Pin (2)-monopole theory I

Transcription

1 Intersection forms with local coefficients Osaka Medical College Dec 12, 2016

2 Pin (2) = U(1) j U(1) Sp(1) H Pin (2)-monopole equations are a twisted version of the Seiberg-Witten (U(1)-monopole) equations. Applications of SW equations Intersection forms Diagonalization theorem The 10/8 inequality for spin [Furuta] SW invariants Exotic structures Adjunction inequalities [Kronheimer-Mrowka] et al. SW=GW [Taubes] Calculation of the Yamabe invariants [LeBrun] et al stable cohomotopy invariants [Bauer-Furuta]

3 Pin (2) = U(1) j U(1) Sp(1) H Pin (2)-monopole equations are a twisted version of the Seiberg-Witten (U(1)-monopole) equations. Applications of Pin (2)-monopole equations Intersection forms with local coefficients Diagonalization theorem in local coefficients 10/8-type inequality (for non-spin) Pin (2)-monopole invariants Exotic structures Adjunction inequalities Calculation of the Yamabe invariants (j/w M. Ishida & S. Matsuo) stable cohomotopy invariants

4 Today Intersection forms with local coefficients Diagonalization theorem in local coefficients - 10/8-type inequality (for non-spin) Z 2 -Froyshov invariants & 4-manifolds with boundary Next talk Pin (2)-monopole invariants stable cohomotopy invariants

5 Intersection forms with local coefficients X: closed oriented 4-manifold with double cover X X l := X {±1} Z, a Z-bundle over X. H (X; l): l-coefficient cohomology. Note l l = Z. The cup product : H 2 (X; l) H 2 (X; l) H 4 (X; Z) = Z, induces the intersection form with local coefficient Q X,l : H 2 (X; l)/torsion H 2 (X; l)/torsion Z. Q X,l is a symmetric bilinear unimodular form.

6 Theorem [Froyshov,2012] X: a closed connected oriented smooth 4-manifold s.t. { H 2 (X; l) contains no element of order 4 b + (X)+ dim Z/2 (torh 1 (X; Z) Z/2) 2. (1) l X: a nontrivial Z-bundle. If Q X,l is definite Q X,l diagonal. The original form of Froyshov s theorem is: If X with X = Y : ZHS 3 satisfies (1) & Q X,l is nonstandard definite δ 0 : HF 4 (Y ; Z/2) Z/2 is non-zero. Y = S 3 HF 4 (Y ; Z/2) = 0 The above result.

7 The proof uses SO(3)-Yang-Mills theory on a SO(3)-bundle V. Twisted reducibles (stabilizer = {±1}) play an important role. V is reduced to λ E, where E is an O(2)-bundle, λ = det E: a nontrivial R-bundle. Cf [Fintushel-Stern 84] s proof of Donaldson s theorem also used SO(3)-Yang-Mills. Abelian reducibles (stabilizer = U(1)) V is reduced to R L, where L is a U(1)-bundle.

8 Theorem 1.(N. 2013) X: a closed connected ori. smooth 4-manifold. l X: a nontrivial Z-bdl. s.t. w 1 (l R) 2 = 0. If Q X,l is definite Q X,l diagonal. Cf. Froyshov s theorem { H 2 (X; l) contains no element of order 4 X: s.t. b + (X)+ dim Z/2 (torh 1 (X; Z) Z/2) 2. l X: a nontrivial Z-bundle. If Q X,l is definite Q X,l diagonal.

9 Theorem 1.(N. 2013) X: a closed connected ori. smooth 4-manifold. l X: a nontrivial Z-bdl. s.t. w 1 (l R) 2 = 0. If Q X,l is definite Q X,l diagonal. The proof uses Pin (2)-monopole equations. Pin (2)-monopole eqns are defined on a Spin c structure. Spin c structure is a Pin (2)-variant of Spin c -structure whose complex structure is twisted along l.

10 The moduli space of Pin (2)-monopoles is compact. Bauer-Furuta theory can be developed. Furuta s theorem Let X be a closed ori. smooth spin 4-manifold with indefinite Q X. b 2 (X) 10 sign(x) Theorem 2(N. 13, Furuta) Let X be a closed connected ori. smooth 4-manifold. For any nontrivial Z-bundle l X s.t. w 1 (l R) 2 = w 2 (X). b l 2(X) 10 sign(x) + 2, 8 where b l 2 (X) = rank H 2(X; l).

