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Lightning Talks IIITech Topology Conference
December 10, 2017
COVERING SPACES, MAPPING CLASS GROUPS, AND THE
SYMPLECTIC REPRESENTATION
Sarah Davis With Laura Stordy, Becca Winarski, Ziyi Zhou Georgia Institute of Technology Tech Topology Conference
3-Fold Branched Cover
= R
Symmetric Mapping Class Group
SMod(S2
) = NMod(S2)
(hRi)
Symplectic Representation
� : Mod(S2) ! Sp(4,Z)
� : R 7! E =
2
664
0 0 �1 00 �1 0 �11 0 �1 00 1 0 0
3
775
Example:
McMullen’s Question
Question: Is finite index in ?
Venkataramana: Yes, if # branch points ≥ 2*degree
�(SMod(Sg))
NSp(2g,Z)(�(hRdi))
Main Theorem:
�(SMod(S2)) = NSp(4,Z)(hEi)�(SMod(S2)) = NSp(4,Z)(hEi)
Main Theorem:
�(SMod(S2)) = NSp(4,Z)(hEi)
Find such that:
M 2 GL(4,Z)
MEM�1 = E±1
Lemma: It is enough to find M such that:
MEM�1 = E�1
�(SMod(S2)) = NSp(4,Z)(hEi)
MATLAB Output
2
664
�z0 z1 � z2 z0 + z3 z1�z4 � z5 �z6 z4 z7 � z6
z3 z1 z0 z2z4 z7 z5 z6
3
775M =
Refine using the symplectic condition
M⌦MT = ⌦
2
664
2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0
3
775 ,
2
664
2x �1 �x 01 0 0 0x 0 �2x 10 0 �1 0
3
775 ,
2
664
0 1 0 0�1 �2x 0 �x
0 0 0 �10 x 1 2x
3
775 ,
2
664
0 �1 0 01 �2x 0 �x
0 0 0 10 x �1 2x
3
775
2
664
x 1 x 10 0 �1 0
2x 1 �x 0�1 0 1 0
3
775 ,
2
664
x �1 x �10 0 1 0
2x �1 �x 01 0 �1 0
3
775 ,
2
664
0 1 0 10 x �1 2x0 1 0 0
�1 x 1 �x
3
775 ,
2
664
0 �1 0 �10 x 1 2x0 �1 0 01 x �1 �x
3
775
2
664
0 x 1 2x0 0 0 11 2x 0 x
0 1 0 0
3
775 ,
2
664
0 x �1 2x0 0 0 �1
�1 2x 0 x
0 �1 0 0
3
775 ,
2
664
0 0 1 0x 0 �2x 11 0 0 0
�2x 1 x 0
3
775 ,
2
664
0 0 �1 0x 0 �2x �1
�1 0 0 0�2x �1 x 0
3
775
MATLAB Output
�!
Ghaswala-Winarski give generators for . SMod(S2)
�(SMod(S2)) = NSp(4,Z)(hEi)
�( eT↵) =
2
664
1 0 0 02 1 �1 00 0 1 0
�1 0 2 1
3
775
How can we obtain this matrix from ?
2
664
2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0
3
775 = �(eh�
) · �( eT↵
)x · �(ehc
)
�(SMod(S2)) = NSp(4,Z)(hEi)
2
664
2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0
3
775 = �(eh�
) · �( eT↵
)x · �(ehc
)�⇣fH�
� fT↵
x
� fHc
⌘
H� Hc = H⌘ �H◆T↵
Main Theorem:
�(SMod(S2)) = NSp(4,Z)(hEi)
Thank you!
Lightning Talks IIITech Topology Conference
December 10, 2017
Salem Number Stretch Factors
Joshua Pankau
University of California, Santa BarbaraAdvisor: Darren Long
12/10/2017
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 1 / 6
I. Background
Definition of pseudo-Anosov map
A homeomorphism φ from a closed, orientable surface S to itself is calledpseudo-Anosov if there are two transverse, measured foliations, Fu andFs , along with a real number λ > 1, such that φ stretches S along Fu bya factor of λ and contracts S along Fs by a factor of λ−1. The number λis known as the stretch factor of φ.
Theorem (Thurston 1974)
If λ is the stretch factor of a pseudo-Anosov homeomorphism of a genus gsurface, then λ is an algebraic unit such that [Q(λ) : Q] ≤ 6g − 6.
Main Question
Which algebraic units can appear as stretch factors?
