z, ∂Z ) g ,g provides a psc-bordism ( Z, ¯ g ) : ( Y , g ) (cid:32) ( Y , g ).We next assume X is a manifold whose boundary ∂X = Y is equipped with the metric g .Furthemore, we assume that both spaces R + ( X, ∂X ) g and R + ( Z, ∂Z ) g ,g are non-empty. Now,by making use of the relevant collars, we glue together X and Z to obtain a smooth manifoldwhich we denote X ∪ Z and which has boundary ∂ ( X ∪ Z ) = Y ; see Fig. 1.In particular, we obtain the space R + ( X ∪ Z, Y ) g of psc-metrics which restrict as g + dt on c ( Y × [0 , ⊂ Z ⊂ X ∪ Z . Then for any metric ¯ g ∈ R + ( Z, ∂Z ) g ,g , we obtain a map: µ Z, ¯ g : R + ( X, ∂X ) g −→ R + ( X ∪ Z, Y ) g h (cid:55)−→ h ∪ ¯ g, (2)where h ∪ ¯ g is the metric obtained on X ∪ Z by the obvious gluing depicted in Fig. 1. ( X, h ) g + dt g + dt g + dt ( Z, ¯ g ) Figure 1.
Attaching (
X, h ) to ( Z, ¯ g ) along a common boundary ∂X = Y Consider the case when the bordism Z : Y (cid:32) Y is an elementary bordism , i.e., when Z isthe trace of a surgery on Y with respect to an embedding φ : S p × D q +1 → Y with p + q + 1 = n − Y . Then we have the following. emma 2.1. (Surgery Lemma, see [7, 17, 8]) Let g ∈ R + ( Y ) be any metric. Assume q ≥ .Then there exist metrics g ∈ R + ( Y ) and ¯ g ∈ R + ( Z, ∂Z ) g ,g such that ( Z, ¯ g ) : ( Y , g ) (cid:32) ( Y , g ) is a psc-bordism. Such a bordism is usually called a
Gromov-Lawson bordism (or
GL-bordism for short). Here isa reformulation of the main technical result from [18]:
Theorem 2.2.
Let Z : Y (cid:32) Y be an elementary bordism as above with p, q ≥ . Then for anymetric g ∈ R + ( Y ) there exist metrics g ∈ R + ( Y ) and ¯ g ∈ R + ( Z, ∂Z ) g ,g such that the map µ Z, ¯ g : R + ( X, ∂X ) g ∼ = −→ R + ( X ∪ Z, Y ) g defined by (2) , is a weak homotopy equivalence. The case of manifolds with fibered singularities.
Let L be a closed manifold with fixedmetric g L of non-negative constant scalar curvature. Definition 2.3.
A link (
L, g L ) is simple if it satisfies either one of the following conditions:(a) the manifold ( L, g L ) is such that s g L is a positive constant;(b) ( L, g L ) = ( S , dθ ) or L = Z k .For each case (a), (b), we fix a subgroup G of the isometry group of the metric g L . Beforegoing forward with our constructions, we would like to clarify why the condition that the scalarcurvature s g L is non-negative constant is important here.In the case (a), s g L is a positive constant. Denote by C ( L ) a cone over L and give the conemetric g C ( L ) = dr + r g L . This is a warped product metric on (0 , R ] × L away from the cone point(where r = 0). Then the scalar curvature of the metric g C ( L ) (away from the vertex) is given as(3) s g C ( L ) = ( s g L − (cid:96) ( (cid:96) − r − , where (cid:96) = dim L . Thus if s g L is a positive constant, we scale the metric g L to achive the idenity s g L = (cid:96) ( (cid:96) − g L satisfies this condition. The case (b), when L = S , has special features. First, according to (3), the scalar curvature of the metric g C ( S ) isidentically zero. Secondly, any smooth S -bundle p : Y → B gives rise a free S -action on themanifold Y such that B = Y /S . Then, according to a result by B´erard-Bergery, [1, TheoremC], a manifold Y admits an S -equivariant psc-metric if and only if the orbit space, the manifold B = Y /S admits a psc-metric. In the case (c) L = Z k . Then the cone C ( Z k ) has a standardEuclidian metric, and we do not make any further assumptions.Hence we assume that the cone C ( L ) is scalar-flat outside of its vertex in the above cases.Let X Σ = X ∪ ∂X N ( βX ) be a pseudomanifold with ( L, G )-singularities, with dim X = n ,dim L = (cid:96) , and f : βX → BG is a corresponding structure map. Assuming that a link ( L, g L ) issimple as above, we define the space of all well-adapted metrics R ( X Σ ) as follows. efinition 2.4. We say that a metric g on a pseudomanifold X Σ = X ∪ ∂X N ( βX ) is a Rimeannian well-adapted metric if it satisfies the following conditions:(i) the restriction g | X is a Riemannian metric such that g | X = g ∂ + dt near the boundary( ∂X, g ∂ );(ii) the Riemannian manifold ( ∂X, g ∂ ) is a total space of a Riemannian submersion ∂X → βX .As we mentioned above, there are two types of surgery that could be performed on X Σ : the firstone on its resolution, the interior of X , and the second one on the structure map f : βX → BG .We consider the latter.Moreover, for now it is convenient to cut the singularity out and to work with a smoothmanifold X whose boundary is fibered over βX with fiber L . We use the notation ( X, βX, f ) forsuch manifold, where the boundary ∂X of X is a total space of the fiber bundle from the diagram: ∂X E ( L ) βX BG (cid:45) ˆ f (cid:63) p (cid:63) p L (cid:45) f Here p L : E ( L ) → BG is the universal fiber bundle with the fiber L and the structure group G .Let ¯ B : B (cid:32) B be an elementary bordism with B = βX and ¯ f : ¯ B → BG a map suchthat ¯ f | B = f = f . We will use the notation ( ¯ B, ¯ f ) : ( B , f ) (cid:32) ( B , f ). Let ¯ p : ¯ E → ¯ B be acorresponding fiber bundle with fiber L and structure group G . By construction, ¯ E | B = E = ∂X ,and ¯ p | E = p . Then the manifold ¯ E gives a bordism ¯ E : E (cid:32) E , where E = E | B . We assumethat the bordism ¯ B is equipped with collars b i : B i × [0 , i = 0 ,
1, along the boundary ∂ ¯ B .Clearly these collars provide collars e i : E i × [0 ,
2) along ∂ ¯ E .In order to study the space R psc ( X Σ ) of well-adapted psc-metrics on X Σ , we first study itsclosest relative, the space R psc ( X, βX, f ) h β , defined as a subspace of R psc ( X, ∂X ) as follows.
Definition 2.5.
Let h β ∈ R psc ( βX ) be a given metric. The space R psc ( X, βX, f ) h β consists ofall Riemannian metric g ∈ R psc ( X, ∂X ) subject to the conditions: • the restriction g | c ( ∂X × [0 , to the collar c ( ∂X × [0 , g ∂X + dt , where g ∂X is apsc-metric on ∂X ; • the metric g ∂X on the total space ∂X of the fiber p : ∂X → βX is given by the psc-metric h β on the base βX and g ∂X restricts to g L (up to isometry from G ) along every fiber L .Consider again an elementary bordism ( ¯ B, ¯ f ) : ( B , f ) (cid:32) ( B , f ) where ( B , f ) = ( βX, f ).Let g βX = h β . We assume that there is a psc-metric ¯ h β ∈ R psc ( ¯ B ) h β ,h β , where h β is a psc-metricon the manifold B . The metric ¯ h β provides a psc-bordism( ¯ B, ¯ f , ¯ h β ) : ( B , f , h β ) (cid:32) ( B , f , h β ) . urthemore, the structure map ¯ f : ¯ B → BG determines a bordism ¯ E : E (cid:32) E , where ¯ E is apull-back of the universal ( L, G )-fibration:¯
E E ( L )¯ B BG (cid:45) ˆ¯ f (cid:63) p (cid:63)(cid:45) ¯ f with ¯ E | B = E and ¯ E | B = E . This provides a psc-bordism ( ¯ E, ¯ g ∂ ) : ( E , g ∂ ) (cid:32) ( E , g ∂ ), wherethe metrics ¯ g ∂ , g ∂ and g ∂ are determined by the metrics ¯ h β , h β and h β respectively on the bases¯ B , B and B and the metric g L on the fiber L . Now we glue the manifolds X and ¯ E (again,making use of collars near their boundaries) to obtain a manifold X = X ∪ ∂X ¯ E with boundary ∂X = E , which is a total space of the ( L, G )-fibration p : ∂X → B . We denote βX = B .Similarly to the case of manifolds with boundary we obtain a map: µ ( ¯ B, ¯ f, ¯ h β ) : R psc ( X, βX, f ) h β −→ R psc ( X , βX , f ) h β g (cid:55)−→ g ∪ ¯ g ∂ , (4)for any fixed metric ¯ h β ∈ R psc ( ¯ B ) h β ,h β . Now we are ready to state our main technical result whichis similar to Theorem 2.2. Theorem 2.6.
