Simplicial volume of surface bundles

Transcription

1 Simplicial volume of surface bundles Caterina Campagnolo Karlsruher Institut für Technologie March 6, 2018

2 Simplicial volume of surface bundles Plan of the presentation:

3 Simplicial volume of surface bundles Plan of the presentation: Surface bundles

4 Simplicial volume of surface bundles Plan of the presentation: Surface bundles Simplicial volume

5 Simplicial volume of surface bundles Plan of the presentation: Surface bundles Simplicial volume Bounded cohomology

6 Simplicial volume of surface bundles Plan of the presentation: Surface bundles Simplicial volume Bounded cohomology Simplicial volume of surface bundles

7 Surface bundles Definition Let Σ h be the closed connected oriented surface of genus h.

8 Surface bundles Definition Let Σ h be the closed connected oriented surface of genus h. We will mostly assume h 2.

9 Surface bundles Definition Let B be a manifold.

10 Surface bundles Definition Let B be a manifold. A Σ h -bundle over B is a continuous map π : E B such that b B there exists an open neighbourhood b U B with φ U : π 1 (U) = U Σ h so that π 1 (b) = {b} Σ h. π U p U

11 Surface bundles Definition Let B be a manifold. A Σ h -bundle over B is a continuous map π : E B such that b B there exists an open neighbourhood b U B with φ U : π 1 (U) = U Σ h π U p U so that π 1 (b) = {b} Σ h. For each U i U j and each b U i U j, the composition φ j φ 1 i is required to be an element of Homeo(Σ h ): φ 1 i {b} Σ h π 1 (b) φ j {b} Σ h.

12 Surface bundles Definition Let B be a manifold. A Σ h -bundle over B is a continuous map π : E B such that b B there exists an open neighbourhood b U B with φ U : π 1 (U) = U Σ h π U p U so that π 1 (b) = {b} Σ h. For each U i U j and each b U i U j, the composition φ j φ 1 i is required to be an element of Homeo(Σ h ): φ 1 i {b} Σ h Notation: Σ h E B. π 1 (b) φ j {b} Σ h.

13 Surface bundles Examples

14 Surface bundles Examples Trivial bundle B Σ h.

15 Surface bundles Examples Trivial bundle B Σ h. Surface bundle over S 1 :

16 Surface bundles Examples Trivial bundle B Σ h. Surface bundle over S 1 : B = S 1 = [0, 1]/{0 1};

17 Surface bundles Examples Trivial bundle B Σ h. Surface bundle over S 1 : B = S 1 = [0, 1]/{0 1}; E = [0, 1] Σ h /{(0, x) (1, φ(x))}

18 Surface bundles Examples Trivial bundle B Σ h. Surface bundle over S 1 : B = S 1 = [0, 1]/{0 1}; E = [0, 1] Σ h /{(0, x) (1, φ(x))} φ E where φ Homeo(Σ h ).

19 Surface bundles Examples Trivial bundle B Σ h. Surface bundle over S 1 : B = S 1 = [0, 1]/{0 1}; E = [0, 1] Σ h /{(0, x) (1, φ(x))} φ E where φ Homeo(Σ h ). If we want E to be orientable, choose f Homeo + (Σ h ).

20 Surface bundles Examples Surface bundle over S 1 : E = [0, 1] Σ h /{(0, x) (1, φ(x))} φ E where φ Homeo(Σ h ). If we want E to be orientable, choose f Homeo + (Σ h ). This is the mapping torus of φ, denoted M φ :

21 Surface bundles Examples Surface bundle over S 1 : E = [0, 1] Σ h /{(0, x) (1, φ(x))} φ E where φ Homeo(Σ h ). If we want E to be orientable, choose f Homeo + (Σ h ). This is the mapping torus of φ, denoted M φ : Σ h M φ = E S 1

22 Surface bundles Examples Surface bundle over S 1 : φ E This is the mapping torus of φ, denoted M φ : Σ h M φ = E S 1 We will be interested in oriented surface bundles over surfaces Σ h E Σ g.

