Introduction to Heegaard Floer homology

Transcription

1 Introduction to Heegaard Floer homology In the viewpoint of Low-dimensional Topology Motoo Tange University of Tsukuba 2016/5/19

2 1 What is Heegaard Floer homology? Theoretical physician says. 4-dimensional gauge theory = 2-dimensional non-linear sigma models Floer s idea(floer s will) This equivalence means mathematically an isomorphism of homology groups!! Consider the Morse theory on Chern-Simons functional. Topologist considers in this situation. This isomorphism can apply to make interesting invariant.

3 Atiyah-Floer conjecture (Yang-Mills theorey) (gauge theory side) Y : homology 3-sphere P Y Y : SU(2), SO(3)-bundle with w 2 (P Y ) 0. Generators: R(Y ) flat connections Differential: M Y R : ASD connections on Y R HF inst (Y, P Y ) (symplectic geometry side) Y = U 0 Σ U 1 : Heegaard decomposition R(Σ): flat connections on Σ SO(3) L 0, L 1 R(Σ): subset of flat connections extending to U 0, U 1. Generators: L 0 L 1 Differential: holomorphic disk connecting two intersection points HF symp (R(Σ), L 0, L 1 ) HF inst (Y, P Y ) = HF symp (R(Σ), L 0, L 1 )

4 Homology theory on Gauge theory = Homology theory on Symplectic theory Theorem 1 (Fukaya 15) This isomorphism of SO(3)-version is true by using bounding cochain b. Theorem 2 (Taubes) χ(hf inst (Y, P Y )) = λ(y ) (Casson invariant) Topological invariant appears!

5 Ozsváth-Szabó s Dream Parallel thinking Seiberg-Witten invariant is Tractable. + Atiyah-Floer isomorphism is mysterious relationship Seiberg-Witten s sympletic side should definitely construct a TRUELY interesting topological invariant!!!! They swore to make the invariants in their mind (about 1998??).

6 Ozsváth and Szabó said in 2001 Look! We did it. And put the following gently. arxiv:math/

7 As expected... The invariants are interesting. This excitement is beyond the expectation.

8 In their mind. (gauge theory side) (Y, g, s): a Spin c Riemannian 3-manifold. Generators: a critical point x of CSD functional=the solution of Seiberg-Witten equation on (Y 3, g, s): CSD : M SW (Y 3, s, g) R. Differential: s(t): a solution of the Seiberg-Witten equation on Y R with lim x(t) s(t) = x, lim s(t) = y x(t) x = y +

9 (symplectic geometry side) Y = U 0 Σ U 1 : Heegaard splitting S Σ : moduli space of the solutions of SW-equation over Σ Generaotrs:?? Differential:?? SWF symp (Y, s) SWF (Y, s) = SWF symp (Y, s)??

10 Their final picture. Definition 3 (Heegaard Floer homology) HF (Y, s) SWF symp (Y, s) This invariant is weak?? No, nevertheless,... χ(hf (Y, s)) = H 1 (Y )

11 2 Floer theory (X, ω): Symplectic manifold ((almost) Complex manifold) L 0, L 1 X : Lagrangian submanifolds (Totally real submanifolds) Lagrangian intersection Floer homology Generator: L 0 L 1 Differential: Counting of holomorphic disk connecting x, y L 0 L 1. x = #( M(ϕ) R )y y L 0 L 1 ϕ π 2 (x,y) CF (X, L 0, L 1 ): Lagrangian intersection Floer homology

12 Differential operator Holomorphic disk D M(ϕ), [D] = ϕ.

13 A -structure The graded map m n There exists the following: m k : CF (T 0, T 1 ) CF (T 1, T 2 ) CF (T n 2, T n 1 ) CF (T 0, T n 1 ) by counting the holomorphic polygon satisfying an A relation. This map is applied to (Dehn) surgery exact sequence.