Let ( X, βX, f ) be a manifold with fibered singularities, i.e. the boundary ∂X isa total space of an ( L, G ) -fibration ∂X → βX given by the structure map f : βX → BG , where dim X = n , dim L = (cid:96) . Furthermore, we assume ( ¯ B, ¯ f ) : ( B , f ) (cid:32) ( B , f ) is an elementarybordism with p, q ≥ , where B = βX . Then for any psc-metric h β on B there exist psc-metrics h β on B and ¯ h β ∈ R psc ( ¯ B ) h β ,h β such that the map µ ( ¯ B, ¯ f, ¯ h β ) : R psc ( X, βX, f ) h β −→ R psc ( X , βX , f ) h β , g (cid:55)−→ g ∪ ¯ g ∂ ,X = X ∪ ∂X = E ¯ E, βX = E , (5) is a weak homotopy equivalence. Here, as above, the psc-metric g ∂ is given by the psc-bordism ( ¯ E, ¯ g ∂ ) : ( E , g ∂ ) (cid:32) ( E , g ∂ ) determined by the psc-bordism ( ¯ B, ¯ f , ¯ h β ) : ( B , f , h β ) (cid:32) ( B , f , h β ) and the metric g L on the fiber L . Proof of Theorem 2.6
Some standard metric constructions.
Here we briefly a recall a couple of standard metricconstructions. These constructions are discussed in detail in [19, Section 5].We fix some constants δ > λ ≥
0. Then a ( δ - λ ) -torpedo metric on the disk D n , denoted g n torp ( δ ) λ , is a psc-metric which roughly takes the form a round hemisphere of radius δ > δ and length λ ≥ D n + denote the upper hemi-disk, we obtain the metric g n torp+ ( δ ) λ := g n torp ( δ ) λ | D n + ; seesecond image in Fig. 2. We call g n torp+ ( δ ) λ a half-torpedo metric . Let λ > ext, we consider the cylinder D n − × [0 , λ ] equipped with the metric g n − ( δ ) λ + dt and attacha half-disk D n + with half-torpedo metric g n torp+ ( δ ) λ along D n − × { } . We denote the resultingRiemannian manifold by ( D n stretch , ˆ g n torp ( δ ) λ ,λ ). This is depicted in the third image in Fig. 2.Typically, we will not care so much about the λ -parameter but only λ which we regard asthe vertical height of this metric. Moreover, we will usually be interested in the case when λ = 1and when δ = 1. With this in mind we make use of the following notational simplifications. g n torp := g n torp (1) . g n torp+ := g n torp+ (1) .ˆ g n torp ( δ ) λ := ˆ g n torp ( δ ) λ ,λ where λ is arbitrary.ˆ g n torp := ˆ g n torp (1) .The following proposition follows immediately from [19, Proposition 3.1.6]. Proposition 3.1.
Let n ≥ , δ > and λ, λ , λ ≥ . (i) The metrics g n torp ( δ ) λ , g n torp+ ( δ ) λ and ˆ g n torp ( δ ) λ ,λ have positive scalar curvature. (ii) For any constant B ≥ and any λ, λ , λ ≥ , there exists δ > so that the scalarcurvature of the metrics g n torp ( δ ) λ , g n torp+ ( δ ) λ and ˆ g n torp ( δ ) λ ,λ is bounded below by B . We now consider product metrics g n − ( δ ) λ + dt on the cylinder D n − × [0 , L ]. It is convenientto allow L to vary bearing in mind that there is an obvious family of rescaling maps (see the map ξ L at the end of section 2 in [19]) which allow us to compare such metrics, for any L >
0, on D n − × I .It is shown in [19, Section 5], provided n ≥
4, that any such product metric g n − ( δ ) λ + dt can λ λ l l l l Figure 2.
The metrics g n torp ( δ ) λ , g n torp+ ( δ ) λ and ˆ g n torp ( δ ) λ ,λ (bottom) on the man-ifolds D n , D n + and D n stretch (top) followed by the boot metric g n boot ( δ ) Λ , ¯ l be moved by isotopy through psc-metrics to a particular psc-metric called a δ -boot metric. We donot provide a precise definition of such a metric here as full details can be found in [19]. However,we would like to explain the basic idea. The metric is denoted g n boot ( δ ) Λ , ¯ l . Here Λ > ossibly large constant and ¯ l = ( l , l , l , l ) ∈ (0 , ∞ ) . This metric should be thought of as definedon D n − × [0 , l ] and, roughly, takes the form: g n − ( δ ) l + dt when t is near l and g n − ( δ ) l + dt when t is near 0 . Importantly, it takes the form ˆ g n torp ( δ ) l on a neighbourhood of (¯0 , ∈ D n − × [0 , L ]; see Fig. 2.We describe this piece as the “toe” of the boot. Remark 3.2.
The constant Λ > δ . This bending arc may need to be quitelarge to maintain positive scalar curvature. In turn, this puts constraints on the components l and l of the vector ¯ l . We will not concern ourselves with this now, except to say that sufficientlylarge Λ , l and l can always be found.3.2. Back to the proof of Theorem 2.6.
The proof follows from that of [18, Theorem A]. Wewill provide a brief review of the main steps of that proof and show that it goes through perfectlywell in our case. The strategy is to decompose the map µ ( ¯ B, ¯ f, ¯ h β ) : R psc ( X, βX, f ) h β −→ R psc ( X , βX , f ) h β into a composition of three maps as shown in the commutative diagram below.(6) R psc ( X, βX, f ) h β µ boot (cid:15) (cid:15) µ ( ¯ B, ¯ f, ¯ hβ ) (cid:47) (cid:47) R psc ( X , βX , f ) h β R pscboot ( X, βX, f ) h β std µ Estd (cid:47) (cid:47) R pscEstd ( X, βX, f ) h β (cid:63)(cid:31) (cid:79) (cid:79) Here the right vertical map denotes inclusion. We will define the spaces R pscboot ( X, βX, f ) h β std and R pscEstd ( X, βX, f ) h β and the maps µ boot and µ Estd in due course. The point is to show that each ofthese maps is a weak homotopy equivalence.We denote by k = n − (cid:96) − βX = B . We consider carefully the elementary bordism( ¯ B, ¯ f ) : ( B , f ) (cid:32) ( B , f ). The manifold ¯ B is given by attaching a handle D p +1 × D q +1 to B along the embeddings φ : S p × D q +1 (cid:44) → B , where p + q + 1 = k , q ≥
2. We would like to havesome flexibility for the embedding φ . We introduce the following family of rescaling maps: σ ρ : S p × D q +1 −→ S p × D q +1 ( x, y ) (cid:55)−→ ( x, ρy ) , where ρ ∈ (0 , φ ρ := φ ◦ σ ρ : S p × D q +1 (cid:44) → B and N ρ := φ ρ ( S p × D q +1 ), abbreviating N := N and φ := φ . Let T φ be the trace of the surgeryon B with respect to φ . We denote by R pscstd ( B ), the space defined as follows: R pscstd ( B ) := { g ∈ R psc ( B ) : φ ∗ g = ds p + g q +1torp on S p × D q +1 } . ccording to Chernysh’s theorem [7, 8], the inclusion R pscstd ( B ) ⊂ R psc ( B )is a weak homotopy equivalence. A major step in the proof of this theorem is the fact (whichfollows easily enough from the original Gromov-Lawson construction in [11]) that for any psc-metric h β ∈ R psc ( B ), there is an isotopy h βt , t ∈ I of metrics in R psc ( B ) connecting h β = h β toa psc-metric h β std ∈ R +std ( Y ). By a well known argument, see [18, Lemma 2.3.2], this isotopy givesrise to a concordance: ¯ h β con on B × [0 , λ + 2] for some λ > h β + dt on B × [ λ + 1 , λ + 2] and g std + dt on B × [0 , . Note that on the slice N × [0 , h β con pulls back to a metric of the form(7) ds p + g q +1torp + dt . Making use of [18, Lemma 5.2.5] we can perform an isotopy of the metric ¯ h β con , adjusting only on N × [0 , g q +1torp + dt factor in (7) with g q +2boot (1) Λ , ¯ l for some appropriately large Λ > l satisfying l = l = 1 . We denote the resulting psc-metric ¯ h β pre on B × [0 , λ + 2]. Weconsider B × [0 , λ + 2] as a long collar of ¯ B and assume that the map ¯ f restricted to B × [0 , λ + 2]is given by ¯ f ( x, t ) = f ( x ). Let ¯ E be a manifold given by pulling back the fiber bundle¯ E E ( L ) B × [0 , λ + 2] BG (cid:45) ˆ¯ f (cid:63) p (cid:63)(cid:45) ¯ f where ¯ f is a restriction of ¯ f . We now use the metric ¯ h β pre on B × [0 , λ + 2] to extend the metric g ∂ from the boundary ∂M to a metric ¯ g ∂ pre on ¯ E by “inserting” the metric g L to the fibers L viathe map ¯ f : B × [0 , λ + 2] → BG .Let ∂X × [ − , ⊂ X be a collar, such that ∂X × { } = ∂X , where, by assumption, every slice ∂X ×{ t } is a total space of the ( L, G )-fibration over B ×{ t } . We consider a manifold X ∪ ¯ E whichwe identify with the original manifold X by deforming linearly the manifold ( ∂X × [ − , ∪ ∂X ¯ E to the collar ∂X × [ − , g ∈ R psc ( X, βX, f ) h β , the metric g ∪ ∂X ¯ g ∂ pre on X ∪ ¯ E (obtained by obvious gluing) is denoted by g std , an element of the space R psc ( X, βX, f ) h β std . Thisgives a map µ ¯ g ∂ pre : R psc ( X, βX, f ) h β → R psc ( X, βX, f ) h β std . We denote R pscboot ( X, βX, f ) h β std := Im( µ ¯ g ∂ pre ). This new metric is depicted in the bottom left of Fig.3, with the original metric g depicted in the top left. For clarity, this figure depicts only the casewhen L is a point. Lemma 6.5.5 of [18], consolidating work from previous sections, shows that inthe case when L is a point, the map µ ¯ g pre is a weak homotopy equivalence.Note that any element of the space R pscboot ( X, βX, f ) h β std := Im( µ ¯ g ∂ pre ) has, near the boundary,a standard piece ¯ g ∂ pre which is determined by the metric ¯ h β pre = ds p + g q +2boot (1) Λ , ¯ l . Replacing this ¯ g ∂ pre ( g ) µ ¯ g ( µ ¯ g ∂ pre ( g )) g A typical element of R psc ( X , βX , f ) h β Figure 3.
Representative elements of the spaces from the commutative diagram(6) above in the case when L is a pointwith g p +1torp + g q +1torp near the boundary, we obtain a map¯ µ : R pscboot ( X, βX, f ) h β std → R pscboot ( X, βX, f ) h β Denoting by R pscEstd ( X, βX, f ) h β ⊂ R pscboot ( X, βX, f ) h β the image of ¯ µ , we obtain the lower horizontalmap µ Estd in diagram (6). A typical element in the image of this map is depicted in the lower rightof Fig. 3. This lower horizontal map is demonstrably a homeomorphism.It remains to show that the inclusion R pscEstd ( X, βX, f ) h β ⊂ R psc ( X , βX , f ) h β is a weak ho-motopy equivalence. Note that the notation “Estd” used in describing the former space (originatingin [18]) is intended to convey the fact that these metrics take a standard form on a much largerregion than typical metrics in R psc ( X , βX , f ) h β and are thus “Extra-standard”.A typical element of R psc ( X , βX , f ) h β (in the case when L is a point) is depicted in theupper right of Fig. 3. Showing that, in the case when L is a point, a compact family of metrics in R psc ( X , βX , f ) h β could be continuously moved to a compact family of extra standard metrics in R pscEstd ( X, βX, f ) h β , without moving already extra-standard metrics out of that space, is the mosttechnically difficult part of the proof of [18, Theorem A]. This is done in [18, section 6.6]. Onceagain however, as it involves making metric adjustments only on the Y factor and as the metric g L on L is scalar flat, the entire argument can be imported to the more general case here. Thiscompletes the proof of Theorem 2.6.4. Proof of Theorem A
Let X Σ = X ∪ ∂X N ( βX ) be a pseudomanifold as above, where X is a manifold with boundary ∂X . We consider the spaces of psc-metrics R psc ( X, ∂X ) and R psc ( ∂X ) which are connected bythe restriction map res : R psc ( X, ∂X ) → R psc ( ∂X ) , res : g (cid:55)→ g | ∂X . This map is very important for us because of the following fact: heorem 4.1. [7, 8] The restriction map res : R psc ( X, ∂X ) → R psc ( ∂X ) is a Serre fiber bundle. Now we consider two pseudomanifolds X Σ = X ∪ ∂X N ( βX ) and X Σ , = X ∪ ∂X N ( βX ),where X = X ∪ ∂X Z , and the manifold Z is given by an elementary psc-bordism ¯ B : βX (cid:32) βX and a structure map ¯ f : ¯ B → BG , so that f = ¯ f | βX and f = ¯ f | βX . Namely, the manifold Z isa total space of the following smooth bundle:(8) Z E ( L )¯ B BG (cid:45) ˆ¯ f (cid:63) p (cid:63)(cid:45) ¯ f Let h β ∈ R psc ( βX ), h β ∈ R psc ( βX ) be metrics as in Theorem 2.6, and g ∂ ∈ R psc ( ∂X ), g ∂ ∈R psc ( ∂X ) be corresponding Riemannian submersion metrics which restrict to g L on each fiber L over βX (respectively, over βX ). It is important to keep in mind that the metrics g ∂ and g ∂ aredetermined by the corresponding metrics h β and h β and by the maps f : βX → BG and f : βX → BG respectively. Now we notice that the spaces R psc ( X, βX, f ) h β and R psc ( X , βX , f ) h β coincide with the fibers R psc ( X, βX, f ) h β = res − ( g ∂ ) , R psc ( X , βX , f ) h β = res − ( g ∂ ) , of the corresponding restriction maps:res : R psc ( X, ∂X ) → R psc ( ∂X ) , res : R