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 2 / 6
I. Background
Definition of pseudo-Anosov map
A homeomorphism φ from a closed, orientable surface S to itself is calledpseudo-Anosov if there are two transverse, measured foliations, Fu andFs , along with a real number λ > 1, such that φ stretches S along Fu bya factor of λ and contracts S along Fs by a factor of λ−1. The number λis known as the stretch factor of φ.
Theorem (Thurston 1974)
If λ is the stretch factor of a pseudo-Anosov homeomorphism of a genus gsurface, then λ is an algebraic unit such that [Q(λ) : Q] ≤ 6g − 6.
Main Question
Which algebraic units can appear as stretch factors?
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 2 / 6
I. Background
Definition of pseudo-Anosov map
A homeomorphism φ from a closed, orientable surface S to itself is calledpseudo-Anosov if there are two transverse, measured foliations, Fu andFs , along with a real number λ > 1, such that φ stretches S along Fu bya factor of λ and contracts S along Fs by a factor of λ−1. The number λis known as the stretch factor of φ.
Theorem (Thurston 1974)
If λ is the stretch factor of a pseudo-Anosov homeomorphism of a genus gsurface, then λ is an algebraic unit such that [Q(λ) : Q] ≤ 6g − 6.
Main Question
Which algebraic units can appear as stretch factors?
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 2 / 6
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s ConstructionRestriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s ConstructionRestriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s Construction
Restriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s ConstructionRestriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s Construction
Restriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s Construction
Restriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s ConstructionRestriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s Construction
Restriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s ConstructionRestriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s ConstructionRestriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
III. Salem numbers
Salem number
A real algebraic unit, λ > 1, is called a Salem number if λ−1 is a Galoisconjugate, and all other conjugates lie on the unit circle.
Theorem A (P. 2017)
Given a Salem number λ, there are positive integers k , g such that λk isthe stretch factor of a pseudo-Anosov homeomorphism φ : Sg → Sg ,where φ arises from Thurston’s construction. Moreover, g depends only onthe degree of λ over Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 4 / 6
III. Salem numbers
Salem number
A real algebraic unit, λ > 1, is called a Salem number if λ−1 is a Galoisconjugate, and all other conjugates lie on the unit circle.
Theorem A (P. 2017)
Given a Salem number λ, there are positive integers k , g such that λk isthe stretch factor of a pseudo-Anosov homeomorphism φ : Sg → Sg ,where φ arises from Thurston’s construction. Moreover, g depends only onthe degree of λ over Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 4 / 6
IV. Connecting Salem numbers to Thurston’s construction
Thurston’s construction requires a collection of curves that cut the surfaceinto disks. The intersection matrix of these curves also plays a crucial role.
Theorem (P. 2017)
Every Salem number λ has a power k such that λk + λ−k is thedominating eigenvalue of an invertible, positive, symmetric, integer matrix.
Theorem (P. 2017)
Given an invertible, positive, integer matrix Q, there is a closed, orientablesurface S along with a collection of curves that cut S into disks, such thatthe intersection matrix of those curves is Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 5 / 6
IV. Connecting Salem numbers to Thurston’s construction
Thurston’s construction requires a collection of curves that cut the surfaceinto disks. The intersection matrix of these curves also plays a crucial role.
Theorem (P. 2017)
Every Salem number λ has a power k such that λk + λ−k is thedominating eigenvalue of an invertible, positive, symmetric, integer matrix.
Theorem (P. 2017)
Given an invertible, positive, integer matrix Q, there is a closed, orientablesurface S along with a collection of curves that cut S into disks, such thatthe intersection matrix of those curves is Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 5 / 6
IV. Connecting Salem numbers to Thurston’s construction
Thurston’s construction requires a collection of curves that cut the surfaceinto disks. The intersection matrix of these curves also plays a crucial role.
Theorem (P. 2017)
Every Salem number λ has a power k such that λk + λ−k is thedominating eigenvalue of an invertible, positive, symmetric, integer matrix.
Theorem (P. 2017)
Given an invertible, positive, integer matrix Q, there is a closed, orientablesurface S along with a collection of curves that cut S into disks, such thatthe intersection matrix of those curves is Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 5 / 6
V. Totally real number fields
Methods and results used to prove Theorem A can be adapted to provethe following:
Theorem B (P. 2017)
Every totally real number field is of the form K = Q(λ+ λ−1) where λ isthe stretch factor of a pseudo-Anosov map coming from Thurston’sconstruction.
Thank you!
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 6 / 6
V. Totally real number fields
Methods and results used to prove Theorem A can be adapted to provethe following:
Theorem B (P. 2017)
Every totally real number field is of the form K = Q(λ+ λ−1) where λ isthe stretch factor of a pseudo-Anosov map coming from Thurston’sconstruction.