14 Low-dimensional applicaction of Heegaard Floer homology Figure: 2007

15 Figure: 2016

16 Knot theory Figure: Red:Strongly related. Cyan: Less known. Magenta: Gently related.

17 Related topics 3-manifold theory (contact structure, foliation, homology cobordism, Dehn surgery, Seiberg-Witten Floer, Casson invariant) 4-manifold theory (cobordism, Seiberg-Witten invariant, exotic structure) Knot theory (Seifert genus, 4-ball genus, concordance, fiberness, ) Categorical action of mapping class group

18 However, it is less known whether the following is related: Fundamental group (expected) Geometric structure, π 1 (Y ), Branched cover or covering Chern-Simons invariant, Quantum invariants (Jones polynomial, Z(K) LMO, Ohtsuki etc.) Purely topological invariant (Heegaard genus, minimal crossing number bridge number, Morse-Novikov number, )

19 The role of low-dimensional topology In the feture, Floer theory Low-dimensional topology Floer theory Low-dimensional topology

20 3 Overview of Heegaard Floer homology SWF (Y ) = HF (Y ) Atiyah-Floer view point. Gauge theory 4-dim SWF HF (3-dim) Symplectic theory 2-dim Isomorphism:Kutluhan(Tr)Lee(Ch)T(US) &Colin(Fr.)Ghiggini(It.)Honda(Jp)

21 Heegaard decomposition Seiberg-Witten solutions on Σ. Moduli space of Seiberg-Witten solution over Σ = Sym g (Σ) (the g-fold Symmetric product of Σ) g: the genus of Σ.

22 α-curves, β-curves

23 pointed Heegaard diagram. pointed Heegaard decomposition pointed Heegaard diagram (Sym g (Σ g ), ω, J, g) Ozsváth-Szabó s Lagrangian submanifolds in Sym g (Σ g ). T g = Tα = α 1 α 2 α g Sym g (Σ g ) T g = Tβ = β 1 β 2 β g Sym g (Σ g )

24 Definition of ĤF n z (ϕ) = #[(disk D) (z Sym g (Σ g ))] [D] = ϕ ĈF (Σ, α, β, s) = CF (Sym g (Σ g ), T α, T β ) Generators: T α T β a g-tuple of points over Σ g σ S g {α 1 β σ(1) } {α 2 β σ(2) } {α g β σ(g) } Differential: counting of holomorphic disks with n z (ϕ) = 0 ˆ x = ( ) M(ϕ) # y R y T α T β ϕ π 2 (x, y), n z (ϕ) = 0 µ(ϕ) = 1 π 2 (x, y): homotopy classes of disks from x to y M(ϕ): the Moduli space of holomorphic curve representing ϕ

25 ĤF (Y ) = H (ĈF (Σ, α, β, s)) Topological invariant ĤF (Y ): H 1 (Y, Z)-module ĤF (Y ) = s Spin c (Y )ĤF (Y, s) Euler number: χ(ĥf (Y )) = H 1(Y ). ŜWF (Y ) = ĤF (Y ) (Atiyah-Floer conjecture)

26 Ozsváth-Szabó s idea D {(x, v) Σ Sym g (Σ g ) x v} Σ g-fold branched cover D Sym g (Σ g ) D: holomorphic curve with boundary Disks in Sym g (Σ g ) =Holomorphic regions on Σ The region is the lift of a disk via the cover.

27 Examples of holomorphic regions

28 Definition of HF z Σ g α 1 β g z Sym g 1 (Σ g ) Sym g (Σ g ) hypersurface Z[U, U 1 ]-module chain complex (CF (Σ, α, β, z), ) Generators: U n x, x T α T β Differential: (U i x) = y T α T β ϕ π 2 (x, y), gr(ϕ) = 1 ( ) M(ϕ) # U i nz (ϕ) y R HF (Y ) = H (CF (Σ, α, β, z)) topological invariant

29 Definition of HF, HF + CF (Σ, α, β) (subchain complex of CF ) Generators: U n x (n > 0 and x T α T β ) Differential: (restricted) Filtered chain complex: Exact sequence {0} CF (Σ, α, β, z) i CF (Σ, α, β, z) 0 CF (Σ, α, β, z) i CF (Σ, α, β, z) π CF + (Σ, α, β, z) 0 (CF, ): sub-chain complex (CF +, + ): quotient complex

