partial hyperbolicity and topology of 3-manifolds Jana Rodriguez - - PowerPoint PPT Presentation

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partial hyperbolicity and topology of 3-manifolds Jana Rodriguez - - PowerPoint PPT Presentation

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panorama conjectures Anosov torus partial hyperbolicity and topology of 3-manifolds Jana Rodriguez Hertz Universidad de la Repblica Uruguay celebrating Mikes work May 8, 2012 panorama conjectures Anosov torus definition setting

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panorama conjectures Anosov torus

partial hyperbolicity and topology of 3-manifolds

Jana Rodriguez Hertz

Universidad de la República Uruguay

celebrating Mike’s work May 8, 2012

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panorama conjectures Anosov torus definition

setting

setting

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panorama conjectures Anosov torus definition

setting

setting

M3 closed Riemannian 3-manifold

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panorama conjectures Anosov torus definition

setting

setting

M3 closed Riemannian 3-manifold f : M → M partially hyperbolic diffeomorphism

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panorama conjectures Anosov torus definition

setting

setting

M3 closed Riemannian 3-manifold f : M → M partially hyperbolic diffeomorphism f conservative (not always, but most of the time)

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panorama conjectures Anosov torus definition

partial hyperbolicity

partial hyperbolicity

f : M3 → M3 is partially hyperbolic

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panorama conjectures Anosov torus definition

partial hyperbolicity

partial hyperbolicity

f : M3 → M3 is partially hyperbolic TM = Es ⊕ Ec ⊕ Eu ↑ ↑ ↑ contracting intermediate expanding

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SLIDE 8

panorama conjectures Anosov torus definition

partial hyperbolicity

partial hyperbolicity

f : M3 → M3 is partially hyperbolic TM = Es ⊕ Ec ⊕ Eu ↑ ↑ ↑ 1-dim 1-dim 1-dim

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panorama conjectures Anosov torus examples

example

example (conservative)

f : T2 × T1 → T2 × T1

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panorama conjectures Anosov torus examples

example

example (conservative)

f : T2 × T1 → T2 × T1 such that f = 2 1 1 1

  • × id
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panorama conjectures Anosov torus examples

example

example (conservative)

f : T2 × T1 → T2 × T1 such that f = 2 1 1 1

  • × id

×

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panorama conjectures Anosov torus examples

example

example (conservative)

f : T2 × T1 → T2 × T1 such that f = 2 1 1 1

  • × Rθ

×

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panorama conjectures Anosov torus examples

example

example (non-conservative)

f : T2 × T1 → T2 × T1 such that f = 2 1 1 1

  • × NPSP

×

xn xs

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panorama conjectures Anosov torus examples

  • pen problems

problems

ergodicity

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panorama conjectures Anosov torus examples

  • pen problems

problems

ergodicity dynamical coherence

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panorama conjectures Anosov torus examples

  • pen problems

problems

ergodicity dynamical coherence classification

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panorama conjectures Anosov torus non-ergodicity

most ph are ergodic

conjecture (pugh-shub)

partially hyperbolic diffeomorphisms ∪ C1-open and Cr-dense set of ergodic diffeomorphisms

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panorama conjectures Anosov torus non-ergodicity

most ph are ergodic

hertz-hertz-ures08

partially hyperbolic diffeomorphisms ∪ C1-open and C∞-dense set of ergodic diffeomorphisms

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panorama conjectures Anosov torus non-ergodicity

  • pen problem
  • pen problem

describe non-ergodic partially hyperbolic diffeomorphisms

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panorama conjectures Anosov torus non-ergodicity

  • pen problem
  • pen problem

describe non-ergodic partially hyperbolic diffeomorphisms

non-ergodicity

ergodic non-ergodic

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panorama conjectures Anosov torus non-ergodicity

  • pen problem
  • pen problem

describe 3-manifolds supporting non-ergodic partially hyperbolic diffeomorphisms

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panorama conjectures Anosov torus non-ergodicity

  • pen problem
  • pen problem

describe 3-manifolds supporting non-ergodic partially hyperbolic diffeomorphisms

3-manifolds

ergodic non-ergodic

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panorama conjectures Anosov torus non-ergodicity