11 Pin (2)-monopole equations Seiberg-Witten equations are defined on a Spin c -structure. Spin c (4) = Spin(4) {±1} U(1) Pin (2)-monopole eqns are defined on a Spin c -structure. Spin c (4) = Spin(4) {±1} Pin (2)

12 Spin c (4) Pin (2) = U(1), j = U(1) j U(1) Sp(1) H. Two-to-one homomorphism Pin (2) O(2) z U(1) Pin (2) z 2 U(1) = SO(2) O(2), ( ) 1 0 j. 0 1 Definition Spin c (4) := Spin(4) {±1} Pin (2). Spin c (4)/ Pin (2) = Spin(4)/{±1} = SO(4) Spin c (4)/ Spin(4) = O(2) The id. compo. of Spin c (4) = Spin(4) {±1} U(1) = Spin c (4) Spin c (4)/ Spin c (4) = {±1}.

13 Spin c -structures X: an oriented Riemannian 4-manifold. F r(x): The SO(4)-frame bundle. X 2:1 X: (nontrivial) double covering, l := X {±1} Z [Furuta,08] A Spin c -structure s on X X is given by P : a Spin c (4)-bundle over X, P/ Spin c (4) = X P/ Pin (2) = F r(x). E = P/ Spin(4) O(2) X: characteristic O(2)-bundle. l-coefficient Euler class c 1 (E) H 2 (X; l).

14 Spin c (4) P J, J 2 = 1 Spin c (4) P/ Spin c (4) = X ι, ι 2 = id X X 2:1 P Spinc (4) X defines a Spin c -structure s on X J = [1, j] Spin(4) {±1} Pin (2) = Spin c (4) J covers ι Involution I on the spinor bundles S ± of s: S ± = P Spin c (4) H ± [J, j] =: I I 2 = 1 & I is antilinear. S ± = S ± /I are the spinor bundles for the Spin c -str. s S ± are not complex bundles.

15 Twisted Clifford multiplication ρ: T X (l 1R) End(S + S ) An O(2)-connection A on E + Levi-Civita Dirac operator D A : Γ(S + ) Γ(S ) Weitzenböck formula DAΦ 2 = A A Φ + s g 4 Φ + ρ(f + A ) Φ 2

16 Pin (2)-monopole equations { DA Φ = 0, ρ(f + A ) = q(φ), where A: O(2)-connection on E & Φ Γ(S + ) F + A Ω+ (l 1R) q(φ) = (Φ Φ) 1 2 Φ 2 id End(S + ) Remark Pin (2)-monopole on X = I-invariant Seiberg-Witten on X

17 Gauge symmetry G Pin =Γ( X {±1} U(1)) ={f Map( X, U(1)) f(ιx) = f(x) 1 } where {±1} U(1) by z z 1. Cf. Ordinary SW (U(1)) case G U(1) = Map(X, U(1))

18 Moduli spaces M Pin = { solutions }/G Pin (A E Γ(S + ))/G Pin where A E = the space of O(2)-connections on E Proposition M Pin is compact. The virtual dimension of M Pin : d = 1 4 ( c 1(E) 2 sign(x)) (b l 0 b l 1 + b l +). where b l = rank H (X; l). - If l is nontrivial & X connected b l 0 = 0.

19 Reducibles Irreducible: (A, Φ), Φ 0 G Pin -action is free. reducible: (A, Φ 0) The stabilizer is {±1} { reducible solutions }/G Pin = T b l 1 MPin.

20 Key difference Ordinary SW(U(1)) case Reducible The stabilizer = U(1). M U(1) \ {reducibles} (A U(1) (Γ(S + ) \ {0}))/G U(1) BG U(1) T b 1 CP. Pin (2)-monopole case Reducible The stabilizer = {±1}. M Pin \ {reducibles} (A (Γ(S + ) \ {0}))/G Pin BG Pin T bl 1 RP.

21 Proof of Theorem 1 Recall Theorem 1. X: a closed connected ori. smooth 4-manifold. l X: a nontrivial Z-bundle s.t. w 1 (l R) 2 = 0. If Q X,l is definite Q X,l diagonal. For simplicity, assume b l 1 = 0.

22 Lemma 1 characteristic elements w of Q X,l, 0 w 2 + b l 2. An element w in a unimodular lattice L is called characteristic if w v v v mod 2 for v L. Lemma 1 & [Elkies 95] Q X,l diagonal.