23 Surface bundles Monodromy

24 Surface bundles Monodromy Fact: φ ψ M φ = Mψ.

25 Surface bundles Monodromy Fact: φ ψ M φ = Mψ. Need only specify [φ] Mod(Σ h ) = Homeo + (Σ h )/Homeo 0 (Σ h ).

26 Surface bundles Monodromy Fact: φ ψ M φ = Mψ. Need only specify [φ] Mod(Σ h ) = Homeo + (Σ h )/Homeo 0 (Σ h ). More generally, one gets a group homomorphism ρ : π 1 (B) Mod(Σ h ),

27 Surface bundles Monodromy Fact: φ ψ M φ = Mψ. Need only specify [φ] Mod(Σ h ) = Homeo + (Σ h )/Homeo 0 (Σ h ). More generally, one gets a group homomorphism the monodromy morphism. ρ : π 1 (B) Mod(Σ h ),

28 Surface bundles Monodromy Remark: The mapping tori M φ are well understood, thanks to Thurston.

29 Surface bundles Monodromy Remark: The mapping tori M φ are well understood, thanks to Thurston. In particular, M φ is hyperbolic φ is pseudo-anosov.

30 Surface bundles Monodromy Remark: The mapping tori M φ are well understood, thanks to Thurston. In particular, M φ is hyperbolic φ is pseudo-anosov. If Σ h E Σ g, it is not known if there exists E admitting a hyperbolic structure.

31 Simplicial volume Definition

32 Simplicial volume Definition Le M be a closed oriented manifold of dimension n.

33 Simplicial volume Definition Le M be a closed oriented manifold of dimension n. H n (M, Z) H n (M, R) [M] [M] = [ i a iσ i ].

34 Simplicial volume Definition Le M be a closed oriented manifold of dimension n. H n (M, Z) H n (M, R) [M] [M] = [ i a iσ i ]. Definition (Gromov, 80 s) The simplicial volume of M is { [ ] M = inf ai ai σ i } = [M] H n (M, R).

35 Simplicial volume Example

36 Simplicial volume Example S 1 = 0.

37 Simplicial volume Example S 1 = 0. 3

38 Simplicial volume Example S 1 =

39 Simplicial volume Example S 1 =

40 Simplicial volume Example S 1 = k

41 Simplicial volume Example S 1 = k = k N, S 1 1 k.

42 Simplicial volume Properties Fact:

43 Simplicial volume Properties Fact: Let f : M N be a degree d map. Then M d N.

44 Simplicial volume Properties Fact: Let f : M N be a degree d map. Then (Equality if f is a covering.) M d N.

45 Simplicial volume Properties Fact: Let f : M N be a degree d map. Then (Equality if f is a covering.) Consequence: M d N.

46 Simplicial volume Properties Fact: Let f : M N be a degree d map. Then (Equality if f is a covering.) Consequence: For n N, M d N. S n = 0, T n = 0, S n M = 0, T n M = 0.

47 Simplicial volume Properties Theorem (Gromov, Thurston)

48 Simplicial volume Properties Theorem (Gromov, Thurston) If n 2 and M n is a closed oriented manifold with negative sectional curvature bounded away from 0, M n > 0.

49 Simplicial volume Properties Theorem (Gromov, Thurston) If n 2 and M n is a closed oriented manifold with negative sectional curvature bounded away from 0, M n > 0. If M n is hyperbolic, M n = vol(mn ) v n, where v n = vol(ideal regular hyperbolic n-simplex).

50 Simplicial volume Properties Theorem (Gromov, Thurston) If n 2 and M n is a closed oriented manifold with negative sectional curvature bounded away from 0, M n > 0. If M n is hyperbolic, M n = vol(mn ) v n, where v n = vol(ideal regular hyperbolic n-simplex). In particular, Σ g = 2π χ(σ g ) π = 2 χ(σ g ) g 2.