Thank you!
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 6 / 6
Lightning Talks IIITech Topology Conference
December 10, 2017
Why are there are so many spectralsequences from Khovanov homology?
Adam Saltz (University of Georgia)December 10, 2017
Georgia TechTech Topology Conference
Spectral sequences galore
Theorem (Ozsvath, Szabo; Bloom; Scaduto; Daemi; Kronhemier, Mrowka)
Let L be a link in S3. Let Σ(L) be the double cover of S3 branchedalong L. There are spectral sequences
Khovanov homology
HeegaardMonopole
Framed instantonPlane
Singular instanton
LeeBar-Natan
Szabo
Floer homologyof Σ(L)
link homology
I am missing a few words like “mirror of” and “reduced.”
Spectral sequences galore
Theorem (Ozsvath, Szabo; Bloom; Scaduto; Daemi; Kronhemier, Mrowka)
Let L be a link in S3. Let Σ(L) be the double cover of S3 branchedalong L. There are spectral sequences
Khovanov homology
HeegaardMonopole
Framed instantonPlane
Singular instanton
LeeBar-Natan
Szabo
Floer homologyof Σ(L)
link homology
I am missing a few words like “mirror of” and “reduced.”
Khovanov-Floer theories
Definition (Baldwin, Hedden, and Lobb)A Khovanov-Floer theory is a gadget:
Dlink diagram
Ei(D)
spectral sequence
D D′ Fi : Ei(D)→ Ei(D′)map of
spectral sequencesone-handleattachment
• E2(D) = Kh(D)
• F2 agrees with the standard map Kh(D)→ Kh(D′).• Kunneth formula, etc.
Khovanov-Floer theories: the good
Theorem (Baldwin, Hedden, Lobb)All of the homology theories from the second slide areKhovanov-Floer theories.
Theorem (Baldwin, Hedden, Lobb)Khovanov-Floer theories are
• link invariants.• functorial: they assign maps to isotopy classes of link
cobordisms in S3 × I.
Everything that works for Khovanov homology works forKhovanov-Floer theories because that’s how maps on spectralsequences work.
Two maps on homology!
one-handle attachment
filtered chain map F
Fi : Ei(D)→ Ei(D′)F∞
F∗
A priori, F∗ 6= F∞!
A different approach
DefinitionA strong Khovanov-Floer theory is a gadget:
Dlink diagram
K(D)
filtered complex
D D′ F : K(D)→ K(D′)filtered chain maphandle attachment
so that
• For a crossingless diagrams, H(K(D)) agrees with Kh(D) (oranother Frobenius algebra).
• Handle attachment maps satisfy some relations (e.g. swappingdistant handles, Bar-Natan’s S, T, and 4Tu)
• Kunneth formula, etc.
Strong Khovanov-Floer theories: the good
DefinitionA strong Khovanov-Floer theory is conic if, for D with crossings,
K = cone(h : D0 → D1)
where h is a one-handle attachment map.
Theorem (S.)Conic strong Khovanov-Floer theories are
• link invariants. (chain homotopy type)• functorial: they assign (chain homotopy types of) maps to
isotopy classes of link cobordisms in S3 × I.
Everything that works for Bar-Natan’s cobordism-theoreticconstruction of link homology works for strong Khovanov-Floertheories.
Strong Khovanov-Floer theories: the good
DefinitionA strong Khovanov-Floer theory is conic if, for D with crossings,
K = cone(h : D0 → D1)
where h is a one-handle attachment map.
Theorem (S.)Conic strong Khovanov-Floer theories are
• link invariants. (chain homotopy type)• functorial: they assign (chain homotopy types of) maps to
isotopy classes of link cobordisms in S3 × I.
Everything that works for Bar-Natan’s cobordism-theoreticconstruction of link homology works for strong Khovanov-Floertheories.
Strong Khovanov-Floer theories: the good
Theorem (S.)Heegaard Floer homology, singular instanton homology, Szabohomology, and Lee/Bar-Natan homology all produce conic strongKhovanov-Floer theories. (The rest probably are, too.)
Theorem (S.)A conic strong Khovanov-Floer theory yields a Khovanov-Floertheory.
Strong Khovanov-Floer theories: what’s next
How does this help us understand invariants of transverse linksand contact structures?
What other link homology theories can we use besides Khovanovhomology? (E.g. Lin has constructed a spectral sequence fromBar-Natan-Lee homology to monopole Floer homology)
Can we understand e.g. Heegaard Floer homology via Morsetheory on surfaces?
Lightning Talks IIITech Topology Conference
December 10, 2017
Least Dilatation of Pure Surface Braids
Marissa LovingUniversity of Illinois at Urbana-Champaign
What?
Why?
How?
Who?
When?
What?
Why?
How?
Who? Me!
When? Now!
What?
Why?
How? Who? Me!
When? Now!
• pure mapping classes
• isotopic to the identity on the closed surface
• denoted PBn(Sg)
What are pure surface braids?
• a real number > 1• associated to a
mapping class 𝑓• denoted 𝜆 𝑓
What is the dilatation?
What did I prove?
Theorem (L., 2017)
𝑐 loglog 𝑔𝑛 + 𝑐 ≤ 𝐿 𝑃𝐵1 𝑆3 ≤ 𝑐4 log
𝑔𝑛 + 𝑐4
Why should we care?
Theorem (Penner, 1991) L(Mod(Sg)) goes to zero as g goes to infinity.
Theorem (Farb-Leininger-Margalit, 2008)L(Ig) is universally bounded between 0.197 and 4.127.
Theorem (Dowdall, Aougab—Taylor)
15 log(2𝑔) ≤ 𝐿 𝑃𝐵9 𝑆3 < 4 log(𝑔) + 2 log(24)
How did I prove it?
The Upper Bound
The Upper Bound
The Upper Bound
The Lower Bound
𝑃𝐵1(𝑆3) ∋ 𝑓 ↷Imayoshi—Ito—Yamamoto:
“𝜆 𝐹@ ≤ 𝜆(𝑓)”
↶ 𝐹@
↶ 𝐹C@𝑥 𝐹C@(𝑥)
Kra: “max
H𝑑(𝑥, 𝐹C@ 𝑥 ) ≤ 𝜆(𝐹C@)”
𝜋
ΗM
𝜋(𝑥) 𝜋(𝐹C@ 𝑥 )
The Lower Bound
𝑃𝐵1(𝑆3) ∋ 𝑓 ↷Imayoshi—Ito—Yamamoto:
“𝜆 𝐹@ ≤ 𝜆(𝑓)”
↶ 𝐹@
↶ 𝐹C@𝑥 𝐹C@(𝑥)
Kra: “max
H𝑑(𝑥, 𝐹C@ 𝑥 ) ≤ 𝜆(𝐹C@)”
𝜋
ΗM
𝜋(𝑥) 𝜋(𝐹C@ 𝑥 )
“maxH
d 𝑥, 𝐹@ 𝑥 ≤ 𝜆(𝑓)”
Theorem (L.—Parlier, 2017)
A filling graph Γ embedded in a surface 𝑆3 has diameter
at least PQR STU
VWX
.
The Lower Bound
Thank you!
𝑃𝐵1(𝑆3) ∋ 𝑓 ↷
↶ 𝐹@
↶ 𝐹C@𝑥 𝐹C@(𝑥)
𝜋
ΗM
𝜋(𝑥) 𝜋(𝐹C@ 𝑥 )
Lightning Talks IIITech Topology Conference
December 10, 2017
Truncated Heegaard Floer homology andconcordance invariants
Linh Truong
Columbia University
Tech Topology Conference, December 2017
Motivation
Our motivation is to better understand knot concordance.
DefinitionK1 and K2 are concordant if they cobound a smooth cylinder inS3 × [0, 1].
DefinitionThe concordance group is C = {knots in S3/ ∼,#}, whereK1 ∼ K2 if K1 is concordant to K2.
Open Questions
There are many open questions about knot concordance.
QuestionIs every slice knot a ribbon knot?
Figure: “Square ribbon knot”; figure by David Eppstein, Wikipedia.
The boundary of a self-intersecting disk with only “ribbonsingularities” is called a ribbon knot.
QuestionIs there any torsion in the concordance group C besides 2-torsion?
Truncated Heegaard Floer homology
Heegaard Floer homology is an invariant for three-manifoldsdefined by Ozsvath and Szabo.
Truncated Heegaard Floer homology, denoted HF n(Y , s)(Ozsvath-Szabo, Ozsvath-Manolescu), is the homology of thekernel CF n(Y , s) of the multiplication map
Un : CF+(Y , s)→ CF+(Y , s)
where n ∈ Z+.
RemarkNote for n = 1, truncated Heegaard Floer homology equalsHF (Y , s).
