Casson towers and filtrations of the smooth knot concordance group - - PowerPoint PPT Presentation

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Casson towers and filtrations of the smooth knot concordance group - - PowerPoint PPT Presentation

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Casson towers and filtrations of the smooth knot concordance group Arunima Ray AMS Central Sectional Meeting Washington University at St. Louis St. Louis, Missouri October 19, 2013 Introduction Goal Casson towers Results Definitions

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Casson towers and filtrations

  • f the smooth knot concordance group

Arunima Ray

AMS Central Sectional Meeting Washington University at St. Louis

  • St. Louis, Missouri

October 19, 2013

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Introduction Goal Casson towers Results

Definitions

Definition A knot is slice if it bounds a smoothly embedded disk ∆ in B4. K S3 B4 ∆ Knots, modulo slice knots, form the smooth knot concordance group, denoted C.

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Introduction Goal Casson towers Results

Definitions

Definition A knot is slice if it bounds a smoothly embedded disk ∆ in B4. K S3 B4 ∆ Knots, modulo slice knots, form the smooth knot concordance group, denoted C. There exist infinitely many smooth concordance classes of topologically slice knots (Endo, Gompf, Hedden–Kirk, Hedden–Livingston–Ruberman, Hom, etc.)

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Introduction Goal Casson towers Results

Approximating sliceness

K S3 B4 ∆ A knot is slice if it bounds a disk in B4.

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Introduction Goal Casson towers Results

Approximating sliceness

K S3 B4 ∆ A knot is slice if it bounds a disk in B4. Two ways to approximate sliceness:

  • knots which bound disks in [[approximations of B4]]
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Introduction Goal Casson towers Results

Approximating sliceness

K S3 B4 ∆ A knot is slice if it bounds a disk in B4. Two ways to approximate sliceness:

  • knots which bound disks in [[approximations of B4]]
  • knots which bound [[approximations of disks]] in B4
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Introduction Goal Casson towers Results

The n–solvable filtration of C

Definition (Cochran–Orr–Teichner, 2003) For any n ≥ 0, a knot K is in Fn (and is said to be n–solvable) if K bounds a smooth, embedded disk ∆ in [[an approximation of B4]]

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Introduction Goal Casson towers Results

The n–solvable filtration of C

Definition (Cochran–Orr–Teichner, 2003) For any n ≥ 0, a knot K is in Fn (and is said to be n–solvable) if K bounds a smooth, embedded disk ∆ in a smooth, compact,

  • riented 4–manifold V with ∂V = S3 such that
  • H1(V ) = 0,
  • there exist surfaces {L1, D1, L2, D2, · · · , Lk, Dk} embedded

in V − ∆ which generate H2(V ) and with respect to which the intersection form is 0 1 1

  • ,
  • π1(Li) ⊆ π1(V − ∆)(n) for all i,
  • π1(Di) ⊆ π1(V − ∆)(n) for all i.
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Introduction Goal Casson towers Results

The n–solvable filtration of C

Definition (Cochran–Orr–Teichner, 2003) For any n ≥ 0, a knot K is in Fn (and is said to be n–solvable) if K bounds a smooth, embedded disk ∆ in a smooth, compact,

  • riented 4–manifold V with ∂V = S3 such that
  • H1(V ) = 0,
  • there exist surfaces {L1, D1, L2, D2, · · · , Lk, Dk} embedded

in V − ∆ which generate H2(V ) and with respect to which the intersection form is 0 1 1

  • ,
  • π1(Li) ⊆ π1(V − ∆)(n) for all i,
  • π1(Di) ⊆ π1(V − ∆)(n) for all i.

Clearly, · · · ⊆ Fn ⊆ Fn−1 ⊆ · · · ⊆ F0 ⊆ C

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Introduction Goal Casson towers Results

The n–solvable filtration of C

  • F0 = {K | Arf(K) = 0}
  • F1 ⊆ {K | K is algebraically slice}
  • F2 ⊆ {K | various Casson–Gordon obstructions vanish}
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Introduction Goal Casson towers Results

The n–solvable filtration of C

  • F0 = {K | Arf(K) = 0}
  • F1 ⊆ {K | K is algebraically slice}
  • F2 ⊆ {K | various Casson–Gordon obstructions vanish}
  • ∀n, Z∞ ⊆ Fn/Fn+1
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Introduction Goal Casson towers Results

The grope filtration of C

Definition For any n ≥ 1, a knot K is in Gn if K bounds a grope of height n in B4.

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Introduction Goal Casson towers Results

The grope filtration of C

Definition For any n ≥ 1, a knot K is in Gn if K bounds a grope of height n in B4. K

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Introduction Goal Casson towers Results

The grope filtration of C

Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0, Gn+2 ⊆ Fn

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Introduction Goal Casson towers Results

Topologically slice knots

Let T denote the set of all topologically slice knots. T ⊆

∞

  • n=0

Fn

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Introduction Goal Casson towers Results

Topologically slice knots

Let T denote the set of all topologically slice knots. T ⊆

∞

  • n=0

Fn How can we use filtrations to study smooth concordance classes of topologically slice knots?