51 Simplicial volume Question:

52 Simplicial volume Question: How does one compute M?

53 Bounded cohomology Definition

54 Bounded cohomology Definition Let X be a topological space.

55 Bounded cohomology Definition Let X be a topological space. 0 C 0 (X, R) δ C 1 (X, R) δ C 2 (X, R) δ...

56 Bounded cohomology Definition Let X be a topological space. 0 C 0 (X, R) δ C 1 (X, R) δ C 2 (X, R) δ... 0 Cb 0(X, R) δ Cb 1(X, R) δ Cb 2(X, R) δ...

57 Bounded cohomology Definition Let X be a topological space. 0 C 0 (X, R) δ C 1 (X, R) δ C 2 (X, R) δ... 0 Cb 0(X, R) δ Cb 1(X, R) δ Cb 2(X, R) δ... where Cb k(x, R) = { f C k (X, R) f < }

58 Bounded cohomology Definition Let X be a topological space. 0 C 0 (X, R) δ C 1 (X, R) δ C 2 (X, R) δ... 0 Cb 0(X, R) δ Cb 1(X, R) δ Cb 2(X, R) δ... where C k b (X, R) = { f C k (X, R) f < } and f = sup σ Sk (X ) f (σ).

59 Bounded cohomology Definition Let X be a topological space. 0 C 0 (X, R) δ C 1 (X, R) δ C 2 (X, R) δ... 0 Cb 0(X, R) δ Cb 1(X, R) δ Cb 2(X, R) δ... where C k b (X, R) = { f C k (X, R) f < } and f = sup σ Sk (X ) f (σ). Hb k (X, R) := Hk (Cb (X, R), δ),

60 Bounded cohomology Definition Let X be a topological space. 0 C 0 (X, R) δ C 1 (X, R) δ C 2 (X, R) δ... 0 Cb 0(X, R) δ Cb 1(X, R) δ Cb 2(X, R) δ... where C k b (X, R) = { f C k (X, R) f < } and f = sup σ Sk (X ) f (σ). the bounded cohomology of X. Hb k (X, R) := Hk (Cb (X, R), δ),

61 Bounded cohomology Semi-norm Hb k (X, R) := Hk (Cb (X, R), δ).

62 Bounded cohomology Semi-norm Hb k (X, R) := Hk (Cb (X, R), δ). The norm on Cb k(x, R) induces a semi-norm on Hk b (X, R):

63 Bounded cohomology Semi-norm Hb k (X, R) := Hk (Cb (X, R), δ). The norm on Cb k(x, R) induces a semi-norm on Hk b (X, R): α = inf { a [a] = α} α Hb k (X, R).

64 Bounded cohomology Semi-norm Hb k (X, R) := Hk (Cb (X, R), δ). The norm on Cb k(x, R) induces a semi-norm on Hk b (X, R): Proposition: α = inf { a [a] = α} α Hb k (X, R).

65 Bounded cohomology Semi-norm Hb k (X, R) := Hk (Cb (X, R), δ). The norm on Cb k(x, R) induces a semi-norm on Hk b (X, R): α = inf { a [a] = α} α Hb k (X, R). Proposition: Let M n be a closed connected oriented n-manifold, let α Hb n (M, R).

66 Bounded cohomology Semi-norm Hb k (X, R) := Hk (Cb (X, R), δ). The norm on Cb k(x, R) induces a semi-norm on Hk b (X, R): α = inf { a [a] = α} α Hb k (X, R). Proposition: Let M n be a closed connected oriented n-manifold, let α Hb n (M, R). Then α, [M] = α M.

67 Simplicial volume of surface bundles over surfaces Question:

68 Simplicial volume of surface bundles over surfaces Question: Let Σ h E Σ g.

69 Simplicial volume of surface bundles over surfaces Question: Let What can we say about E? Σ h E Σ g.

70 Simplicial volume of surface bundles over surfaces What is known Let Σ h E Σ g denote an orientable surface bundle over a surface with g, h 1.

