dehn surgery problem

8/11/2019 dehn surgery problem

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The Dehn Surgery Problem

Marc Lackenby

19 June 2012
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Dehn surgery

This is a method for building3-manifolds:
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Dehn surgery


Start with a knot or link K in 3-sphere.
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Dehn surgery



Remove an open regular neighbourhoodof K , creating a 3-manifold M withboundary.
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Dehn surgery




Re-attach solid tori to M , but in adifferent way.
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Dehn surgery




Re-attach solid tori to M , but in adifferent way.

The resulting manifold is obtained byDehn surgery on K .
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Slopes

The essentially different ways of doingthis are parametrised by the slopes on N (K ).
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Slopes


Each solid torus S 1 D 2 has a meridiandisc D 2.
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Slopes


Each solid torus S 1 D 2 has a meridiandisc D 2.Its boundary is attached to a simpleclosed curve on N (K ).
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Slopes



An isotopy class of essential simpleclosed curves on N (K ) is a slope.


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Slopes



An isotopy class of essential simpleclosed curves on N (K ) is a slope.

The slopes on each component of N (K ) are parametrised by fractionsp / q Q{}.

-1
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Lickorishs and Kirbys theorems

Theorem: [Lickorish, Wallace] Any closed orientable 3-manifold isobtained by Dehn surgery on some link in S 3.


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Theorem: [Kirby] Any two surgery descriptions of a 3-manifolddiffer by a sequence of Kirby moves.
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Theorem: [Kirby] Any two surgery descriptions of a 3-manifolddiffer by a sequence of Kirby moves.


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Dehn lling

LetM = a compact orientable 3-manifold, with M = a collection of toris 1, . . . , s n = a collection slopes on M , one on each component.

ThenM (s 1, . . . , s n ) = the manifold obtained by attaching solid tori to M along the slopes s 1, . . . , s n .

It is obtained from M by Dehn lling.

General theme: Properties of M should be inherited byM (s 1, . . . , s n ), for generic slopes s 1, . . . , s n .
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Geometrisation

A major revolution in 3-manifold theory was the formulation (byThurston) and proof (by Perelman) of:

Geometrisation Conjecture: Any closed orientable 3-manifold has acanonical decomposition into geometric pieces.
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Geometrisation

A major revolution in 3-manifold theory was the formulation (byThurston) and proof (by Perelman) of:

Geometrisation Conjecture: Any closed orientable 3-manifold has acanonical decomposition into geometric pieces.

Of these geometries, by the far the most important is hyperbolicgeometry.
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Hyperbolic structures

M is hyperbolic if M M = H 3/ for some discrete group of isometries acting freely on H

3.

Example: Seifert-Weber dodecahedral space
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Hyperbolic structures

M is hyperbolic if M M = H 3/ for some discrete group of isometries acting freely on H

3.

Example: Seifert-Weber dodecahedral space
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The gure-eight knot

h h k
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The gure-eight knot

S 3 K =
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We can realise these as (ideal) tetrahedra in H3:

This induces a hyperbolic structure on S 3 K .

H b li t t k t l t
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Hyperbolic structures on knot complements

Theorem: [Thurston] Let K be a knot in S 3. Then S 3 K admitsa hyperbolic structure if and only if K is not a torus knot or asatellite knot:

satellite knot torus knot

H b li D h g
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Hyperbolic Dehn surgery

Suppose that M has a hyperbolic structure, and that M is asingle torus.

Theorem: [Thurston] For slopes s on M , with at most nitely

many exceptions, M (s ) is also hyperbolic.

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many exceptions, M (s ) is also hyperbolic.The slopes s for which M (s ) is not hyperbolic are calledexceptional.

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many exceptions, M (s ) is also hyperbolic.The slopes s for which M (s ) is not hyperbolic are calledexceptional.

The Dehn Surgery Problem: How many exceptional slopes can

there be?

The gure eight knot
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The gure-eight knot

The gure eight knot
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The gure-eight knot

Let M = the exterior of the gure-eight knot.

Theorem: [Thurston] M has 10 exceptional slopes:

{4, 3, 2, 1, 0, 1, 2, 3, 4, }.

The gure-eight knot
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The gure-eight knot

Let M = the exterior of the gure-eight knot.

Theorem: [Thurston] M has 10 exceptional slopes:

{4, 3, 2, 1, 0, 1, 2, 3, 4, }.

Conjecture: [Gordon] For any hyperbolic 3-manifold M with M asingle torus, M has at most 10 exceptional slopes. Moreover, thegure-eight knot exterior is the unique manifold with precisely 10.

The maximal number
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The maximal number

Theorem: [L-Meyerhoff] For any hyperbolic 3-manifold M with M a single torus, M has at most 10 exceptional slopes.

The maximal number
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The maximal number

Theorem: [L-Meyerhoff] For any hyperbolic 3-manifold M with M a single torus, M has at most 10 exceptional slopes.

But we still dont know whether the gure-eight knot exterior isthe unique such manifold.

History
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History

This problem has received a lot of attention ...Certainly > 200 papers

History
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History

This problem has received a lot of attention ...Certainly > 200 papersM. Culler, C. Gordon, J. Luecke, P. Shalen, Dehn surgery on knots , Ann. of Math. (2) 125 (1987) 237300.D. Gabai, Foliations and the topology of 3-manifolds. III. J. Differential Geom.26 (1987) 479536.S. Boyer, X. Zhang, Finite Dehn surgery on knots. J. Amer. Math. Soc. 9(1996) 10051050S. Boyer, X. Zhang, On Culler-Shalen seminorms and Dehn lling, Ann. of Math. (2) 148 (1998) 737801.M. Lackenby, Word hyperbolic Dehn surgery , Invent. math. 140 (2000)243282.I. Agol, Bounds on exceptional Dehn lling , Geom. Topol. 4 (2000) 431449.

S. Boyer, X. Zhang, A proof of the nite lling conjecture, J. DifferentialGeom. 59 (2001) 87176.C. Hodgson, S. Kerckhoff, Universal bounds for hyperbolic Dehn surgery. Ann.of Math. (2) 162 (2005) 367421.I. Agol, Bounds on exceptional Dehn lling II. Geom. Topol. 14 (2010)19211940.

Some approaches
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pp

Approach 1: Quantify Thurstons arguments.

Some approaches
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pp


Very difficult.

Some approaches
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pp


Very difficult.

Theorem: [Hodgson-Kerckhoff, Annals 2006] There are at most 60exceptional slopes.This was the rst universal bound (ie independent of M ) on thenumber of exceptional slopes.

They used the theory of cone manifolds.

Topological methods
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p g

Perelmans proof of geometrisation A closed orientable 3-manifold is hyperbolic if and only if it isirreducible, atoroidal, not Seifert bred.

Topological methods


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Perelmans proof of geometrisation A closed orientable 3-manifold is hyperbolic if and only if it isirreducible, atoroidal, not Seifert bred.

Irreducible: any embedded 2-sphere bounds a 3-ball.

Atoroidal: no 1-injective embedded torus.

Seifert bred: is foliated by circles.

Topological methods


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Approach 2: Bound the intersection number of slopes s 1 and s 2where M (s 1) and M (s 2) are reducible/toroidal/Seifert bred.

Topological methods
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For example . . .

Theorem: [Gordon-Luecke, Topology 1997] If M (s 1) and M (s 2) arereducible, then s 1 and s 2 have intersection number 1. Hence, M (s )can be reducible for at most 3 slopes s .

Topological methods
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For example . . .

Theorem: [Gordon-Luecke, Topology 1997] If M (s 1) and M (s 2) arereducible, then s 1 and s 2 have intersection number 1. Hence, M (s )can be reducible for at most 3 slopes s .

But analysing when M (s 1) and M (s 2) are both Seifert bred is

hard.

Negatively curved metrics
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Approach 3: Construct a negatively curved metric on M (s ).

Theorem: [Gromov-Thurston] For all slopes s on M with at most48 exceptions, M (s ) admits a negatively curved Riemannian metric.

Perelman M (s ) then admits a hyperbolic structure.

The ends of a hyperbolic 3-manifold
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Suppose M is a non-empty collection of tori.

The ends of M M are of the form M [1, ).Geometrically, they are obtained from {(x , y , z ) : z 1} inupper-half space, quotiented out by the action of Z Z acting byparabolic isometries.

H 3

{( x , y , z ) : z 1}

The ends of a hyperbolic 3-manifold
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Suppose M is a non-empty collection of tori.

The ends of M M are of the form M [1, ).Geometrically, they are obtained from {(x , y , z ) : z 1} inupper-half space, quotiented out by the action of Z Z acting byparabolic isometries.

H 3

{( x , y , z ) : z 1}

This is a called a horoball neighbourhood of the end.

Example: the gure-8 knot
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Boundary of N

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H3

{( x , y , z ) : z 1}

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Horoball diagrams
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Looking down from innity in hyperbolic space, one gets a view of horoballs:

Horoball diagrams


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Looking down from innity in hyperbolic space, one gets a view of horoballs:

This is actually from the hyperbolic structure on the Borromeanrings.

It was produced by the computer program Snappea.

The length of a slope
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Let N = a maximal horoball neighbourhood of the end.

N is a Euclidean torus.Each slope s on N is realised by a geodesic in N .

The length L(s ) of s is the length of this geodesic.

The length of a slope
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Let N = a maximal horoball neighbourhood of the end.

N is a Euclidean torus.Each slope s on N is realised by a geodesic in N .

The length L(s ) of s is the length of this geodesic.

One can visualise this as the translation length of thecorresponding parabolic acting on {(x , y , z ) : z = 1}.

H 3

{( x , y , z ) : z 1}

The 2 theorem
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Theorem: [Gromov-Thurston] If L(s ) > 2, then M (s ) admits anegatively curved Riemannian metric.

The proof is very clever, but surprisingly straightforward!

The 2 theorem
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Theorem: [Gromov-Thurston] If L(s ) > 2, then M (s ) admits anegatively curved Riemannian metric.

The proof is very clever, but surprisingly straightforward!So, to bound the number of exceptional slopes, all we need is . . .

Theorem: [Gromov-Thurston] At most 48 slopes s have L(s ) 2.

Bounding slope length
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Lemma: [Thurston] Each slope has length 1.Proof:

Let N be the inverse image of N in H3.

We may arrange that one component of N is {(x , y , z ) : z 1}.Since N is maximal, another component of N touches it.

Then parabolic translations much translate by at least one,otherwise the interior of two components of N would intersect.

H 3

{( x , y , z ) : z 1}

The number of slopes with length at most 2
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From Thurstons Lemma, it is clear that there is an upper boundon the number of slopes with length at most 2.

1

2

Intersection numbers


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For two slopes s 1 and s 2, let ( s 1, s 2) denote the modulus of theirintersection number.

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Lemma: For two slopes s 1 and s 2,

( s 1, s 2) L(s 1) L(s 2)

Area ( N ) .

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( s 1, s 2) L(s 1) L(s 2)Area ( N ) .

s 1

s 2

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( s 1, s 2) L(s 1) L(s 2)Area ( N ) .

In our case,( s

1, s

2)

(2 )2

3/ 2 < 46.

s 1

s 2

The number of slopes with length at most 2
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Lemma: [Agol] Let S be a collection of slopes, such that any twohave intersection number at most . Let p be any prime more

than . Then |S

| p

+ 1.Setting p = 47 gives our bound.

How to improve the bound?
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Recall

( s 1, s 2) L(s 1) L(s 2)

Area ( N ) .

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Recall

( s 1, s 2) L(s 1) L(s 2)

Area ( N ) .

1. Increase the lower bound on A = Area

( N ).2. Decrease the critical slope length below 2.



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Recall

( s 1, s 2) L(s 1) L(s 2)

Area ( N ) .



[Adams, 1987] A 31.732



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Recall

( s 1, s 2) L(s 1) L(s 2)

Area ( N ) .



[Adams, 1987] A 31.732[Cao-Meyerhoff, 2001] A

3.35

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Recall

( s 1, s 2) L(s 1) L(s 2)

Area ( N ) .



[Adams, 1987] A 31.732[Cao-Meyerhoff, 2001] A

3.35[Gabai-Meyerhoff-Milley, 2009] A 3.7 (under some hypotheses)

The 6 theorem
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Theorem: [L, Agol] If L(s ) > 6, then M (s ) is irreducible, atoroidal,not Seifert bred and has innite, word hyperbolic fundamentalgroup.

Perelman M (s ) is then hyperbolic.So, the critical slope length is reduced from 2 to 6.

Proving the 6 theorem
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Suppose that M (s ) is reducible, and so contains a 2-sphere S thatdoesnt bound a ball.

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We may arrange that S intersects the surgery solid torus inmeridian discs.

Then S M is a planar surface P , with each boundary componenthaving slope s .

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Then S M is a planar surface P , with each boundary componenthaving slope s .With some pushing and pulling of S , we may ensure that P isessential (which means 1-injective and a boundary version of this).

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Then S

M

is a planar surface P

, with each boundary componenthaving slope s .

With some pushing and pulling of S , we may ensure that P isessential (which means 1-injective and a boundary version of this).

Then P can isotoped to a minimal surface in M .

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For any minimal surface, its intrinsic curvature is at most thecurvature of the ambient space. So, 1.



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For any minimal surface, its intrinsic curvature is at most thecurvature of the ambient space. So, 1.Gauss-Bonnet:

2 (P ) = P P 1 = Area (P ).



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2 (P ) = P P 1 = Area (P ).ie.

Area (P ) 2(| P | 2).



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2 (P ) = P P 1 = Area (P ).ie.

Area (P ) 2(| P | 2).If we can show that each boundary component of P contributes atleast 2 to the area of P , well have a contradiction.


Consider the inverse image of P in H3.


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Part of this lies in the horoball

{(x , y , z ) : z

1

}.




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{(x , y , z ) : z

1

}.

H 3

Area L( s )




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{(x , y , z ) : z

1

}.

H 3

Area L( s )

So, each boundary component of P contributes at least L(s ) to thearea of P .


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{(x , y , z ) : z

1

}.

H 3

Area L( s )

So, each boundary component of P contributes at least L(s ) to thearea of P .

So, if L(s ) > 2, we have a contradiction.


We havent used all the area of P !
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We haven t used all the area of P !

Geodesic spine S

Boundary of N


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Geodesic spine S

Boundary of N

In fact, each component of P contributes at least (2 / 6) L(s ) tothe area of P .

So, if L(s ) > 6, then M (s ) is irreducible.


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Geodesic spine S

Boundary of N

In fact, each component of P contributes at least (2 / 6) L(s ) tothe area of P .

So, if L(s ) > 6, then M (s ) is irreducible.

The other conclusions (atoroidality etc) are proved similarly.
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Down to 12 exceptional slopes


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Using that cusp area is at least 3.35 [Cao-Meyerhoff] and thatcritical slope length is 6, we proved:

Theorem: [Agol, L 2000] For any hyperbolic 3-manifold M with M a single torus, M has at most 12 exceptional slopes.

But reducing 12 down to 10 is very hard . . .

A difficulty

Unfortunately, there is an example, due to Agol, of a manifold M t
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s.t.

1. It has an exceptional slope s with length 6.
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Work of Gabai, Meyerhoff and Milley


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Fortunately, this manifold falls into a well-understood family of

exceptions.Theorem: [Gabai, Meyerhoff, Milley] The cusp area of M is at least3.7 unless M is obtained by Dehn lling some of the boundarycomponents of one of the manifolds

m412 s 596 s 647 s 774 s 776 s 780 s 785 s 898 s 959

The notation is from the census of hyperbolic 3-manifolds.



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Fortunately, this manifold falls into a well-understood family of

exceptions.Theorem: [Gabai, Meyerhoff, Milley] The cusp area of M is at least3.7 unless M is obtained by Dehn lling some of the boundarycomponents of one of the manifolds

m412 s 596 s 647 s 774 s 776 s 780 s 785 s 898 s 959

The notation is from the census of hyperbolic 3-manifolds.

Using this and the 6 theorem, you can get a bound of 11exceptional surgeries.

Work of Cao and Meyerhoff They consider the horoball diagram of M :
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They let e 2 1 be the (Euclidean diameter)1/ 2 of the secondlargest horoballs.If e

2 is close to 1, then we get lots of large horoballs and so large

cusp area.

Work of Cao and Meyerhoff They consider the horoball diagram of M :


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They let e 2 1 be the (Euclidean diameter)1/ 2 of the secondlargest horoballs.If e

2 is close to 1, then we get lots of large horoballs and so large

cusp area.

If e 2 is large, then the largest horoballs must be well-separated.

A computer calculation, ranging over all values of e 2, gives thatA

3 35

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They consider the second, third and fourth largest horoballs!

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They consider the second, third and fourth largest horoballs!One can deduce that A 3.7 unless certain congurations of horoballs arise.

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They consider the second, third and fourth largest horoballs!One can deduce that A 3.7 unless certain congurations of horoballs arise.In these cases, one can show that M is obtained by Dehn llingsome of the boundary components of one of the manifolds

m412 s 596 s 647 s 774 s 776 s 780 s 785 s 898 s 959

How to solve the Dehn surgery problem

Improve the 6 theorem:
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Improve the 6 theorem:

Theorem: Suppose that

L(s ) > e 2

arcsin(e 2/ 2)

if e 2 2, and thatL(s ) >

2 1 e 22arcsin( 1 e 22 )

if e 2 > 2. Then, M (s ) is hyperbolic.

h1

2

e2-1

e2-1

e2-2

1- h2 2

2(1/e ) - h

B1

P 1

B2

{ z=h }

B

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Repeat the analysis of Gabai, Milley and Meyerhoff, but specicallysearch for the number of slopes with length less than the givenbound.

Must also use a range of geometric arguments to rule out certainhoroball congurations.

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Repeat the analysis of Gabai, Milley and Meyerhoff, but specicallysearch for the number of slopes with length less than the givenbound.

Must also use a range of geometric arguments to rule out certainhoroball congurations.The computer calculation takes about 2 days.

It only just works!

Can we trust the computer?
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Yes!

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Yes!The computer code can be checked by hand, much like checking aproof.

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The possibility that errors arise due to the use of oating pointarithmetic can be ruled out. One replaces each real number by aninterval, which is effectively the number with an error bar.

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The possibility that errors arise due to the use of oating pointarithmetic can be ruled out. One replaces each real number by aninterval, which is effectively the number with an error bar.

Use is also made of the computer program Snap which is a versionof Snappea that uses exact arithmetic.
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dehn surgery problem

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