Invertible Dirac operators and handle attachments

Transcription

1 Invertible Dirac operators and handle attachments Nadine Große Universität Leipzig (joint work with Mattias Dahl) Rauischholzhausen,

2 Motivation Not every closed manifold admits a metric of positive scalar curvature. In contrast, on every closed manifold the space of metric wirth negative scalar curvature is nonempty and contractable. Topological obstruction for psc-metrics: (M, g) closed spin, Dirac operator D g Lichnerowicz formula scal g > 0 D g is invertible (D g ) 2 = g + scal g 4

3 Motivation Not every closed manifold admits a metric of positive scalar curvature. Topological obstruction for psc-metrics: (M, g) closed spin, Dirac operator D g Lichnerowicz formula scal g > 0 D g is invertible (D g ) 2 = g + scal g 4 Metr(M) psc Metr(M) inv Metr(M)

4 Obstruction for psc metrics From index theory Â(M) if n 0 mod 4 dim ker D g 1 if n 1 mod 8, α(m) 0 2 if n 2 mod 8, α(m) 0 0 otherwise where  and α are determined only by the topology of the underlying manifold. E.g. if Â(M 4 ) 0, Metr(M 4 ) psc Metr(M 4 ) inv =. Metr(M) psc Metr(M) inv Metr(M)

5 Construction of manifolds admitting psc-metrics - Review Surgery S k B n k f B k+1 S n k 1 M M embedding f : S k B n k M S := f (S k {0}) - surgery sphere (M \ f (S k B n k )) = S k 1 S n k 1 M = (M \ f (S k B n k )) B k+1 S n k 1 M is obtained from M by a surgery of dim k / codim n k.

6 Construction of manifolds admitting psc-metrics - Review Surgery B k+1 B n k 1 W W View the cylinder W := M [0, 1] as a bordism from M to itself Attach B k+1 B n k 1 to M {1} W is a bordism from M to M - the trace of the surgery. W is obtained from W by attaching a (k + 1)-handle.

7 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. The torus is obtained as:

8 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. The torus is obtained as: B 2 + B 2

9 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. The torus is obtained as: B 1 B 1 B 2 B 2 + a 1-handle

10 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. The torus is obtained as: B 1 B 1 B 2 B 2 + a 1-handle

11 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. The torus is obtained as: B 2 + a 1-handle

12 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. B 1 B 1 The torus is obtained as: B 2 + a 1-handle + a 1-handle

13 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. B 1 B 1 The torus is obtained as: B 2 + a 1-handle + a 1-handle

14 Construction of manifolds admitting psc-metrics - Review Surgery Each closed manifold has a handle decomposition. B 1 B 1 The torus is obtained as: B 2 + a 1-handle + a 1-handle + B 2 = T 2

15 Construction of manifolds admitting psc-metrics Theorem (Gromov, Lawson / Schoen, Yau; 80 ) Let (M, g) be a closed Riemannian manifold with g Metr(M) psc. Let M be obtained from M by a surgery of codimension 3. Then, M admits a psc-metric g. (M, g) (M, g ) g can be chosen such that it coincides with g outside a small neighbourhood around the surgery sphere.

16 Construction of manifolds admitting psc-metrics (M, g) (M, g ) Intuition psc is a local property codim n k 3 = gluing in B k+1 S n k 1 2 standard product structure on B k+1 S n k 1 2 has psc

17 Psc-metrics and handle attachments Theorem (Carr 88 / Gajer 87 ) Let (M n+1, g) be a compact Riemannian manifold with closed boundary M, g Metr(M) psc and g having product structure near M. Let M be obtained from M by adding a (k + 1)-handle of codimension n k 3. Then, M admits a psc-metric g that is again product near the (new) boundary. (M n+1, g) (M, g )

18 Psc-metrics and handle attachments (M n+1, g) (M, g ) Intuition On the boundary: surgery of codim n k 3

19 Psc-metrics and handle attachments Intuition On the boundary: surgery of codim n k 3 On the double: surgery of codim n k 3

20 Psc-metrics and handle attachments (M n+1, g) (M, g ) Intuition On the boundary: surgery of codim n k 3 On the double: surgery of codim n k 3 Implication Metr psc (S 4k 1 ) has infinitely many components (k 2) (Metr psc (S 3 ) is connected (Marques, 2011))