Truncated Concordance Invariants
Motivated by the constructions of the Ozsvath-Szabo ν(K ) andHom-Wu ν+(K ), we construct a sequence of knot invariantsνn(K ), n ∈ Z:
DefinitionFor n > 0, define
νn(K ) = min{s ∈ Z | vns : CF n(S3N(K ), ss)→ CF n(S3)
induces a surjection on homology},
where N is sufficiently large so that the Ozsvath-Szabo largeinteger surgery formula holds, and ss denotes the restriction toS3N(K ) of a Spinc structure t on the corresponding 2-handle
cobordism such that
〈c1(t), [F ]〉+ N = 2s,
where F is a capped-off Seifert surface for K .
Truncated Concordance Invariants, continued...
DefinitionFor n < 0, define
νn(K ) = max{s ∈ Z | vns : CF−n(S3)→ CF−n(S3−N(K ), ss)
induces an injection on homology},
where N is sufficiently large so that the Ozsvath-Szabo largeinteger surgery formula holds, and ss denotes the restriction toS3−N(K ) of a Spinc structure t on the corresponding 2-handle
cobordism such that
〈c1(t), [F ]〉 − N = 2s,
where F is a capped-off Seifert surface for K .For n = 0, we define ν0(K ) = τ(K ).
Properties of νn(K )
The knot invariants νn(K ), n ∈ Z, satisfy the following properties:
• νn(K ) is a concordance invariant.
• ν1(K ) = ν(K ).
• νn(K ) ≤ νn+1(K ).
• For sufficiently large n, νn(K ) = ν+(K ).
• νn(−K ) = −ν−n(K ), where −K is the mirror of K .
• νn(K ) ≤ g4(K ).
Homologically thin knots are knots with HFK supported in a singleδ = A−M grading.
TheoremLet K be a homologically thin knot with τ(K ) = τ .
(i) If τ = 0, νn(K ) = 0 for all n.
(ii) If τ > 0,
νn(K ) =
0, for n ≤ −(τ + 1)/2,
τ + 2n + 1, for − τ/2 ≤ n ≤ −1,
τ, for n ≥ 0.
(iii) If τ < 0,
νn(K ) =
τ, for n ≤ 0,
τ + 2n − 1, for 1 ≤ n ≤ −τ/2,
0, for n ≥ (−τ + 1)/2.
Large Gaps
In fact, the difference between νn(K ) and νn+1(K ) can bearbitrarily big.
TheoremLet Tp,p+1 denote the (p, p + 1) torus knot. For p > 3,
ν−1(Tp,p+1)− ν−2(Tp,p+1) = p.
Thank you!
Lightning Talks IIITech Topology Conference
December 10, 2017
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations and Immersed Exact LagrangianFillings
Yu Pan
MIT
Tech Topology ConferenceDec. 10th, 2017
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Exact Lagrangian fillings
An embedded exact Lagrangian filling of Λ is a 2-dimensionalembedded surface L in (Rt × R3, ω = d(etα)) such that
L is cylindrical over Λ when t is bigenough;
there exists a function f : L→ R such
that etα∣∣∣TL
= df and f is constant on
Λ.
t
Λ
L
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Augmentations
By [Ekholm-Honda-Kalman, ’12],an exact Lagrangian filling L =⇒ an augmentation ε of A(Λ)
Λ
L
(A(Λ), ∂
)y(Z2, 0)
ε
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Correspondence
Derived Fukaya Category Augmentation Category
Objects: Exact Lagrangian Fillings Augmentations
However, not all the augmentations of A(Λ) are induced fromembedded exact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Correspondence
Derived Fukaya Category Augmentation Category
Objects: Exact Lagrangian Fillings Augmentations
However, not all the augmentations of A(Λ) are induced fromembedded exact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Correspondence
Derived Fukaya Category Augmentation Category
Objects: Exact Lagrangian Fillings Augmentations
However, not all the augmentations of A(Λ) are induced fromembedded exact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Immersed Exact Lagrangian fillings
t
Λ
Σ
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Augmentations induced from immersed exact Lagrangianfillings
Suppose that Σ can be lifted to an embedded Legendrian surface Lin R× R× R3 and A(L) has an augmentation εL.
t
Λ
Σ
(A(Λ), ∂
)(A(L), ∂
)y f
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Augmentations induced from immersed exact Lagrangianfillings
Suppose that Σ can be lifted to an embedded Legendrian surface Lin R× R× R3 and A(L) has an augmentation εL.
t
Λ
Σ
(A(Λ), ∂
)(A(L), ∂
)yy
(Z2, 0)
εL
fThus ε = εL ◦ f is anaugmentation of A(Λ).
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Result
Theorem (P.-D. Rutherford)
All the augmentations of A(Λ) are induced from possibly immersedexact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
Lightning Talks IIITech Topology Conference
December 10, 2017
Lightning Talks IIITech Topology Conference
December 10, 2017