  • pen problem
  • pen problem

describe 3-manifolds supporting non-ergodic partially hyperbolic diffeomorphisms

3-manifolds

ergodic non-ergodic ergodic

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panorama conjectures Anosov torus non-ergodicity

conjecture

subliminal conjecture

most 3-manifolds

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panorama conjectures Anosov torus non-ergodicity

conjecture

subliminal conjecture

most 3-manifolds do not support non-ergodic partially hyperbolic diffeomorphisms

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panorama conjectures Anosov torus non-ergodicity

evidence

hertz-hertz-ures08

N 3-nilmanifold,

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panorama conjectures Anosov torus non-ergodicity

evidence

hertz-hertz-ures08

N 3-nilmanifold, then either

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panorama conjectures Anosov torus non-ergodicity

evidence

hertz-hertz-ures08

N 3-nilmanifold, then either N = T3,

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panorama conjectures Anosov torus non-ergodicity

evidence

hertz-hertz-ures08

N 3-nilmanifold, then either N = T3, or {partially hyperbolic } ⊂ { ergodic }

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panorama conjectures Anosov torus non-ergodicity

evidence

hertz-hertz-ures08

N 3-nilmanifold, then either N = T3, or {partially hyperbolic } ⊂ { ergodic }

nilmanifolds

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panorama conjectures Anosov torus non-ergodicity

evidence

hertz-hertz-ures08

N 3-nilmanifold, then either N = T3, or {partially hyperbolic } ⊂ { ergodic }

nilmanifolds

ergodic

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panorama conjectures Anosov torus non-ergodicity

non-ergodic conjecture

non-ergodic conjecture (hertz-hertz-ures)

the only 3-manifolds supporting non-ergodic PH diffeomorphisms are:

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panorama conjectures Anosov torus non-ergodicity

non-ergodic conjecture

non-ergodic conjecture (hertz-hertz-ures)

the only 3-manifolds supporting non-ergodic PH diffeomorphisms are:

1

the 3-torus,

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panorama conjectures Anosov torus non-ergodicity

non-ergodic conjecture

non-ergodic conjecture (hertz-hertz-ures)

the only 3-manifolds supporting non-ergodic PH diffeomorphisms are:

1

the 3-torus,

2

the mapping torus of −id : T2 → T2

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panorama conjectures Anosov torus non-ergodicity

non-ergodic conjecture

non-ergodic conjecture (hertz-hertz-ures)

the only 3-manifolds supporting non-ergodic PH diffeomorphisms are:

1

the 3-torus,

2

the mapping torus of −id : T2 → T2

3

the mapping tori of hyperbolic automorphisms of T2

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panorama conjectures Anosov torus non-ergodicity

non-ergodic conjecture

1 the 3-torus

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panorama conjectures Anosov torus non-ergodicity

non-ergodic conjecture

2 the mapping torus of −id

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panorama conjectures Anosov torus non-ergodicity

non-ergodic conjecture

3 the mapping torus of a hyperbolic automorphism

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panorama conjectures Anosov torus non-ergodicity

stronger non-ergodic conjecture

stronger non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic diffeomorphism, then

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panorama conjectures Anosov torus non-ergodicity

stronger non-ergodic conjecture

stronger non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic diffeomorphism, then ∃ torus tangent to Es ⊕ Eu

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panorama conjectures Anosov torus dynamical coherence

integrability

integrability

f : M3 → M3 is partially hyperbolic TM = Es ⊕ Ec ⊕ Eu

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panorama conjectures Anosov torus dynamical coherence

integrability

integrability

f : M3 → M3 is partially hyperbolic TM = Es ⊕ Ec ⊕ Eu ↑ ↑ Fs Fu

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panorama conjectures Anosov torus dynamical coherence

integrability

integrability

f : M3 → M3 is partially hyperbolic TM = Es ⊕ Ec ⊕ Eu ↑ ↑ ↑ Fs ? Fu

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panorama conjectures Anosov torus dynamical coherence

dynamical coherence

dynamical coherence

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panorama conjectures Anosov torus dynamical coherence

dynamical coherence

dynamical coherence

1

∃ invariant Fcs tangent to Es ⊕ Ec

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panorama conjectures Anosov torus dynamical coherence

dynamical coherence

dynamical coherence

1

∃ invariant Fcs tangent to Es ⊕ Ec

2

∃ invariant Fcu tangent to Ec ⊕ Eu

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panorama conjectures Anosov torus dynamical coherence

dynamical coherence

dynamical coherence

1

∃ invariant Fcs tangent to Es ⊕ Ec

2

∃ invariant Fcu tangent to Ec ⊕ Eu

remark

⇒ ∃ invariant Fc tangent to Ec

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panorama conjectures Anosov torus dynamical coherence

  • pen question

longstanding open question

f : M3 → M3 partially hyperbolic

?