30 Module structure F = Z/2Z HF (Y, s): F[U, U 1 ]-module HF + (Y, s): F[U]-module HF (Y, s): F[U]-module π : HF (Y, s) HF + (Y, s) Im(π ) is a finite direct sum of T + : T + = F 1 F U 1 F U 2 = F[U, U 1 ]/U F[U] (Infinite dimensional F-module with U-action)

31 HF red (Y, s) HF red (Y, s) := HF + (Y, s)/π (HF (Y, s)) T + [n] = F 1 F U 1 F U 2 F U n+1 HF red (Y, s) is a finite sum of T + [n]. Definition 4 (L-space) Let Y be a QHS 3. We define Y to be L-space HF red (Y, s) = {0} for any s Spin c (Y ).

32 TQFT invariant (Y i, s i ): two Spin c 3-dim manifolds (i = 0, 1) (X, t): a 4-dimensional cobordism between (Y i, s i ) HF (Y 0, s 0 ) F X,t HF (Y 1, s 1 ) (homomorphism)

33 For any torsion spin c structure s, HF admits an absolute Q-grading ( = ±, ). x i HF (Y i, s i ) HF + (Y 0, s 0 ) F W,t HF + (Y 1, s 1 ) gr(f W,t (x 0 )) gr(x 0 ) = c2 1 (t) 2χ(W ) 3σ(W ) 4 d-invariant (correction term) d(y, s) = min{gr(x) Q 0 x Image(π )} λ(y, s) = χ(hf red (Y, s)) 1 2d(Y, s) λ(y ) = s d(y, s) (the Casson-Walker invariant formula)

34 d-invariant (I) d(y, s) is spin c rational homology cobordism invariant, i.e., (Y 0, s 0 ) (W,t) (Y 1, s 1 ) (Spin c cobordism) Then we have H (Y i, Q) inclusion = H (W, Q), t Yi = s i d(y 0, s 0 ) = d(y 1, s 1 ) Θ 3 Z : the group of the homology spheres d : Θ 3 Z 2Z (homo.) Y : QHS 3 W : negative definite bounding with W = Y c 2 1 (t) + b 2 (W ) 4d(Y, s)

35 d-invariant (II) Y : QHS 3 HF + (M, s) = T + d(m,s) HF red(m, s) d(l(p, q), i) = 3s(q, p) 1 p = 1 4p p 1 r=1 ( +2 cos (i q 1 2 ) 2πr p ) 2p i p + 2 i j=1 ( cot πr qπr p cot p cosec πr p cosec qπr p ) (( )) q j p (T.) Are there relationship between hyperbolic structure and d(y, s) and HF red? Y : graph manifold (with at most one bad vertex) HF + (Y, s) is algorithmically computable (Nemethi s algorighm)

36 HF red (Y, s) Differential topology complexity? If 4-manifolds X 0, X 1 have distinct mixed invariants and X 1 = (X 0 W ) Y W, then HF red (Y ) = {0}. L-space: HF red (Y, s) = 0 (e.g., lens spaces) Y admits no taut foliation on Y. If Sp 3 (K) = Y, then K is fibered and K (t) has flat & alternating. NON Left orderable

37 Dehn surgery Dehn surgery K Y : null-homologous knot Y p (K) := [Y N(K)] φ S 1 D 2 exact triangle

38 Y : 3-manifold Y = (Σ, T, T ), Y 0 = (Σ, T, T 0 ), Y 1 = (Σ, T, T 1 ) Y 0 = Y 0 (K): 0-surgery Y 1 = Y 1 (K) m 2 : CF (Σ, T, T ) CF (Σ, T, T 0 ) CF (Σ, T, T 0 ) Here (Σ, T, T 0 ) = # g 1 S 2 S 1 Exact sequence Applying the m 2 -map, we have 0 CF (Σ, T, T ) CF (Σ, T, T 0 ) CF (Σ, T, T 0 ) 0

39 L-spaces HF red (Y, s) = 0 dim ĤF = H 1 Example 5 (L-spaces) S 3 Lens spaces Elliptic geometry Double branched cover of (quasi-)alternating knot Sp(K) 3 p >> 0, K is L-space knot. Seifert fibered rational homology spheres with no taut foliation taut closed transversal

40 The cases which two in the exact triangle are L-spaces s i{ Z/pZ Y : L-space ZHS 3 (I ) HF + red Y p (K) are L-space (Y 0, i) = T + [n i ] { Y : L-space ZHS 3 (II ) Y 0 (K) L-space HF red (Y, s) = {0} HF + red { (Y p(k), i) = T + [n i ] Y 0 (K) an L-space (III ) HF + red Y p (K) an L-space (Y, i) = Z[U]/U n

41 (I) Y, Y p L-spaces Theorem 6 (Ozsváth-Szabó) K (t) = ( 1) m + m ( 1) m j (t n j + t n j ) j=1 0 < n 1 < n 2 < < n m In Y = S 3 case, Greene classified a simple lens space knot. In Y general L-space ZHS 3 case, T. classified a simple lens space knot (TBA).

42 Other cases (II) Y, Y 0 L-spaces (III) Y 0, Y p L-spaces g(k) = 1 Problem 7 Classify these surgeries.

43 Gordon conjecture Theorem 8 (Kronheimer-Mrowka-Ozsváth-Szabó (2003)) Let U be the unknot. If S 3 p (K) = S 3 p (U), then K = U. q q Seiberg-Witten Floer Problem 9 Prove it by using the Heegaard Floer homology.

44 Geometric aspect Hyperbolic Dehn surgery (W.Thurston) Geom. of Knots Geom. of 3-manifolds Torus knots Satellite knots Hyperbolic knots Spherical manifolds Elliptic manifolds. Hyperbolic manifolds In fact some hyperbolic knots K yield lens spaces.

45 Fundamental group Y is an L-space. Theorem 10 If Y is L-space, then Y does not admit taut foliation. Conjecture 11 If Y is irreducible rational homology sphere, then this converse is true. Theorem 12 (Gabai) If Y admit taut foliation, then the universal cover of Y is homeomorphic to R 3. L-space condition gives some kind of the restriction to the fundamental group.

46 Left-orderability Definition 13 (Left-orderable) G {e}: group order < on G g, h, k π 1 (Y ) g < h kg < kh If b 1 (Y ) > 0 then π 1 (Y ) is left-orderable. In particular left-orderable G is non-torsion. Conjecture 14 Y : QHS 3. π 1 (Y ) is not left-orderable Y : L-space.

47 Evidences Theorem 15 (Many topologists) Well-known L-spaces admit no left-orderable. Theorem 16 (Strong L-space) If Y is strong L-space then Y is not left-orderable. Strong L-space dim ĈF = H 1. Theorem 17 If Y is an L-space Seifert fibered QHS 3 Y is not left-orderable. Y is not taut foliation.

48 Mixed 4-manifold invariant X : closed 4-manifold X = W 1 N W 2 :b + 2 (W i) > 1 N: admissible cut. ( δh 1 (N) H 2 (X ) is trivial) 1 HF (S 3 ) F W p HF (N) HF red (N) HF + (N) p F + W + HF + (S 3 ) Φ X,s Z/2Z HF (S 3 ) FW HF (N) HF (N) FW HF (S 3 ) are 0-map. OS X,s = Φ X,s s s Spin c (X )

49 Theorem 18 OS X,s is smooth 4-manifold invariant. Conjecture 19 Let X be a closed oriented 4-manifold. Then [SW X,s ] = OS X,s

50 Contact invariant Definition 20 ((Y, ξ) contact structure) (Y, ξ): everywhere non-integrable plane field Definition 21 c(ξ) HF + (Y, s(ξ)) isotopy invariant of (Y, ξ). Furthermore, if ξ is over-twisted, then c(ξ) = 0. Thus we have c(ξ) 0 (Y, ξ) is tight.