23 Lemma 2 If w 1 (l R) 2 = 0 characteristic element w, Spin c -str. s.t. c 1 (E) = w. Lemma 3 If b l + = b l 1 = 0 dim M Pin 0 for Spin c -str. Lemma 2 & 3 Lemma 1 0 dim M Pin = 1 4 ( c 1(E) 2 sign(x)) (b l 0 b l 1 + b l +) = 1 4 ( w2 + b l 2)

24 The structure of M Pin when b + (X; l) = 0 b 1 (X, l) = 0 1 reducible class ρ 0 M Pin. Perturb the Pin (2)-monopole equations by adding η Ω + (iλ) to the curvature equation. F + A = q(ϕ) + η. For generic η, M Pin \ {ρ 0 } is a d-dimensional manifold. Fix a small neighborhood N(ρ 0 ) of {ρ 0 }. N(ρ 0 ) = R d /{±1} = a cone of RP d 1 Then M Pin := M Pin \ N(ρ 0 ) is a compact d-manifold & M Pin = RP d 1.

25

26 Note M Pin (A (Γ(S + ) \ {0})) /G =: B. Recall B T b1(x;l) RP. h.e. Lemma 3 If b l + = 0 & b l 1 = 0 d = dim M Pin 0. Proof Suppose d > 0. Recall M Pin is a compact d-manifold s.t. M Pin = RP d 1. C H d 1 (B ; Z/2) = H d 1 (RP ; Z/2) s.t. C, [ M Pin ] = 0. Contradiction.

27 Another proof of Lemma 3 Monopole map - For simplicity, we assume b l 1 = dim H 1 (X; l) = 0. Fix a reference O(2)-connection A on E µ: Γ(S + ) Ω(l ir) Γ(S ) (Ω + Ω 0 )(l ir) (a, ϕ) (D A+a ϕ, F + A+a q(ϕ), d a) µ is {±1}-equivariant µ 1 (ball) ball

28 Finite dimensional approximation [Furuta, 95] Decompose µ = D + Q as D: linear & Q: quadratic Fix λ 1. ( ) eigenspaces of D D V λ = Span eigenvalues < λ ( ) eigenspaces of DD W λ = Span eigenvalues < λ p λ : Γ(S ) (Ω + Ω 0 )(l ir) W λ, L 2 -projection Finite dim approx. f = D + p λ Q: V λ W λ f is {±1}-equivariant, proper

29 f has the following form: f : R x+a R y R x R y+b, a = ind R D A, b = b l +(X) { R by multiplication {±1} acts on R trivially R : spinor part, R : form part [Fact] f {0} R y is a linear inclusion Suppose b = b l +(X) = 0. Consider the diagram ( Rx+a R y ) + f + i 1 ( Rx R y ) + i 2 (R y ) + = (R y ) + ( ) + means the 1-point compactifications.

30 Let G = {±1}. Apply H G ( ; F 2) to the diagram. (f + ) H G (( R x+a R y ) + ; F 2 ) H G (( R x R y ) + ; F 2 ) i 1 i 2 H G ( (R y ) + ) = ( ; F 2 H G (R y ) + ) ; F 2 H G ( (R y ) + ; F 2 ) = H (BG; F 2 ) = H (RP ; F 2 ) = F 2 [v], (deg v = 1) Im i 1 = e( R x+a ) = v x+a, Im i 2 = e( R x ) = v x. Commutativity Im i 1 = vx+a Im i 2 = vx 0 a = ind R D A = 1 4 ( w2 + b l 2 )

31 Definite 4-manifold with boundary Recall: [Elkies] Q: Definite standard characteristic w, 0 w 2 + rank Q. ( ) For closed X, if Q X, Q X,l : definite ( ). However if X has a boundary Y : ZHS 3, even when Q X or Q X,l : definite, ( ) may not be true. Instead, we can estimate how false ( ) is by some quantities related with Y. For Q X Froyshov invariant For Q X,l Z 2 -Froyshov invariant

32 Froyshov invariants Y : QHS 3 with Spin c -structure [Froyshov invariant] δ U(1) (Y ) = h(y ) Q δ U(1) (Y ): Manolescu s convention h(y ): Froyshov, Kronheimer-Mrowka Theorem [Froyshov] δ U(1) (Y 1 #Y 2 ) = δ U(1) (Y 1 ) + δ U(1) (Y 2 ), δ U(1) ( Y ) = δ U(1) (Y ) δ U(1) (Y ) is a Spin c -homology cobordism invariant. W : compact Spin c 4-manifold s.t. W = Y 1 Y k, Y i : QHS 3. b + (W ) = 0 k δ U(1) (Y i ) 1 8 (c 1(L) 2 + b 2 (W )) i=1 where L is the determinant line bundle.