71 Simplicial volume of surface bundles over surfaces What is known Let Σ h E Σ g denote an orientable surface bundle over a surface with g, h 1. Theorem (Hoster-Kotschick, 2001) E 4χ(E).

72 Simplicial volume of surface bundles over surfaces What is known Let Σ h E Σ g denote an orientable surface bundle over a surface with g, h 1. Theorem (Hoster-Kotschick, 2001) Theorem (Bucher, 2009) E 4χ(E). E 6χ(E).

73 Simplicial volume of surface bundles over surfaces What is known Let Σ h E Σ g denote an orientable surface bundle over a surface with g, h 1. Theorem (Hoster-Kotschick, 2001) Theorem (Bucher, 2009) E 4χ(E). E 6χ(E). The inequality is sharp for the trivial bundle (Bucher, 2008).

74 Simplicial volume of surface bundles over surfaces Observation:

75 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ).

76 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E).

77 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E). Proof:

78 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E). Proof: Ker(ρ) π 1 (Σ g ) is of finite index, say d.

79 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E). Proof: Ker(ρ) π 1 (Σ g ) is of finite index, say d. induces a covering f : Σ g Σ g of degree d.

80 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E). Proof: Ker(ρ) π 1 (Σ g ) is of finite index, say d. induces a covering f : Σ g Σ g of degree d. Then E Σ g

81 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E). Proof: Ker(ρ) π 1 (Σ g ) is of finite index, say d. induces a covering f : Σ g Σ g of degree d. Then f (E) Σ g d f d E Σ g

82 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E). Proof: Ker(ρ) π 1 (Σ g ) is of finite index, say d. induces a covering f : Σ g Σ g of degree d. Then Σ h Σ g f (E) d E Σ g f d Σ g

83 Simplicial volume of surface bundles over surfaces Observation: Recall the monodromy ρ : π 1 (Σ g ) Mod(Σ h ). If Im(ρ) <, E = 6χ(E). Proof: Ker(ρ) π 1 (Σ g ) is of finite index, say d. induces a covering f : Σ g Σ g of degree d. Then Σ h Σ g f (E) d E Σ g f d Σ g d E = f (E) = Σ h Σ g = 6χ(Σ h Σ g ) = 6dχ(E).

84 Simplicial volume of surface bundles over surfaces Let Question: Σ h E Σ g.

85 Simplicial volume of surface bundles over surfaces Let Question: Can we have E > 6χ(E)? Σ h E Σ g.

86 Simplicial volume of surface bundles over surfaces Let Question: Can we have E > 6χ(E)? Theorem (Bucher-C.) Yes! Σ h E Σ g.

87 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002)

88 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) E f Σ Σ Σ p Σ

89 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) E f Σ Σ Here f is a ramified covering of degree d > 1 and Σ Σ is a degree d 2g covering, where g is the genus of Σ. Σ p Σ

90 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) Strategy of proof:

91 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) Strategy of proof: E f Σ Σ Σ p Σ

92 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) Strategy of proof: E f Σ Σ Σ p Σ Notice that E f Σ Σ gives a second bundle structure to E. Σ p Σ

93 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) Strategy of proof: E f Σ Σ Σ p Σ Notice that E f Σ Σ gives a second bundle structure to E. Σ = 2 Euler classes e Σ, e Σ H 2 (E, Z), bounded. p Σ

94 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) Strategy of proof: E f Σ Σ Σ p Σ Notice that E f Σ Σ gives a second bundle structure to E. Σ = 2 Euler classes e Σ, e Σ H 2 (E, Z), bounded. Compute e Σ e Σ, [E] p Σ

95 A special family of surface bundles Kodaira (1967), Atiyah (1969), Morita (1999), Bryan-Donagi (2002) Strategy of proof: E f Σ Σ Σ p Σ Notice that E f Σ Σ gives a second bundle structure to E. Σ = 2 Euler classes e Σ, e Σ H 2 (E, Z), bounded. Compute e Σ e Σ, [E] and use the fact that e Σ e Σ 1 6. p Σ

96 Thank you for your attention.