⇒ f dynamically coherent

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panorama conjectures Anosov torus dynamical coherence

  • pen question

longstanding open question

f : M3 → M3 partially hyperbolic

?

⇒ f dynamically coherent

hertz-hertz-ures10

NO

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panorama conjectures Anosov torus dynamical coherence

counterexample

hertz-hertz-ures10

∃f : T3 → T3 partially hyperbolic

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panorama conjectures Anosov torus dynamical coherence

counterexample

hertz-hertz-ures10

∃f : T3 → T3 partially hyperbolic non-dynamically coherent

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panorama conjectures Anosov torus dynamical coherence

counterexample

hertz-hertz-ures10

∃f : T3 → T3 partially hyperbolic non-dynamically coherent non-conservative

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panorama conjectures Anosov torus dynamical coherence

counterexample

hertz-hertz-ures10

∃f : T3 → T3 partially hyperbolic non-dynamically coherent non-conservative “robust”

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panorama conjectures Anosov torus dynamical coherence

  • pen problem
  • pen problem

describe 3-manifolds supporting non-dynamically coherent examples

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panorama conjectures Anosov torus dynamical coherence

  • pen problem
  • pen problem

describe 3-manifolds supporting non-dynamically coherent examples

3-manifolds

DC non DC

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panorama conjectures Anosov torus dynamical coherence

  • pen problem
  • pen problem

describe 3-manifolds supporting non-dynamically coherent examples

3-manifolds

DC non DC DC

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panorama conjectures Anosov torus dynamical coherence

non-dynamically coherent conjecture

non-dynamically coherent conjecture (hertz-hertz-ures)

f : M3 → M3 non-dynamically coherent,

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panorama conjectures Anosov torus dynamical coherence

non-dynamically coherent conjecture

non-dynamically coherent conjecture (hertz-hertz-ures)

f : M3 → M3 non-dynamically coherent, then M is either:

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panorama conjectures Anosov torus dynamical coherence

non-dynamically coherent conjecture

non-dynamically coherent conjecture (hertz-hertz-ures)

f : M3 → M3 non-dynamically coherent, then M is either:

1

T3

3-manifolds

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panorama conjectures Anosov torus dynamical coherence

non-dynamically coherent conjecture

non-dynamically coherent conjecture (hertz-hertz-ures)

f : M3 → M3 non-dynamically coherent, then M is either:

1

T3

2

the mapping torus of −id

3-manifolds

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panorama conjectures Anosov torus dynamical coherence

non-dynamically coherent conjecture

non-dynamically coherent conjecture (hertz-hertz-ures)

f : M3 → M3 non-dynamically coherent, then M is either:

1

T3

2

the mapping torus of −id

3

the mapping torus of a hyperbolic automorphism

3-manifolds

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panorama conjectures Anosov torus dynamical coherence

stronger conjecture

stronger non-dynamically coherent conjecture

f : M3 → M3 non-dynamically coherent, then either

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panorama conjectures Anosov torus dynamical coherence

stronger conjecture

stronger non-dynamically coherent conjecture

f : M3 → M3 non-dynamically coherent, then either ∃ torus tangent to Ec ⊕ Eu, or

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panorama conjectures Anosov torus dynamical coherence

stronger conjecture

stronger non-dynamically coherent conjecture

f : M3 → M3 non-dynamically coherent, then either ∃ torus tangent to Ec ⊕ Eu, or ∃ torus tangent to Es ⊕ Ec

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panorama conjectures Anosov torus dynamical coherence

intermediate conjecture

intermediate conjecture

f volume preserving ⇒ f dynamically coherent

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panorama conjectures Anosov torus dynamical coherence

evidence

potrie11

f : T3 → T3 non-dynamically coherent, then

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panorama conjectures Anosov torus dynamical coherence

evidence

potrie11

f : T3 → T3 non-dynamically coherent, then ∃ torus tangent to Ec ⊕ Eu, or

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panorama conjectures Anosov torus dynamical coherence

evidence

potrie11

f : T3 → T3 non-dynamically coherent, then ∃ torus tangent to Ec ⊕ Eu, or ∃ torus tangent to Es ⊕ Ec

