Rohlins invariant and periodic-end Dirac operators Daniel Ruberman 1 - PowerPoint PPT Presentation

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Rohlin’s invariant and periodic-end Dirac operators Daniel Ruberman 1 Nikolai Saveliev 2 1 Department of Mathematics Brandeis University 2 Department of Mathematics University of Miami Indiana University, April 2008

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Outline Rohlin’s theorem Rohlin invariant of spin 3-manifolds Rohlin invariant of certain 4-manifolds Periodic-end manifolds and operators Fredholm properties Generic metrics theorem Analytical interpretation of Rohlin invariant Application to scalar curvature Seiberg-Witten theory Positive scalar curvature

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Theorem 1 (Rohlin’s theorem) Let X be a smooth, closed, oriented spin 4 -manifold. Then the signature σ ( X ) is divisible by 16 . Analytic proof: Spin structure and metric on X give spinor bundles S ± . Dirac operator D + : C ∞ ( S + ) → C ∞ ( S − ) is Fredholm. ind ( D + ) = dim C ker ( D + ) − dim C coker ( D + ) = σ ( X ) 8 . But D + is quaternionic-linear, so ind ( D + ) is even.

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Let ( M , s ) be a 3-manifold with a spin structure. Then M = ∂ W where the spin structure s extends over W . Define ρ ( M , s ) = σ ( W ) ∈ Q / 2 Z . 8 For M = homology sphere, ρ ( M ) ∈ Z / 2 Z . Analytical versions give Z -valued lift(s). Euler characteristic of Floer homology Instanton, Seiberg-Witten, Heegaard-Floer.

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators For X 4 with homology of S 1 × S 3 , and generator γ ∈ H 1 ( X ) dual to M 3 ⊂ X , define ρ ( X , γ ) = ρ ( M ) ∈ Q / 2 Z . Previous work with Saveliev, using Donaldson theory Analytically defined integer λ Don ( X , γ ) : count ( × 1 4 ) of irreducible solutions to Yang-Mills equations. Independent of metrics and perturbations. Vanishes if π 1 X = Z . Conjectured to lift ρ ( X , γ ) ∈ Z / 2 Z if X is a Z [ Z ] -homology S 1 × S 3 . Potential applications to triangulation, homotopy S 1 × S 3 ... New approach via Seiberg-Witten theory and analysis of Dirac operator on periodic-end manifolds.

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators For metric g on X , count irreducible solutions to Seiberg-Witten equations D + ( g ) ψ = 0 F + A + q ( ψ ) = 0 Count = SW ( X , g ) depends on g : Consider behavior of SW ( X , g t ) for 1-parameter family g t . Since b + 2 ( X ) = 0, may have solutions ( A t , ψ t ) with ψ t → 0 as t → t 0 . For metric g t 0 , must have ker ( D + A t 0 ( X , g t 0 )) � = { 0 } . Need some other metric-dependent term with similar jump. For X = S 1 × M 3 , done by Chen (1997) and Lim (2000).

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Periodic Dirac operators In non-product case, use Periodic-end Dirac operator. Setup: closed manifold X with a map f : X → S 1 (equivalently γ ∈ H 1 ( X ) ). This gives A Z -cover ˜ X → X , and lift t : ˜ X → R of f . If X is spin, Dirac operator ˜ D + : C ∞ (˜ S + ) → C ∞ (˜ S − ) . For any regular value θ ∈ S 1 for f , a submanifold f − 1 θ = M ⊂ X . D + a Fredholm operator? Question: When is ˜

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Periodic Dirac operators To make sense of this, need to complete C ∞ (˜ S ± ) in some norm. Pick δ ∈ R , and define � e t δ | s | 2 < ∞} L 2 S ± ) = { s | δ (˜ X ˜ as well as Sobolev spaces L 2 k , δ (˜ S ± ) . Should really ask if the dimensions of the kernel/cokernel of D + : L 2 ˜ k , δ (˜ S ± ) → L 2 k − 1 , δ (˜ S ± ) D + is Fredholm on L 2 are finite. If so we’ll be sloppy and say ˜ δ . The most useful case for us is δ = 0.

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Periodic Dirac operators Taubes’ idea: Fourier-Laplace transform converts to family of problems on compact X . For each c ∈ C , have the twisted Dirac operator D c : C ∞ ( S ) → C ∞ ( S ) given by D c s = Ds − ic dt · s . Theorem 2 (Taubes, 1987) Fix δ ∈ R . Suppose that ker D c = { 0 } for all c ∈ C ∗ with D + is Fredholm on L 2 2 . Then ˜ | c | = e δ δ . Corollary 3 If X has a Riemannian metric of positive scalar curvature, then D + is Fredholm on L 2 δ for any δ . ˜

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Periodic Dirac operators This theorem, originally proved (more directly) by Gromov-Lawson (1983), is not per se an obstruction to existence of PSC metrics. Theorem 4 (R-Saveliev, 2006) D + is Fredholm on L 2 . For a generic metric on X, the operator ˜ Idea (building on Ammann-Dahl-Humbert (2006)) Invertibility of D c ∀ c ∈ S 1 can be pushed across a cobordism.

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators End-periodic manifolds End-periodic manifolds are periodic in finitely many directions, each modeled on a Z covering ˜ X → X . Let M ⊂ X be non-separating; it lifts to a compact submanifold M 0 ⊂ ˜ X . M 0 X ˜ X 0 ˜ M X

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators End-periodic manifolds Let ˜ X 0 be everything to the right of M 0 , and choose a compact oriented spin manifold W with (oriented) boundary − M . From these pieces, form the end-periodic manifold with end modeled X : on ˜ Z = W ∪ M 0 ˜ X 0 M 0 W X 0 ˜

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators End-periodic manifolds Excision principle: Everything we said about Dirac operators X holds for Dirac operators on Z . on ˜ For metric g on X , extending to metric on Z , get Dirac operator D + ( Z , g ) . Fredholm on L 2 for PSC metric g . Fredholm on L 2 for generic metric g . ind ( D + ( Z , g )) depends on choice of W . Unlike compact case, ind ( D + ( Z , g )) depends on g. Could jump in family g t if ker ( D + c ( X , g 0 )) � = { 0 } for c ∈ S 1 .

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators From above, ind ( D + ( Z , g )) jumps at the same place as SW ( X , g ) . This suggests that we try to use one to balance the other. Have to get rid of dependence of ind ( D + ( Z , g )) on compact manifold W . Provisional definition: Consider the quantity λ SW ( X , g ) = SW ( X , g ) − ind ( D + ( Z , g )) − 1 8sign ( W ) Conjecture 5 λ SW ( X , g ) is metric-independent and equals λ Don ( X , γ ) .

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Will discuss approach to independence part of Conjecture 5 shortly. Properties of λ SW Independence from various choices 1 Choice of slice M ⊂ X and lift M 0 ⊂ ˜ X . Choice of W with ∂ W = M , and extension of metric over W . Reduction mod 2 of λ SW is ρ ( X ) . 2 Item 1: excision principle. Item 2: two ingredients. Involution in Seiberg-Witten theory makes SW ( X , g ) even, and quaternionic nature of Dirac operator makes ind ( D + ( Z , g )) even.

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators Have seen that in a family g t , the invariants SW ( X , g t ) and ind ( D + ( Z , g t )) jump at the same t . Change in SW ( X , g ) not that hard to understand. This is the wall-crossing phenomenon in gauge theory; in generic family, SW ( X , g ) jumps by ± 2. Sign can be worked out. If X = S 1 × M 3 , then change in index is ‘spectral flow’ of Dirac operators on M , studied by Atiyah-Patodi-Singer. Conjecture 5 proved in this situation independently by Chen and Lim. General periodic case more subtle; there’s no operator on M or spectrum to flow.

Seiberg-Witten lift of ρ ( X , s ) Rohlin’s invariant Seiberg-Witten invariants Periodic manifolds and Dirac operators What we know so far: (joint with Tom Mrowka). Somewhat easier, but basically equivalent to fix metric g , and vary the exponential weight. Then we want to consider the operators D + ( Z , g ) on L 2 δ as δ runs over the interval [ δ 0 , δ 1 ] . When Fredholm, denote its index by ind δ ( D + ( Z , g )) . Denote by S ( δ 0 , δ 1 ) the set of z ∈ C with ker ( D z ) � = 0 and e δ 0 / 2 < | z | < e δ 1 / 2 . By Taubes’ theorem 2, this is a finite set. To each z ∈ S ( δ 0 , δ 1 ) , we associate a ‘multiplicity’ d ( z ) . Definition of d ( z ) complicated; count of solutions to some system of equations. But we can show Lemma 6 If dim ker ( D z ) = 1 , then d ( z ) = ± 1 .