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Introduction Goal Casson towers Results

Positive and negative filtrations of C

Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0, a knot K is in Pn (and is said to be n–positive) if K bounds a smooth, embedded disk ∆ in [[an approximation of B4]]

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Introduction Goal Casson towers Results

Positive and negative filtrations of C

Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0, a knot K is in Pn (and is said to be n–positive) if K bounds a smooth, embedded disk ∆ in a smooth, compact,

  • riented 4–manifold V with ∂V = S3 such that
  • π1(V ) = 0,
  • there exist surfaces {Si} embedded in V − ∆ which generate

H2(V ) and with respect to which the intersection form is 1

  • ,
  • π1(Si) ⊆ π1(V − ∆)(n) for all i,
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Introduction Goal Casson towers Results

Positive and negative filtrations of C

Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0, a knot K is in Nn (and is said to be n–negative) if K bounds a smooth, embedded disk ∆ in a smooth, compact,

  • riented 4–manifold V with ∂V = S3 such that
  • π1(V ) = 0,
  • there exist surfaces {Si} embedded in V − ∆ which generate

H2(V ) and with respect to which the intersection form is −1

  • ,
  • π1(Si) ⊆ π1(V − ∆)(n) for all i,
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Introduction Goal Casson towers Results

Positive and negative filtrations of C

Definition (Cochran–Harvey–Horn, 2012) For any n ≥ 0, a knot K is in Nn (and is said to be n–negative) if K bounds a smooth, embedded disk ∆ in a smooth, compact,

  • riented 4–manifold V with ∂V = S3 such that
  • π1(V ) = 0,
  • there exist surfaces {Si} embedded in V − ∆ which generate

H2(V ) and with respect to which the intersection form is −1

  • ,
  • π1(Si) ⊆ π1(V − ∆)(n) for all i,

These filtrations can be used to distinguish smooth concordance classes of topologically slice knots

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Introduction Goal Casson towers Results

Goal

Prove a version of the result relating the grope filtration and n–solvable filtration, for the positive/negative filtrations

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Introduction Goal Casson towers Results

Casson towers

Any knot bounds a kinky disk in B4, i.e. a disk with transverse self-intersections.

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Introduction Goal Casson towers Results

Casson towers

Any knot bounds a kinky disk in B4, i.e. a disk with transverse self-intersections. Any knot which bounds such a kinky disk with only positive self-intersections lies in P0.

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Introduction Goal Casson towers Results

Casson towers

Any knot bounds a kinky disk in B4, i.e. a disk with transverse self-intersections. Any knot which bounds such a kinky disk with only positive self-intersections lies in P0. A Casson tower is built using layers of kinky disks, so they are natural objects to study in this context.

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Introduction Goal Casson towers Results

Casson towers

K

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Introduction Goal Casson towers Results

Casson towers

K

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Introduction Goal Casson towers Results

Casson towers

K

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Introduction Goal Casson towers Results

Casson towers

K A Casson tower of height n consists of n layers of kinky disks.

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Introduction Goal Casson towers Results

Casson towers

K A Casson tower of height n consists of n layers of kinky disks. A Casson tower T is of height (2, n) if it has two layers of kinky disks, and each member of a standard set of generators

  • f π1(T) is in

π1(B4 − T)(n).

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Introduction Goal Casson towers Results

Casson towers

Definition (R.)

  • A knot is in Cn if it bounds a Casson tower of height n in B4
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Introduction Goal Casson towers Results

Casson towers

Definition (R.)

  • A knot is in Cn if it bounds a Casson tower of height n in B4
  • A knot is in C+

n if it bounds a Casson tower of height n in B4

such that all the kinks at the initial disk are positive

  • A knot is in C−

n if it bounds a Casson tower of height n in B4

such that all the kinks at the initial disk are negative

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Introduction Goal Casson towers Results

Casson towers

Definition (R.)

  • A knot is in C2, n if it bounds a Casson tower of height (2, n)

in B4

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Introduction Goal Casson towers Results

Casson towers

Definition (R.)

  • A knot is in C2, n if it bounds a Casson tower of height (2, n)

in B4

  • A knot is in C+

2, n if it bounds a Casson tower of height (2, n)

in B4 such that all the kinks at the initial disk are positive

  • A knot is in C−

2, n if it bounds a Casson tower of height (2, n)

in B4 such that all the kinks at the initial disk are negative

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Introduction Goal Casson towers Results

Results

Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0, Gn+2 ⊆ Fn

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Introduction Goal Casson towers Results

Results

Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0, Gn+2 ⊆ Fn Theorem (R.) For all n ≥ 0,

  • C+

n+2 ⊆ Pn

  • C−

n+2 ⊆ Nn

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Results

Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0, Gn+2 ⊆ Fn Theorem (R.) For all n ≥ 0,

  • C+

n+2 ⊆ Pn

  • C−

n+2 ⊆ Nn

  • C+

2, n ⊆ Pn

  • C−

2, n ⊆ Nn

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Results

Model Theorem (Cochran–Orr–Teichner, 2003) For all n ≥ 0, Gn+2 ⊆ Fn Theorem (R.) For all n ≥ 0,

  • C+

n+2 ⊆ Pn

  • C−

n+2 ⊆ Nn

  • C+

2, n ⊆ Pn

  • C−

2, n ⊆ Nn

  • Cn+2 ⊆ Gn+2 ⊆ Fn
  • C2, n ⊆ Fn
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Results

Proposition (R.) For m–component links, let Cn(m), C2, n(m), Fn(m), Pn(m), and Nn(m) denote the Casson tower, n–solvable, n–positive and n–negative filtrations respectively. For all n and m ≥ 2n+2, Z ⊆ Fn(m)/Cn+2(m) Z ⊆ Pn(m)/C+

n+2(m)

Z ⊆ Nn(m)/C−

n+2(m)

Z ⊆ Fn(m)/C2, n(m) Z ⊆ Pn(m)/C+

2, n(m)

Z ⊆ Nn(m)/C−

2, n(m)

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Results

Proposition (R.) Let T denote the set of all topologically slice knots. Then T ⊆

∞

  • n=1

Gn