33 Corollary [Froyshov] W : compact 4-manifold, W = Y 1 Y k, Y i : ZHS 3. If b + (W ) = 0, for characteristic element w of Q W k δ U(1) (Y i ) 1 8 ( w2 + b 2 (W )) i=1

34 Z 2 -Froyshov invariant Y : QHS 3 with Spin c -structure Theorem 3 [N, 16] We can define a topological invariant δ Z 2 (Y ) Q δ Z 2 (Y ) is a homology cobordism invariant (for ZHS 3 ). δ Z 2 (Y ) 2δ U(1) (Y ) W : cobordism from Y 0 to Y 1 with a Z bundle l W s.t. b l +(W ) = 0 & l Y0, l Y1 : trivial For a Spin c structure on l, δ Z 2 (Y 1 ) δ Z 2 (Y 0 ) ( c 1(E) 2 + b l 2(W )) [Remark] l Yi : trivial Spin c str on W induces a Spin c str on Y i

35 Corollary [N, 16] W : compact 4-manifold, W = Y 1 Y k, Y i : ZHS 3. If l W is a Z bundle with w 1 (l R) 2 = 0 & b l +(W ) = 0, for characteristic element w of Q W,l k δ U(1) (Y i ) 1 2 δz 2 (Y 1 # #Y K ) 1 8 ( w2 + b l 2(W )) i=1 Conjecture δ Z 2 (Y 1 #Y 2 ) = δ Z 2 (Y 1 ) + δ Z 2 (Y 2 ) δ Z 2 ( Y ) = δ Z 2 (Y )

36 A definition of δ U(1) (Y ) Let X = SWF(Y) be the Seiberg-Witten-Floer homotopy type constructed by Manolescu. Here we assume X as a space. U(1) acts on X & X U(1) = (R s ) +. Apply H U(1) ( ; F) to the inclusion i: XU(1) X Note i : H U(1) (X; F) H U(1) (XU(1) ; F) H s U(1) (XU(1) ; F) = H (CP ; F) = F[u], deg u = 2. Via this identification, d, Im i = u d. δ U(1) (Y ) = d + (some grading shift)

37 Definition of δ Z 2 (Y ) Since X = SWF(Y) is a U(1)-space & U(1) {±1} = Z 2, Z 2 acts on X & X Z 2 = (R s ) +. Apply H Z 2 ( ; F 2 ) to the inclusion i: X Z 2 X Note i : H Z2 (X; F 2 ) H Z 2 (X Z 2 ; F 2 ) H s Z 2 (X Z 2 ; F 2 ) = H (RP ; F 2 ) = F 2 [v], deg v = 1. Via this identification, d 2, Im i = v d 2. δ Z 2 (Y ) = d 2 + (some grading shift)

38 Idea of the proof of Theorem 3 Let W be the Spin c cobordism in Theorem 3. Let X i = SWF(Y i ) (i = 0, 1). Pin (2)-monopole map on W induces a Z 2 -equivariant map We have a diagram f : Σ X 0 Σ X 1 Σ X 0 f Σ X 1 (Σ X 0 ) Z 2 (Σ X 1 ) Z 2 Applying H Z 2 ( ; F 2 ), we obtain the inequality in Theorem 3.

39 The outline of the proof of Theorem 2 Recall Theorem 2 Let X be a closed connected ori. smooth 4-manifold. For any nontrivial Z-bundle l X s.t. w 1 (l R) 2 = w 2 (X). where b l 2 (X) = rank H 2(X; l). b l 2(X) 10 sign(x) + 2, 8

40 w 1 (λ) 2 = w 2 (X) E = R (l R) Spin c -structure on (X, E) has a larger symmetry - For simplicity, assume b l 1(X) = 0. G Pin = Γ( X {±1} Pin (2)). By taking a finite dimensional approximation of the monopole map, we obtain a proper Z 4 -equivariant map f : R y C x+k 1 R y+b C x 1, k = ind C D A, b = b l +(X) { C1 by multiplication of i Z/4 acts on R via Z/4 Z/2 = {±1} R Here, Z 4 is generated by the constant section j G = Γ( X {±1} Pin (2)).

41 By using the techniques of equivariant homotopy theory, e.g., tom Dieck s character formula, we can see that any proper Z 4 -map of the form, should satisfy b > k. That is, f : R y C x+k 1 R y+b C x 1, b l +(X) 1 sign(x)