Topology of the space of metrics with positive scalar curvature

Size: px

Start display at page:

Download "Topology of the space of metrics with positive scalar curvature"

Neal Herbert Norris
5 years ago
Views:

1 Topology of the space of metrics with positive scalar curvature Boris Botvinnik University of Oregon, USA November 11, 2015 Geometric Analysis in Geometry and Topology, 2015 Tokyo University of Science

2 Notations: W is a compact manifold, dim W = d, R(W ) is the space of all Riemannian metrics, if W, we assume that a metric g = h + dt 2 near W ; R g is the scalar curvature for a metric g, R + (W ) is the subspace of metrics with R g > 0; if W, and h R + ( W ), we denote R + (W ) h := {g R + (W ) g = h + dt 2 near W }. psc-metric = metric with positive scalar curvature.

3 Existence Question: For which manifolds R + (W )? It is well-known that for a closed manifold W, Yamabe invariant R + (W ) Y (W ) > 0. Assume R + (W ). More Questions: What is the topology of R + (W )? In particular, what are the homotopy groups π k R + (W )?

4 Let (W, g) be a spin manifold. Then there is a canonical real spinor bundle S g W and a Dirac operator D g acting on the space L 2 (W, S g ). Theorem. (Lichnerowicz 60) D 2 g = s g R g. In particular, if R g > 0 then D g is invertible. For a manifold W, dim W = d, we obtain a map (W, g) D g Fred d,0, Dg where Fred d,0 is the space of Cl d -linear Fredholm operators. The space Fred d,0 also classifies the real K-theory, i.e., π q Fred d,0 = KO d+q

5 It gives the index map α : (W, g) ind(d g ) = [D g ] π 0 Fred d,0 = KO d. The index ind(d g ) does not depend on a metric g. Magic of Index Theory gives a map: α : Ω Spin d KO d. Thus α(m) := ind(d g ) gives a topological obstruction to admitting a psc-metric. Theorem. (Gromov-Lawson 80, Stolz 93) Let W be a spin simply connected closed manifold with d = dim W 5. Then R + (W ) if and only if α(w ) = 0 in KO d. Magic of Topology: There are enough examples of psc-manifolds to generate Ker α Ω Spin d, then we use surgery.

6 Surgery. Let W be a closed manifold, and S p D q+1 W. We denote by W the manifold which is the result of the surgery along the sphere S p : W = (W \ (S p D q+1 )) S p S q (Dp+1 S q ). Codimension of this surgery is q + 1. S p D q+2 + D p+1 D q+1 W S p D q+1 I 1 V 0 W I 0 V W I 0

7 Surgery Lemma. (Gromov-Lawson) Let g be a metric on W with R g > 0, and W be as above, where the q Then there exists a metric g on W with R g > 0. S p D q+2 + D p+1 D q+1 W g + dt 2 V 0 W I 0 V W I 0

8 Surgery Lemma. (Gromov-Lawson) Let g be a metric on W with R g > 0, and W be as above, where the q Then there exists a metric g on W with R g > 0. S p D q+2 + D p+1 D q+1 W standard metric V 0 W I 0 V W I 0

9 Surgery Lemma. (Gromov-Lawson) Let g be a metric on W with R g > 0, and W be as above, where the q Then there exists a metric g on W with R g > 0. S p D q+2 + D p+1 D q+1 W V 0 W I 0 V W I 0

10 Surgery Lemma. (Gromov-Lawson) Let g be a metric on W with R g > 0, and W be as above, where the q Then there exists a metric g on W with R g > 0. D p+1 D q+1 (W, g ) V 0 W I 0 V W I 0 (W, g)

11 Conclusion: Let W and W be simply connected cobordant spin manifolds, dim W = dim W = d 5. Then Assume R + (W ). R + (W ) R + (W ). Theorem. (Chernysh, Walsh) Let W and W be simply connected cobordant spin manifolds, dim W = dim W 5. Then R + (W ) = R + (W ). Let W = W, and h R + ( W ), then Questions: R + (W ) h = R + (W ) h. What is the topology of R + (W )? In particular, what are the homotopy groups π k R + (W )?