psc ( X , ∂X ) → R psc ( ∂X ) . We denote by R psc ( ∂X, g L ; f ) the space of psc-metrics g ∂ which are submersion metrics on thetotal space ∂X given by some psc-metric h β on βX and by the metric g L on the fiber (whichgiven by the map f : βX → BG ). We have the inclusion map ι : R psc ( ∂X, g L ; f ) → R psc ( ∂X ) . Now, by definition, we obtain the space R psc ( X Σ ) as a pull-back in the following diagram R psc ( X Σ ) R psc ( ∂X, g L ; f ) R psc ( X, ∂X ) R psc ( ∂X ) (cid:45) res Σ (cid:63) i (cid:63) ι (cid:45) res Since we fixed the map f : βX → BG , it follows that the space R psc ( ∂X, g L ; f ) is homeomorphic tothe space R psc ( βX ). Let g β ∈ R psc ( βX ) and g ∂ ∈ R psc ( ∂X, g L ; f ) be a corresponding submersionmetric. Clearly, we can identify the fiber res − ( g ∂ ) ⊂ R psc ( X Σ ) with the space R psc ( X, βX, f ) h β .We obtain the following diagram of fiber bundles:(9) R psc ( X, βX, f ) h β R psc ( X Σ ) R psc ( βX ) R psc ( X, ∂X ) g ∂ R psc ( X, ∂X ) R psc ( ∂X ) (cid:45) i Σ (cid:63) ∼ = (cid:45) res Σ (cid:63) i (cid:63) ι (cid:45) i (cid:45) res14 et ( ¯ B, ¯ h β ) : ( βX, h β ) (cid:32) ( βX , h β ) be an elementary psc-bordism (with p, q ≥ f : ¯ B → BG such that f = ¯ f | βX and f = ¯ f | βX . Let Z be amanifold given by (8) equipped with corresponding Riemannian submersion metrics g ∂ ∈ R psc ( ∂X ), g ∂ ∈ R psc ( ∂X ) determined by the given data. Then the psc-bordism ( ¯ B, ¯ h β ) determines a psc-bordism ( Z, ¯ g ∂ ) : ( ∂X, g ∂ ) (cid:32) ( ∂X , g ∂ ). Theorem 2.2 and Theorem 2.6 give us the followinghomotopy equivalences:(10) µ ¯ B, ¯ g β : R psc ( βX ) R psc ( βX ) µ Z, ¯ g β : R psc ( ∂X ) R psc ( ∂X ) µ Z, ¯ g ∂ : R psc ( X, ∂X ) g ∂ R psc ( X , ∂X ) g ∂ µ ¯ B, ¯ f, ¯ h β : R psc ( X, βX, f ) h β R psc ( X , βX , f ) h β (cid:45) ∼ = (cid:45) ∼ = (cid:45) ∼ = (cid:45) ∼ = We obtain the following commutative diagram: R psc ( X, βX, f ) h β i Σ (cid:47) (cid:47) (cid:15) (cid:15) µ ¯ B, ¯ f, ¯ hβ (cid:127) (cid:127) R psc ( X Σ ) (cid:15) (cid:15) res Σ (cid:47) (cid:47) ∼ = (cid:127) (cid:127) R psc ( βX ) ι (cid:15) (cid:15) µ ¯ B, ¯ gβ (cid:127) (cid:127) R psc ( X , βX , f ) h β (cid:15) (cid:15) i Σ (cid:47) (cid:47) R psc ( X Σ , ) (cid:15) (cid:15) res Σ (cid:47) (cid:47) R psc ( βX ) ι (cid:15) (cid:15) R psc ( X, ∂X ) g ∂ i (cid:47) (cid:47) µ Z, ¯ g∂ (cid:127) (cid:127) R psc ( X, ∂X ) res (cid:47) (cid:47) ∼ = (cid:127) (cid:127) R psc ( ∂X ) µ Z, ¯ gβ (cid:127) (cid:127) R psc ( X , ∂X ) g ∂ i (cid:47) (cid:47) R psc ( X , ∂X ) res (cid:47) (cid:47) R psc ( ∂X )where all horizontal rows are Serre fiber bundles. Thus, it is evident that the circled maps aboveare weak homotopy equivalences. This proves Theorem A. (cid:3) Some further developments
In this section we would like to emphasize that recent results concerning homotopy groups ofthe spaces R psc ( M ) (of psc-metrics (see [3, 9, 10, 12, 13]) could be applied directly and indirectlyto the case of manifolds with ( L, G )-fibered singularities. In particular, we would like to attractthe attention of topologically-minded experts to relevant conjectures and results from the recentwork [5, 6]. .1. Index-difference map.
We mentioned earlier that the homotopy-invariance of various spacesof psc-metrics is a crucial property in helping detect their non-trivial homotopy groups. With thisin mind, there is a secondary index invariant, the index-difference map(11) inddiff g : R psc ( M ) → Ω ∞ + n +1 KO , which is defined as follows. Let g ∈ R psc ( M ) be a base point. Then for any psc-metric g on M ,there is an interval g t = (1 − t ) g + tg of metrics such that a corresponding curve of the Diracoperators D g t starts and ends at the subspace ( Fred n ) × ⊂ Fred n of invertible Dirac operators.Since the subspace ( Fred n ) × is contractible, the curve D g t is a loop in the space Fred n of all Diracoperators. This space, in turn, is homotopy equivalent to the loop space Ω ∞ + n KO representinga shifted KO -theory, i.e. π q (Ω ∞ + n KO ) = KO n + q . Thus the curve D g t gives an element inΩ ∞ + n +1 KO , well-defined up to homotopy, to determine the map (11). Theorem 5.1. (Botvinnik–Ebert–Randal-Williams [3], and Perlmutter [12, 13])
Assume M is aspin manifold with dim M ≥ and R psc ( M ) (cid:54) = ∅ with a base point g ∈ R psc ( M ) . Then theindex-diffence map (11) induces a non-trivial homomorphism in the homotopy groups (12) ( inddiff g ) ∗ : π q ( R psc ( M )) → KO q + n +1 when the target group KO q + n +1 is non-trivial. Results and conjectures.
The reader should note that much is also known about the spacesof psc-metrics for non-simply connected manifolds; see [9, 10]. We will however return to the sameexamples we considered above. We have the following conjectures concerning examples (1) and (2):
Conjecture 5.2.
Let X Σ be a spin ( (cid:104) k (cid:105) - fb) -manifold. Assume dim X ≥ and X and βX (cid:54) = ∅ are simply-connected and R psc ( X Σ ) (cid:54) = ∅ with a base point g ∈ R psc ( X Σ ) . Then there is anindex-difference map (13) inddiff (cid:104) k (cid:105) g : R psc ( X Σ ) → Ω ∞ + n +1 KO (cid:104) k (cid:105) which induces a non-trivial homomorphism in the homotopy groups (14) ( inddiff (cid:104) k (cid:105) g ) ∗ : π q ( R psc ( X Σ )) → KO (cid:104) k (cid:105) n + q +1 when the target group KO (cid:104) k (cid:105) n + q +1 ( KO with Z k -coefficients) is non-trivial. Conjecture 5.3.
Let X Σ be a spin manifold with ( η - fb) -singularity of dimension n ≥ . Assume βX (cid:54) = ∅ , and R psc ( X Σ ) (cid:54) = ∅ with a base point g ∈ R psc ( X Σ ) . Then there is an index-differencemap (15) inddiff η - fb g : R psc ( X Σ ) → Ω ∞ + n +1 KO η - fb , which induces a non-trivial homomorphism in the homotopy groups (16) ( inddiff ηg ) ∗ : π q ( R psc ( X Σ )) → KO η - fb q + n +1 when the target group KO η - fb q + n +1 = KO q + n +1 ( CP ∞ ) is non-trivial. t turns out that the above examples (3) and (4) (and many others, see [5]) lead to particularresults concerning the homotopy groups of the spaces R psc ( X Σ ). Let X Σ = X ∪ ∂X − N ( βX ) bea spin manifold with ( L, G )-singularities. Let g ∈ R psc ( X Σ ) be a well-adapted metric. Then g determines the metrics g ∂X ∈ R psc ( ∂X ) and g βX ∈ R psc ( βX ) such that the bundle ∂X → βX isa Riemannian submersion. We fix the metric g βX, . This gives rise to a Serre fiber bundleres Σ : R psc ( X Σ ) → R psc ( βX )with fiber R psc ( X Σ ) g βX , , where R psc ( X Σ ) g βX is the space of all metrics g ∈ R psc ( X Σ ) whichrestrict to g βX, on R psc ( βX ). Since the metric g ∂X, on ∂X is determined by the metric g βX, ,the fiber R psc ( X Σ ) g βX , coincides with the space R psc ( X ) g ∂X , . Here is the result we need: Theorem 5.4. (see [6, Theorem 6.1])
Let M Σ be an ( L, G ) -fibered compact pseudomanifold with L a simply connected homogeneous space of a compact semisimple Lie group. Assume R psc ( X Σ ) (cid:54) = ∅ .Then there exists a section s : R psc ( βM ) → R psc ( M Σ ) to res Σ . In particular, there is a split shortexact sequence: (17) 0 → π q ( R + w ( M Σ ) g βM ) i ∗ −→ π q ( R + w ( M Σ )) (res Σ ) ∗ −−−−→ π q ( R + ( βM )) → , q = 0 , , . . . . Here is one of the conclusions we would like to emphasize:
Corollary 5.5. (see [5, Corollary 6.7])
Let M Σ be an ( L, G ) -fibered compact pseudomanifold with L a simply connected homogeneous space of a compact semisimple Lie group, and n − (cid:96) − ≥ ,where dim M = n , dim L = (cid:96) . Let g ∈ R psc ( M Σ ) (cid:54) = ∅ be a base point giving corresponding basepoints, the metrics g βM, ∈ R psc ( βM ) , g ∂M, ∈ R psc ( ∂M ) and g M, ∈ R psc ( M ) g ∂M, .If M Σ is spin and simply connected, then we have the following commutative diagram: → π q R psc ( M Σ ) g βM, π q R psc ( M Σ ) π q R psc ( βM ) → −→ KO q + n +1 KO q + n +1 ⊕ KO q + n − (cid:96) KO q + n − (cid:96) → (cid:45) j ∗ (cid:63) inddiff gM, (cid:45) (res Σ ) ∗ (cid:63) inddiff g (cid:63) inddiff gβM, (cid:45) (cid:45) where the homomorphisms inddiff g M, and inddiff g βM, are both nontrivial whenever the targetgroups are. In particular, the homomorphism inddiff g : π q R psc ( M Σ ) → KO q + n +1 ⊕ KO q + n − (cid:96) is surjective rationally and surjective onto the torsion of KO q + n +1 ⊕ KO q + n − (cid:96) . There are much more general results concerning the homotopy groups of the space π q R psc ( M Σ )if M Σ is not simply-connecetd, see [5, Section 6]. References [1] L. B´erard-Bergery, Scalar curvature and isometry group, Spectra of Riemannian Manifolds (Tokyo),Kaigai Publications, 1983, Proc. Franco-Japonese seminar on Riemannian geometry (Kyoto, 1981), pp.9–28.2] B. Botvinnik, Manifolds with singularities accepting positive scalar curvature metrics, Geom. & Topol,(2001) v.5, 683-718.[3] B. Botvinnik, J. Ebert, O. Randal-Williams, Infinite loop spaces and positive scalar curvature. Invent.Math. 209 (2017), no. 3, 749-835.[4] B. Botvinnik, J. Rosenberg, Positive scalar curvature on manifolds with fibered singularities.arXiv:1808.06007.[5] B. Botvinnik, P. Piazza, J. Rosenberg, Positive scalar curvature on stratified spaces, I, the simplyconnected case, arXiv:1908.04420.[6] B. Botvinnik, P. Piazza, J. Rosenberg, Positive scalar curvature on stratified spaces, II: the effect of thefundamental group, Preprint.[7] V. Chernysh, On the homotopy type of the space R + ( M ). math.DG.0405235.[8] J. Ebert, G. Frenk, The Gromov-Lawson-Chernysh surgery theorem. arXiv:1807.06311[9] J. Ebert, O. Randal-Williams, Infinite loop spaces and positive scalar curvature in the presence of afundamental group. Geom. Topol. 23 (2019), no. 3, 1549-1610.[10] J. Ebert, O. Randal-Williams, The positive scalar curvature cobordism category. arXiv:1904.12951[11] M. Gromov and H. B. Lawson, Jr., The classification of simply-connected manifolds of positive scalarcurvature,
Ann. of Math. (1980), 423–434.[12] N. Perlmutter, Parametrized Morse Theory and Positive Scalar Curvature. arXiv:1705.02754.[13] N. Perlmutter, Cobordism Categories and Parametrized Morse Theory. Nathan Perlmutter.arXiv:1703.01047[14] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) (1992), 511-540.[15] W. Tuschmann and D. Wraith,
Moduli Spaces of Riemannian Metrics,
Birkhauser, Oberwolfach Semi-nars Vol. , (2010)[16] M. Walsh, Metrics of positive scalar curvature and generalised Morse functions, Part I,
Memoirs of theAmerican Mathematical Society. Volume 209, No. 983, January 2011[17] M. Walsh, Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics.Proc. Amer. Math. Soc. 141 (2013), no. 7, 2475-2484.[18] M. Walsh, The Space of Positive Scalar Curvature Metrics on a Manifold with Boundary.arXiv:1411.2423[19] M. Walsh, H-spaces, loop spaces and the space of positive scalar curvature metrics on the sphere. Geom.Topol. 18 (2014), no. 4, 2189-2243.
Department of Mathematics, University of Oregon, Eugene, OR, 97405, USA
E-mail address : [email protected] Department of Mathematics and Statistics, Maynooth University, Maynooth, Ireland