21 Metr inv (M) What can be done for metrics with invertible Dirac operators?

22 Metr inv (M) What can be done for metrics with invertible Dirac operators? From now on: Let all manifolds be spin.

23 Surgery for Metr inv (M) (M n, g) (M, g ) After the surgery the manifold should still be spin!

24 Surgery for Metr inv (M) (M n, g) (M, g ) After the surgery the manifold should still be spin! S k B n k - induces spin structure on S k S n k 1 glue in B k+1 S n k 1 Its boundary should carry same spin structure. For k > 1, the spin structure on S k is unique and bounds the disk. - No Problem here. For k = 1, two spin structures on S 1 - we only allow the one that bounds the disk.

25 Surgery for Metr inv (M) (M n, g) (M, g ) f : S k B n k M spin-preserving embedding.

26 Surgery for Metr inv (M) (M n, g) (M, g ) Intuition Invertible Dirac operator is a global condition. codim n k 3 = gluing in B k+1 S n k 1 2 standard product structure on R k+1 S n k 1 2 has invertible Dirac operator

27 Surgery for Metr inv (M) (M n, g) (M, g ) Intuition Invertible Dirac operator is a global condition. codim n k 2 = gluing in B k+1 S n k 1 1 standard product structure on R k+1 S n k 1 1 has invertible Dirac operator ( When taking the right S 1 )

28 Surgery for Metr inv (M) (M n, g) (M, g ) Intuition Invertible Dirac operator is a global condition. codim n k 2 = gluing in B k+1 S n k 1 1 standard product structure on R k+1 S n k 1 1 has invertible Dirac operator ( When taking the right S 1 ) If the inserted cylinder is large enough, invertibility survives.

29 Construction for manifolds admitting inv-metrics Theorem (Ammann, Dahl, Humbert; 2009 ) Let (M n, g) be a closed Riemannian spin manifold with g Metr(M) inv. Let M be obtained from M by a surgery of codimension 2. Then, M admits an inv-metric g. Moreover, g can be chosen such that it coincides with g outside a small neighbourhood around the surgery sphere. Consequences (Ammann, Dahl, Humbert; 2009) For a generic metric g, dim ker D g is no larger than forced by the index theorem.

30 Inv-metrics on manifolds with boundary ( M [ ɛ, 0], g + dt 2 ) (M n, g) When do we call D g invertible?

31 Inv-metrics on manifolds with boundary ( M [0, ), g + dt 2 ) ( M [ ɛ, 0], g + dt 2 ) (M, g ) g Metr(M) inv iff D g is invertible as operator on L 2 (M, S)

32 Inv-metrics on manifolds with boundary (M 1, g 1 ), ( M 1, g 1 ) = (N +, h), g 1 Metr(M 1 ) inv (M 2, g 2 ), ( M 2, g 2 ) = (N, h), g 2 Metr(M 2 ) inv If M 1 and M 2 are glued together using a large enough cylinder (N [ R, R], h + dt 2 ), the resulting metric has again invertible Dirac operator.

33 Inv-metrics and handle attachments Theorem (Dahl, G. 2012) Let (M n+1, g) be a compact Riemannian spin manifold with closed boundary M, g Metr(M) inv and g having product structure near M. Let M be obtained from M by adding a (k + 1)-handle of codimension n k 2. Then, M admits an inv-metric g that is again product near the (new) boundary. (M n+1, g) (M, g )

34 Inv-metrics and handle attachments Theorem (Dahl, G. 2012) Let (M n+1, g) be a compact Riemannian spin manifold with closed boundary M, g Metr(M) inv and g having product structure near M. Let M be obtained from M by adding a (k + 1)-handle of codimension n k 2. Then, M admits an inv-metric g that is again product near the (new) boundary. Implication Metr(S 4k 1 ) inv has infinitely many components for all k 1

35 Strategy and Methods Topological strategy - Decompose the handle attachment surgery of codim n k half surgery of codim n k + 1 glue in 1 2 Bk+1 S n k

36 Metric strategy Approximation by double product metrics near the surgery sphere (U δ (S [ ɛ, )), g S + ξ R n k + dt 2 ) ( M [ ɛ, ),, g + dt 2 ) M {0} If δ small enough, still g δ Metr(M) inv. ( C 1 -continuity of the spectrum )

37 Metric strategy First surgery {C} {B} {A} {B} {A} {C } (M, g) (M, g ρ) Parameter for tuning : ρ - diameter of {B} For ρ small enough, gρ Metr(M ) inv - proof by contradiction

38 Metric strategy First surgery {C} {B} {A} {B} {A} {C } (M, g) (M, g ρ) For ρ small enough, g ρ Metr(M ) inv - proof by contradiction ρ i 0, g ρi Metr(M ) inv g ρi has a harmonic spinor: D gρ i ϕ i = 0, ϕ i L 2 (M,g ρi ) = 1 (regularity) φ i φ in Cloc 1 (M \ (S [ ɛ, )) (removal of singularities) D g φ = 0 on M, φ L2 (M,g) 1 a priori estimates on the L 2 -norm of φ i on {A} vs {C}.

39 Metric strategy Again approximating by double product metrics

40 Metric strategy Second surgery

41 One Application Theorem (Dahl, G.; 2012) Let M be a closed 3-dimensional Riemannian spin manifold and g Metr(M) inv. Then there are metrics g i Metr(M) inv, i Z, such that g i is bordant to g but g i is not concordant to g j for i j. In particular, Metr(M) inv has infinitely many connected components.

42 Notations g 0, g 1 Metr(M) inv are isotopic if smooth family g t Metr(M) inv with g t = g 0 for t 0, g t = g 1 for t 1. g t g 1 g 0 Metr(M) (M [0, 1], g) g 0, g 1 Metr(M) inv are concordant if g Metr(M [0, 1]) inv with g M {i} = g i. (M, g 0 ) (M, g 1 ) g i Metr(M i ) inv (i = 0, 1) are bordant if (W, g) with W = M 0 (M 1 ), g Metr(W ) inv, g Mi = g i. (W, g) (M 0, g 0 ) (M 1, g 1 )

43 An application Theorem (Dahl, G.; 2012) Let M be a closed 3-dimensional Riemannian spin manifold and g Metr(M) inv. Then there are metrics g i Metr(M) inv, i Z, such that g i is bordant to g but g i is not concordant to g j for i j. In particular, Metr(M) inv has infinitely many connected components. Lemma There exist 4-manifolds (Y i, h i ) (i Z) with h i Metr(Y i ) inv, Y i = S 3 such that α(y i S 3 (Y j ) ) = c(i j) for a constant c 0.

44 An application Lemma There exist 4-manifolds (Y i, h i ) (i Z) with h i Metr(Y i ) inv, Y i = S 3 such that α(y i S 3 (Y j ) ) = c(i j) for a constant c 0. Construction: Y 0 - B 4 with a torpedo metric h 0 Metr(B 4 ) psc and h 0 S 3 = standard metric Y i = (K3#K3# #K3) \B 4 = Y 0 + several 2-handles }{{} i times α(y i S 3 (Y j ) ) = α(# (i j) K3) = (i j)α(k3) 0 for i j h i := h i S 3

45 Constructions of g i Start with (M 3, g Metr(M) inv ) - construct g i bordant to g. (M, g) (M, g i ) (S 3, h i ) attachment of a 1-handle

46 Constructions of g i (M, g) Y i (M, g i ) (S 3, h i ) (M, g) and (M, g i ) are bordant. Bordism (W i, g i ) Metr(W i ) inv

47 Constructions of g i assume concordance (M, g) (M, g) W i (W j ) (M, g i ) (M, g j )

48 Constructions of g i (M, g) (M, g) W i (W j ) (M, g i ) (M, g j ) Closed manifold (W, g) with g Metr(W ) inv and α(w ) = c(i j).

49 One Application Theorem (Dahl, G.; 2012) Let M be a closed 3-dimensional Riemannian spin manifold and g Metr(M) inv. Then there are metrics g i Metr(M) inv, i Z, such that g i is bordant to g but g i is not concordant to g j for i j. In particular, Metr(M) inv has infinitely many connected components.

50 Thank you for your attention.