51 4 Knot Floer homology

52 Knot filtration CFK (Σ, α, β, z, w): Filtered chain complex of CF (Σ, α, β, z) For generators it is the identity.

53 Example 22 (The trefoil knot) x 1, x 2, x 3 generators of ĈF (Σ, α, β, z, w)

54 Differentials x 1 = 0, x 2 = Ux 1 + x 3, x 3 = 0 ˆ x 1 = 0, ˆ x 2 = x 3, ˆ x 3 = 0

55 Alexander Filtration x 1 = 0, x 2 = Ux 1 + x 3, x 3 = 0 ˆ x 1 = 0, ˆ x 2 = x 3, ˆ x 3 = 0 Alexander filtration Alex : T α T β Z For a disk ϕ π 2 (x, y) Alex(x) Alex(y) = n z (ϕ) n w (ϕ) Alex(x 2 ) Alex(x 1 ) = 0 1 = 1 Alex(x 2 ) Alex(x 3 ) = 1 0 = 1

56 Alexander filtration Thus, x 1 = 0, x 2 = Ux 1 + x 3, x 3 = 0 ˆ x 1 = 0, ˆ x 2 = x 3, ˆ x 3 = 0 Alex(x 1 ) = 1, Alex(x 2 ) = 0, Alex(x 3 ) = 1 Actually generators are Symmetric via this Alexander filtration: ( ) Generators Kauffman s states Poincaré polynomial of ĈFK Alexander polynomial

57 Another grading x 1 = 0, x 2 = Ux 1 + x 3, x 3 = 0 ˆ x 1 = 0, ˆ x 2 = x 3, ˆ x 3 = 0 Maslov index M(x) M(y) = µ(ϕ) 2n w (ϕ) M(x 2 ) M(x 1 ) = = 1 M(x 2 ) M(x 3 ) = = 1 Thus, M(x 1 ) = 0, M(x 2 ) = 1, M(x 3 ) = 2 (The grading of ĈF (S3 ) is zero.)

58 x 1 = 0, x 2 = Ux 1 + x 3, x 3 = 0 ˆ x 1 = 0, ˆ x 2 = x 3, ˆ x 3 = 0

59 (ĈKF, ˆ K ) x 1 = 0, x 2 = Ux 1 + x 3, x 3 = 0 ˆ x 1 = 0, ˆ x 2 = x 3, ˆ x 3 = 0 Definition 23 (Differential operator ˆ K of ĈFK) ˆ K x = #( M(ϕ) R )y y T α T β ϕ π 2 (x,y),ν(ϕ)=1,n z (ϕ)=n z (ϕ)=0 ˆ K x 1 = 0, ˆ K x 2 = 0, ˆ K x 3 = 0 Z (0) j = 1 Z ĤFK(3 1, j) = ( 1) j = 0 Z ( 2) j = 1 0 otherwise 1 χ(ĥfk(3 1, j))t j = 1 t + ( 1) t 1 j= 1 = t t = K (t)

60 z, w Σ g α 1 β g z Sym g 1 (Σ g ), w Sym g 1 (Σ g ) Sym g (Σ g ) (CFK (Σ, α, β, z, w), ) (ĈKF, ˆ ) Definition 24 ((ĈFK, ˆ K )) Generators: a lift of T α T β. Differential: Counting holomorphic disks x = ( M(ϕ) # R T α T β ϕ π 2 (x,y),µ(ϕ)=0 ) U nw (ϕ) y Euler number: s χ(ĥfk(k, s))t s = K (t) Alexander polynomial

61 Interpretation of (ĈKF, ˆ K ) in low-dimensional topology Generators: ĈFK(K) {Kauffman states} Differential : clockwise exchange

62 Definition 25 (Alternating knot) K admits an alternating knot diagram. Theorem 26 (Ozsváth-Szabó) Let K be a alternating knot. ĤFK(K, s) = Z as (s+ σ 2 ) d(s1 3 σ(k) (K)) = 2 min(0, 2 ) Theorem 27 (Ozsváth-Szabó) Alternating knot ĤFK(KinoTera, s) = ĤFK(Conw, s) for some s Problem 28 Clarify the mutant effect to ĤFK.

63 5 Knot Concordance c :concordant equivalent C := {knots}/ c K 0 c K 1 φ : S 1 I S 3 I, K i = φ(s 1 i) C is an abelian group. C [K] = 0 K is slice knot Definition 29 (Zero element) K is slice knot if embedding D 2 B 4 with D 2 = K.