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panorama conjectures Anosov torus classification

examples of ph dynamics

known ph dynamics in dimension 3

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panorama conjectures Anosov torus classification

examples of ph dynamics

known ph dynamics in dimension 3

perturbations of time-one maps of Anosov flows

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panorama conjectures Anosov torus classification

examples of ph dynamics

known ph dynamics in dimension 3

perturbations of time-one maps of Anosov flows certain skew-products

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panorama conjectures Anosov torus classification

examples of ph dynamics

known ph dynamics in dimension 3

perturbations of time-one maps of Anosov flows certain skew-products certain DA-maps

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panorama conjectures Anosov torus classification

examples of ph dynamics

known ph dynamics in dimension 3

perturbations of time-one maps of Anosov flows certain skew-products certain DA-maps

new example

non-dynamically coherent example

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panorama conjectures Anosov torus classification

question

question

are there more examples?

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panorama conjectures Anosov torus classification

conjecture pujals

classification conjecture (pujals01)

If f : M3 → M3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either

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panorama conjectures Anosov torus classification

conjecture pujals

classification conjecture (pujals01)

If f : M3 → M3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either

1

a perturbation of a time-one map of an Anosov flow

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panorama conjectures Anosov torus classification

conjecture pujals

classification conjecture (pujals01)

If f : M3 → M3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either

1

a perturbation of a time-one map of an Anosov flow

2

a skew-product

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panorama conjectures Anosov torus classification

conjecture pujals

classification conjecture (pujals01)

If f : M3 → M3 is a transitive partially hyperbolic diffeomorphism, then f is (finitely covered by) either

1

a perturbation of a time-one map of an Anosov flow

2

a skew-product

3

a DA-map

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panorama conjectures Anosov torus classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent, then f is

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SLIDE 80

panorama conjectures Anosov torus classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent, then f is

1

a perturbation of a time-one map of an Anosov flow,

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panorama conjectures Anosov torus classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent, then f is

1

a perturbation of a time-one map of an Anosov flow,

2

a skew-product, or

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SLIDE 82

panorama conjectures Anosov torus classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent, then f is

1

a perturbation of a time-one map of an Anosov flow,

2

a skew-product, or

3

a DA-map.

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SLIDE 83

panorama conjectures Anosov torus classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent, then f is

1

leafwise conjugate to an Anosov flow

2

a skew-product, or

3

a DA-map.

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SLIDE 84

panorama conjectures Anosov torus classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent, then f is

1

leafwise conjugate to an Anosov flow

2

leafwise conjugate to a skew-product with linear base

3

a DA-map.

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SLIDE 85

panorama conjectures Anosov torus classification

conjecture

classification conjecture (hhu)

If f : M3 → M3 is partially hyperbolic and dynamically coherent, then f is

1

leafwise conjugate to an Anosov flow

2

leafwise conjugate to a skew-product with linear base

3

leafwise conjugate to an Anosov map in T3.

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panorama conjectures Anosov torus Anosov torus

problems

ergodicity

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panorama conjectures Anosov torus Anosov torus

problems

ergodicity dynamical coherence

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panorama conjectures Anosov torus Anosov torus

problems

ergodicity dynamical coherence classification

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panorama conjectures Anosov torus Anosov torus

problems

ergodicity dynamical coherence classification                        → Anosov torus

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panorama conjectures Anosov torus Anosov torus

Anosov torus

Anosov torus

T embedded 2-torus

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panorama conjectures Anosov torus Anosov torus

Anosov torus

Anosov torus

T embedded 2-torus ∃f : M → M s.t.

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panorama conjectures Anosov torus Anosov torus

Anosov torus

Anosov torus

T embedded 2-torus ∃f : M → M s.t.

1

f(T) = T

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panorama conjectures Anosov torus Anosov torus

Anosov torus

Anosov torus

T embedded 2-torus ∃f : M → M s.t.

1

f(T) = T

2

f|T isotopic to Anosov

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panorama conjectures Anosov torus Anosov torus

invariant tori in ph dynamics

invariant tori in PH dynamics

T invariant torus tangent to

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panorama conjectures Anosov torus Anosov torus

invariant tori in ph dynamics

invariant tori in PH dynamics

T invariant torus tangent to Es ⊕ Eu

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panorama conjectures Anosov torus Anosov torus

invariant tori in ph dynamics

invariant tori in PH dynamics

T invariant torus tangent to Es ⊕ Eu Ec ⊕ Eu

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panorama conjectures Anosov torus Anosov torus

invariant tori in ph dynamics

invariant tori in PH dynamics

T invariant torus tangent to Es ⊕ Eu Ec ⊕ Eu Es ⊕ Eu

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panorama conjectures Anosov torus Anosov torus

invariant tori in ph dynamics

invariant tori in PH dynamics

T invariant torus tangent to Es ⊕ Eu Ec ⊕ Eu Es ⊕ Eu ⇒ T Anosov torus

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panorama conjectures Anosov torus Anosov torus

conjectures

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panorama conjectures Anosov torus Anosov torus

conjectures

non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic

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panorama conjectures Anosov torus Anosov torus

conjectures

non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic

non-dyn. coh. conjecture

f : M → M non-dyn. coherent partially hyperbolic

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panorama conjectures Anosov torus Anosov torus

conjectures

non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic

non-dyn. coh. conjecture

f : M → M non-dyn. coherent partially hyperbolic

then M is either

1

T3

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panorama conjectures Anosov torus Anosov torus

conjectures

non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic

non-dyn. coh. conjecture

f : M → M non-dyn. coherent partially hyperbolic

then M is either

1

T3

2

the mapping torus of −id : T2 → T2

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SLIDE 104

panorama conjectures Anosov torus Anosov torus

conjectures

non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic

non-dyn. coh. conjecture

f : M → M non-dyn. coherent partially hyperbolic

then M is either

1

T3

2

the mapping torus of −id : T2 → T2

3

the mapping torus of a hyperbolic map of T2

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SLIDE 105

panorama conjectures Anosov torus Anosov torus

conjectures

non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic

non-dyn. coh. conjecture

f : M → M non-dyn. coherent partially hyperbolic

then M is either

1

T3

2

the mapping torus of −id : T2 → T2

3

the mapping torus of a hyperbolic map of T2

stronger conjecture

∃ Anosov torus tangent to Es ⊕ Eu

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panorama conjectures Anosov torus Anosov torus

conjectures

non-ergodic conjecture

f : M → M non-ergodic partially hyperbolic

non-dyn. coh. conjecture

f : M → M non-dyn. coherent partially hyperbolic

then M is either

1

T3

2

the mapping torus of −id : T2 → T2

3

the mapping torus of a hyperbolic map of T2

stronger conjecture

∃ Anosov torus tangent to Es ⊕ Eu

stronger conjecture

∃ Anosov torus tangent to Ec ⊕ Eu or Es ⊕ Ec

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SLIDE 107

panorama conjectures Anosov torus Anosov torus

why stronger conjectures?

hertz-hertz-ures11

M irreducible contains an Anosov torus,

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SLIDE 108

panorama conjectures Anosov torus Anosov torus

why stronger conjectures?

hertz-hertz-ures11

M irreducible contains an Anosov torus, then M is either

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SLIDE 109

panorama conjectures Anosov torus Anosov torus

why stronger conjectures?

hertz-hertz-ures11

M irreducible contains an Anosov torus, then M is either

1

T3

3-manifolds

slide-110
SLIDE 110

panorama conjectures Anosov torus Anosov torus

why stronger conjectures?

hertz-hertz-ures11

M irreducible contains an Anosov torus, then M is either

1

T3

2

the mapping torus of −id : T2 → T2

3-manifolds

slide-111
SLIDE 111

panorama conjectures Anosov torus Anosov torus

why stronger conjectures?

hertz-hertz-ures11

M irreducible contains an Anosov torus, then M is either

1

T3

2

the mapping torus of −id : T2 → T2

3

a mapping torus of a hyperbolic automorphism of T2

3-manifolds

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SLIDE 112

panorama conjectures Anosov torus Anosov torus

remark

remark

f : M3 → M3 partially hyperbolic ⇒ M irreducible

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panorama conjectures Anosov torus Anosov torus

remark

remark

f : M3 → M3 partially hyperbolic ⇒ M irreducible

why

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SLIDE 114

panorama conjectures Anosov torus Anosov torus

remark

remark

f : M3 → M3 partially hyperbolic ⇒ M irreducible

why

Rosenberg68

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panorama conjectures Anosov torus Anosov torus

remark

remark

f : M3 → M3 partially hyperbolic ⇒ M irreducible

why

Rosenberg68 Burago-Ivanov08

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panorama conjectures Anosov torus Anosov torus

reduced theorem

reduced theorem

N3 irreducible manifold with boundary

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panorama conjectures Anosov torus Anosov torus

reduced theorem

reduced theorem

N3 irreducible manifold with boundary ∂N consists of Anosov tori

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panorama conjectures Anosov torus Anosov torus

reduced theorem

reduced theorem

N3 irreducible manifold with boundary ∂N consists of Anosov tori ⇒ N = T2 × [0, 1]

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

JSJ-decomposition

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

JSJ-decomposition

N3 in our hypotheses

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

JSJ-decomposition

N3 in our hypotheses ∃ T1, . . . , Tn incompressible tori

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SLIDE 122

panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

JSJ-decomposition

N3 in our hypotheses ∃ T1, . . . , Tn incompressible tori such that each component of M \ T1 ∪ · · · ∪ Tn

1

atoroidal, or

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SLIDE 123

panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

JSJ-decomposition

N3 in our hypotheses ∃ T1, . . . , Tn incompressible tori such that each component of M \ T1 ∪ · · · ∪ Tn

1

atoroidal, or

2

Seifert manifold

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SLIDE 124

panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

JSJ-decomposition

N3 in our hypotheses ∃ T1, . . . , Tn incompressible tori such that each component of M \ T1 ∪ · · · ∪ Tn

1

atoroidal, or

2

Seifert manifold

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

remark

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

remark

N Seifert manifold

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

remark

N Seifert manifold ∂N are Anosov tori

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

remark

N Seifert manifold ∂N are Anosov tori ⇒ ∃ 2 Seifert fibrations non-isotopic on ∂N

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

classic lemma

∃ 2 Seifert fibrations non-isotopic on ∂N

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

classic lemma

∃ 2 Seifert fibrations non-isotopic on ∂N ⇒ N is

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

classic lemma

∃ 2 Seifert fibrations non-isotopic on ∂N ⇒ N is

1

D2 × S1 the solid torus

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

classic lemma

∃ 2 Seifert fibrations non-isotopic on ∂N ⇒ N is

1

D2 × S1 the solid torus

2

S1 × S1 × [0, 1] the twisted I-bundle over the Klein bundle

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

classic lemma

∃ 2 Seifert fibrations non-isotopic on ∂N ⇒ N is

1

D2 × S1 the solid torus

2

S1 × S1 × [0, 1] the twisted I-bundle over the Klein bundle

3

T2 × [0, 1] the torus cross the interval

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

1 N = solid torus

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

1 N = solid torus

∂N = T

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

2 NS = twisted I-bundle over K

c Ken Baker

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

2 NS = twisted I-bundle over K

c Ken Baker ∂N = T

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

3 N = T × [0, 1]

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

3 N = T × [0, 1]

∂N = T ⊔ T

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

final lemma

∂N consist of Anosov tori

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

why

i : H1(∂N) ֒ → H1(N)

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

why

i : H1(∂N) ֒ → H1(N) rank(ker(i)) = 1

2 rank(H1(∂N))

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panorama conjectures Anosov torus Anosov torus

brief sketch of the proof

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

why

i : H1(∂N) ֒ → H1(N) rank(ker(i)) = 1

2 rank(H1(∂N))

rank

rank=“dimension"

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panorama conjectures Anosov torus Anosov torus

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

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panorama conjectures Anosov torus Anosov torus

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

possibilities

∂N = T

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panorama conjectures Anosov torus Anosov torus

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

possibilities

∂N = T ∂N = T

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panorama conjectures Anosov torus Anosov torus

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

possibilities

∂N = T ∂N = T ∂N = T ⊔ T

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panorama conjectures Anosov torus Anosov torus

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

possibilities

∂N = T ⊔ T

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panorama conjectures Anosov torus Anosov torus

final lemma

∂N consist of Anosov tori ⇒ ∂N = T

possibilities

∂N = T ⊔ T √

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panorama conjectures Anosov torus Anosov torus

thank you

thank you

THANK YOU MIKE!