$Let B := B \ (D 8 1 D8 2 ): (S 7, g 0 ) B (S 7, g 1 ) Thus Z π 0 R + (S 7 ).$

12 Example. Let us show that Z π 0 R + (S 7 ). Let B be a Bott manifold, i.e. B is a simply connected spin manifold, dim B = 8, with α(b 8 ) = Â(B) = 1. Let B := B \ (D 8 1 D8 2 ): (S 7, g 0 ) B (S 7, g 1 ) Thus Z π 0 R + (S 7 ).

invertibe operators (Fred d,0 ) + g g 0 R(W ) R + (W ) Fred d,0 Fact: The space (Fred

13 Index-difference construction (N.Hitchin): Let g 0 R + (W ) be a base point, and g R + (W ). Let g t = (1 t)g 0 + tg. invertibe operators (Fred d,0 ) + g g 0 R(W ) R + (W ) Fred d,0 Fact: The space (Fred d,0 ) + is contractible. The index-difference map: A g0 : R + (W ) ΩFred d,0. We obtain a homomorphism: A g0 : π k R + (W ) π k ΩFred d,0 = KO k+d+1

14 Index-difference construction (N.Hitchin): Let g 0 R + (W ) be a base point, and g R + (W ). Let g t = (1 t)g 0 + tg. invertibe operators (Fred d,0 ) + g g 0 R(W ) R + (W ) Fred d,0 Fact: The space (Fred d,0 ) + is contractible. The index-difference map: A g0 : R + (W ) ΩFred d,0. We obtain a homomorphism: A g0 : π k R + (W ) π k ΩFred d,0 = KO k+d+1

15 Index-difference construction (N.Hitchin): Let g 0 R + (W ) be a base point, and g R + (W ). Let g t = (1 t)g 0 + tg. invertibe operators (Fred d,0 ) + g g 0 R(W ) R + (W ) Fred d,0 Fact: The space (Fred d,0 ) + is contractible. The index-difference map: A g0 : R + (W ) ΩFred d,0. We obtain a homomorphism: A g0 : π k R + (W ) π k ΩFred d,0 = KO k+d+1

16 There is different way to construct the index-difference map. Let g 0 R + (W ) be a base point, and g R + (W ), and g t = (1 t)g 0 + tg. Then we have a cylinder W I with the metric ḡ = g t + dt 2 : g 0 ḡ = g t + dt 2 g W I It gives the Dirac operator Dḡ with the Atyiah-Singer-Patodi boundary condition. We obtain the second map ind g0 : R + (W ) ΩFred d,0, g Dḡ D 2 ḡ +1 Fredd+1,0 ΩFred d,0. Magic of the Index Theory: ind g0 A g0.

17 The classifying space BDiff (W ). Let W be a connected spin manifold with boundary W. Fix a collar W ( ε 0, 0] W. Let Diff (W ) := {ϕ Diff(W ) ϕ = Id near W }. We fix an embedding ι : W ( ε 0, 0] R m and consider the space of embeddings Emb (W, R m+ ) = {ι : W R m+ ι W ( ε0,0] = ι } The group Diff (W ) acts freely on Emb (W, R m+ ) by re-parametrization: (ϕ, ι) (W ϕ W ι R m+ ). Then BDiff (W ) = Emb (W, R m+ )/Diff (W ). The space BDiff (W ) classifies smooth fibre bundles with the fibre W.

18 The classifying space BDiff (W ). Let W be a connected spin manifold with boundary W. Fix a collar W ( ε 0, 0] W. Let Diff (W ) := {ϕ Diff(W ) ϕ = Id near W }. We fix an embedding ι : W ( ε 0, 0] R m and consider the space of embeddings Emb (W, R m+ ) = {ι : W R m+ ι W ( ε0,0] = ι } The group Diff (W ) acts freely on Emb (W, R m+ ) by re-parametrization: (ϕ, ι) (W ϕ W ι R m+ ). E(W ) W BDiff (W ) E(W ) = Emb (W, R m+ ) Diff (W ) W

19 The classifying space BDiff (W ). Let W be a connected spin manifold with boundary W. Fix a collar W ( ε 0, 0] W. Let Diff (W ) := {ϕ Diff(W ) ϕ = Id near W }. We fix an embedding ι : W ( ε 0, 0] R m and consider the space of embeddings Emb (W, R m+ ) = {ι : W R m+ ι W ( ε0,0] = ι } The group Diff (W ) acts freely on Emb (W, R m+ ) by re-parametrization: (ϕ, ι) (W ϕ W ι R m+ ). E E(W ) W f ˆf W B BDiff (W )

20 Moduli spaces of metrics. Let W be a connected spin manifold with boundary W, h 0 R + ( W ). Recall: R(W ) h0 := {g R(W ) g = h 0 + dt 2 near W }, Diff (W ) := {ϕ Diff(W ) ϕ = Id near W }. The group Diff (W ) acts freely on R(W ) h0 and R + (W ) h0 : M(W ) h0 = R(W ) h0 /Diff (W ) = BDiff (W ), M + (W ) h0 = R + (W ) h0 /Diff (W ). Consider the map M + (W ) h0 BDiff (W ) as a fibre bundle: R + (W ) h0 M + (W ) h0 BDiff (W )