64 Concordance invariants g 4 (K) = min{g(σ) Σ = K, Σ B 4 } g 4 : C Z 0 ω S 1 σ ω (K) = σ((1 ω)v + (1 ω) t V ) σ(x ) 2g 4 (K) σ ω : C Z (homo.) ds 1 (K) = d(s 3 1 (K)) C Z(not homo.)

65 Heegaard Floer concordance invariant δ(k) = 2d(Σ 2 (K), s 0 ) Z (homo.) F(K, s) F(K, s + 1) F(K, s + 2) τ : C Z (Ozsváth-Szabó s τ-invariant (homo.)) τ(k) = min{s i s : H (F(K, s)) ĤF (S 3 ) non-trivial}

66 Theorem 30 τ(k) g 4 (K) The τ is independent of σ(k) and Rasmussen s s(k). c.f. σ(k) 2g 4 (K) (Murasugi) Theorem 31 (Ozsváth-Szabó) If K is alternating knot, then τ(k) = 1 2 σ(k) Theorem 32 (Another formula (T.)) g τ(k) = χ(f(k, s)) s= g

67 Dehn surgery formula g = g(k), n 2g 1 Theorem 33 (Ozsváth-Szabó) ĈF (Sn 3 (K), [s]) = ĈFK(K){max{i, j s} = 0} ĈF (S n(k), 3 [s]) = ĈFK(K){min{i, j s} = 0}

68 F s : ĤF (S3 ) ĤF (S3 N (K), [s]), G s : ĤF (S3 N (K), [s]) ĤF (S 3 ) Definition 34 (Hom s ϵ) τ = τ(k). 1 F τ trivial ϵ(k) = 1 G τ trivial 0 F τ, G τ non-trivial Theorem 35 (Hom) ϵ(k) is concordant invariant and stronger than τ. [K] C then ϵ(k) = 0. ϵ(k) = 0 then τ(k) = 0 ϵ( K) = ϵ(k) The cable τ(k p,q ) is computable by τ(k) and ϵ(k). Z C top

69 Other invariants in terms of ĈFK V k -invariant ν(k) ν (K) = ν + (K) g c (K) g c (C) Theorem 36 (Wu-Hom) Strictly sharper bounds: τ(k) ν + (K) g 4 (K)

70 Υ-invariant Livingston s interpretation. t, s R C = CFK (C, F t ) s = {x C t ( 2 Alex(x) + 1 t ) } Alg(x) s 2 ν(c, F t ) = min{s Im(H 0 ((C, F t ) s ) H 0 (C) is surjective} R For s 1 s 2 (C, F t ) s1 (C, F t) s2

71 0 t 2 Definition 37 (Ozsváth-Stipsicz-Szabó) Υ(K) : C C([0, 2]) (homo.) Υ K (t) = 2ν(C, F t )

72 (3, 4)-torus knot 3t 0 t 2 3 Υ K (t) = t t 6 3 t

73 Properties Knot concordance invariant Υ K (0) = 0 Υ K (0) = τ(k) Υ K (t) = Υ K (2 t) Υ K ( m n ) 1 n Z 0 < t < 1, Υ K (t) t g 4 (K) max{slopes of Υ K (t) } g c (K) K is alternating: Υ K (t) = (1 t 1 )σ

74 Theorem 38 (Ozsváth-Szabó) K: a torus knot. Then Problem 39 K (t) = n ( 1) i t n i i=0 0 j = n δ i = δ i+1 2(n i+1 n i ) + 1 n i : odd δ i+1 1 n i : even Υ K (t) = max {δ 2i tn 2i } {i 0 2i n} Let K be a torus knot. Give an explicit formula of Υ K (t).

75 6 Future s problem For given chain complex C, construct a knot with C = CFK. Give combinatorial description of knot (link) Floer homology Z-coefficient. Consider torsion part of HF (Y, K) with Z-coefficient. Classify the categorical action on Bordered Floer homology. Connection with Khovanov homology. Deal (branched) covering. Deal hyperbolic structure. L-space & Left-orderable. Relate to Quantum invariants. Relate to Twisted Alexander polynomial. Increase people who study Heegaard Floer homology in Japan.