21 Let g 0 R + (W ) h0 be a base point. We have the fibre bundle: M + (W ) h0 R+ (W ) h0 BDiff (W ) Let ϕ : I BDiff (W ) be a loop with ϕ(0) = ϕ(1) = g 0, and ϕ : I M + (W ) h0 We obtain: its lift. ϕ 1 (g 0 ) g 0 g 0 ϕ t (g 0 ) g 0 ḡ = ϕ 1 (g t ) + dt 2 ϕ 1 (g 0 ) W I ΩBDiff e (W ) R + ind g0 (W ) h0 ΩFred d,0

22 Let W be a spin manifold, dim W = d. Consider again the index-difference map: ind g0 : R + (W ) ΩFred d,0, where g 0 R + (W ) is a base-point. In the homotopy groups: (ind g0 ) : π k R + (W ) π k ΩFred d,0 = KO k+d+1. Theorem. (BB, J. Ebert, O.Randal-Williams 14) Let W be a spin manifold with dim W = d 6 and g 0 R + (W ). Then π k R + (W ) (indg 0 ) KO k+d+1 = is non-zero whenever the target group is non-zero. Z k + d + 1 0, 4 (8) Z 2 k + d + 1 1, 2 (8) 0 else Remark. This extends and includes results by Hitchin ( 75), by Crowley-Schick ( 12), by Hanke-Schick-Steimle ( 13).

For example: = W = D 2n W = D 2n ((S 2n 1 I )#(S n S n )) Then we have: R + (D 2n ) h0 = R +

24 Let dim W = d = 2n. Assume W is a manifold with boundary W, and W is the result of an admissible surgery on W. For example: = W = D 2n W = D 2n ((S 2n 1 I )#(S n S n )) Then we have: R + (D 2n ) h0 = R + (W ) h0, where h 0 is the round metric on S 2n 1. Observation: It is enough to prove the result for R + (D 2n ) h0 any manifold obtained by admissible surgeries from D 2n. or

25 We need a particular sequence of surgeries: V 0 V 1 V k 1 V k S 2n 1 S 2n 1 S 2n 1 S 2n 1 S 2n 1 Here V 0 = (S n S n ) \ D 2n, V 1 = (S n S n ) \ (D 2n D 2n + ),..., V k = (S n S n ) \ (D 2n D 2n + ). Then W k := V 0 V 1 V k = # k (S n S n ) \ D 2n. We choose psc-metrics g j on each V j which gives the standard round metric h 0 on the boundary spheres.

26 We need a particular sequence of surgeries: h 0 h 0 h 0 h 0 h 0 (V 0,g 0 ) (V 1,g 1 ) (V k 1,g k 1 ) (V k,g k ) S 2n 1 S 2n 1 S 2n 1 S 2n 1 S 2n 1 Here V 0 = (S n S n ) \ D 2n, V 1 = (S n S n ) \ (D 2n D 2n + ),..., V k = (S n S n ) \ (D 2n D 2n + ). Then W k := V 0 V 1 V k = # k (S n S n ) \ D 2n. We choose psc-metrics g j on each V j which gives the standard round metric h 0 on the boundary spheres.

27 W k h 0 h 0 h 0 h 0 h 0 (V 0,g 0 ) (V 1,g 1 ) (V k 1,g k 1 ) (V k,g k ) S 2n 1 S 2n 1 S 2n 1 S 2n 1 S 2n 1 W k 1 We have the composition map R + (W k 1 ) h0 R + (V k ) h0,h 0 R + (W k ) h0. Gluing metrics along the boundary gives the map: m : R + (W k 1 ) h0 R + (W k ) h0, g g g k

28 W k h 0 h 0 h 0 h 0 h 0 (V 0,g 0 ) (V 1,g 1 ) (V k 1,g k 1 ) (V k,g k ) S 2n 1 S 2n 1 S 2n 1 S 2n 1 S 2n 1 W k 1 We have the composition map R + (W k 1 ) h0 R + (V k ) h0,h 0 R + (W k ) h0. Gluing metrics along the boundary gives the map: m : R + (W k 1 ) h0 R + (W k ) h0, g g g k Magic of Geometry: The map m : R + (W k 1 ) h0 R + (W k ) h0 is homotopy equivalence.

29 W k h 0 h 0 h 0 h 0 h 0 (V 0,g 0 ) (V 1,g 1 ) (V k 1,g k 1 ) (V k,g k ) S 2n 1 S 2n 1 S 2n 1 S 2n 1 S 2n 1 W k 1 Let and s : W k W k+1 be the inclusion. It induces the stabilization maps Diff (W 0 ) Diff (W k ) Diff (W k+1 ) BDiff (W 0 ) BDiff (W k ) BDiff (W k+1 )

30 W k h 0 h 0 h 0 h 0 h 0 (V 0,g 0 ) (V 1,g 1 ) (V k 1,g k 1 ) (V k,g k ) S 2n 1 S 2n 1 S 2n 1 S 2n 1 S 2n 1 W k 1 Let and s : W k W k+1 be the inclusion. It induces the stabilization maps Diff (W 0 ) Diff (W k ) Diff (W k+1 ) BDiff (W 0 ) BDiff (W k ) BDiff (W k+1 ) Topology-Geometry Magic: the space BDiff (W k ) is the moduli space of all Riemannian metrics on W k which restrict to h 0 + dt 2 near the boundary W k.

31 W k h 0 h 0 h 0 h 0 h 0 (V 0,g 0 ) (V 1,g 1 ) (V k 1,g k 1 ) (V k,g k ) S 2n 1 S 2n 1 S 2n 1 S 2n 1 S 2n 1 W k 1 Let and s : W k W k+1 be the inclusion. It gives the fiber bundles: M + (W 0 ) h0 M + (W 1 ) h0,h 0 M + (W k ) h0,h 0 = = = R + (W 0 ) h0 R + (W 0 ) h0,h 0 R + (W k ) h0,h 0 BDiff (W 0 ) BDiff (W 1 ) BDiff (W k ) with homotopy equivalent fibers R + (W 0 ) h0 = = R + (W k ) h0,h 0

32 We take a limit to get a fiber bundle: M + R + B = lim k M + (W k ) h0,h 0 R + (W k ) h0,h 0 BDiff (W 0 ) where R + is a space homotopy equivalent to R + (W k ) h0,h 0. Remark. We still have the map: ΩB which is consistent with the maps ΩBDiff (W k ) e R + ind ΩFred d,0 e R + (W k ) h0,h 0 ind g0 ΩFred d,0 Magic of Topology: the limiting space B := lim k BDiff (W k ) has been understood.

33 About 10 years ago, Ib Madsen, Michael Weiss introduced new technique, parametrized surgery, which allows to describe various Moduli Spaces of Manifolds. Theorem. (S. Galatius, O. Randal-Williams) There is a map B η Ω 0 MTθ n inducing isomorphism in homology groups. This gives the fibre bundles: M + ˆM + R + B η R + Ω 0 MTΘ n Again, it gives a holonomy map e : ΩΩ 0 MTΘ n R +

34 The space Ω 0 MTΘ n is the moduli space of (n 1)-connected 2n-dimensional manifolds. In particular, there is a map (spin orientation) ˆα : Ω 0 MTΘ n Fred 2n,0 sending a manifold W to the corresponding Dirac operator. ΩΩ 0 MTΘ n ΩFred 2n,0 e Ωˆα R + ind

35 The space Ω 0 MTΘ n is the moduli space of (n 1)-connected 2n-dimensional manifolds. In particular, there is a map (spin orientation) ˆα : Ω 0 MTΘ n Fred 2n,0 sending a manifold W to the corresponding Dirac operator. ΩΩ 0 MTΘ n ΩFred 2n,0 e Ωˆα R + D 0 D t D 1 ind g 0 ϕ 1 (g t ) ϕ 1 (g 0 ) W I ΩBDiff e (W ) R + ind g0 (W ) h0 ΩFred d,0

36 The space Ω 0 MTΘ n is the moduli space of (n 1)-connected 2n-dimensional manifolds. In particular, there is a map (spin orientation) ˆα : Ω 0 MTΘ n Fred 2n,0 sending a manifold W to the corresponding Dirac operator. Index Theory Magic gives commutative diagram ΩΩ Ωˆα 0 MTΘ n ΩFred 2n,0 e ind R + Then we use algebraic topology to compute the homomorphism (Ωˆα) : π k (ΩΩ 0 MTΘ n ) π k (ΩFred 2n,0 ) = KO k+2n+1 to show that it is nontrivial when the target group is non-trivial.

38 THANK YOU!

THE SPACE OF POSITIVE SCALAR CURVATURE METRICS ON A MANIFOLD WITH BOUNDARY

THE SPACE OF POSITIVE SCALAR CURVATURE METRICS ON A MANIFOLD WITH BOUNDARY MARK WALSH Abstract. We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary.