liam watson universit e de sherbrooke july 26, 2017liam/files/talk-mca-v2.pdf · liam watson...
TRANSCRIPT
![Page 1: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/1.jpg)
Bordered Floer homology via immersed curvesJoint work with Jonathan Hanselman and Jake Rasmussen
Liam WatsonUniversite de Sherbrooke
MCA MontrealJuly 26, 2017
![Page 2: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/2.jpg)
Heegaard Floer homology
Let M be a compact, connected, oriented three-manifold withtorus boundary; fix a marked point ? ∈ ∂M.
Theorem (Hanselman-Rasmussen-W.)
The Heegaard Floer homology HF (M) can be interpreted as a setof immersed curves inT = ∂M \ ?, up to regular homotopy.
![Page 3: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/3.jpg)
Heegaard Floer homology
Let M be a compact, connected, oriented three-manifold withtorus boundary; fix a marked point ? ∈ ∂M.
Theorem (Hanselman-Rasmussen-W.)
The Heegaard Floer homology HF (M) can be interpreted as a setof immersed curves (possibly decorated with local systems) inT = ∂M \ ?, up to regular homotopy.
A local system is a finite dimensional vector space V (in our case,over F = Z/2Z), together with an endomorphism Φ: V → V .
![Page 4: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/4.jpg)
Some examples (in the cover R2 \ Z2 → T )
![Page 5: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/5.jpg)
The pairing theorem
Suppose further that Y = M0 ∪h M1 where h : ∂M1 → ∂M0 is anorientation reversing homeomorphism for which h(?1) = h(?0).
Theorem (Hanselman-Rasmussen-W.)
HF (Y ) ∼= HF (γ0,γ1)
Here, HF (γ0,γ1) computes the Lagrangian intersection Floerhomology of
γ0 = HF (M0) and γ1 = h!(HF (M1)
)in the punctured torus T = ∂M0 \ ?0. The function h! composes hwith the hyperelliptic involution on T .
![Page 6: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/6.jpg)
Example: splicing right-handed trefoils
Most of the time, this boils down to counting minimal intersection.
HF (M0 ∪h M1) ∼= F7
h(µ1) = λ0h(λ1) = µ0 µi
λiµ0
λ0
See Hedden-Levine to compare this with a direct bordered Floercalculation of this particular splice.
![Page 7: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/7.jpg)
Example: splicing right-handed trefoils
Most of the time, this boils down to counting minimal intersection.
HF (M0 ∪h M1) ∼= F7
h(µ1) = λ0h(λ1) = µ0 µi
λiµ0
λ0
See Hedden-Levine to compare this with a direct bordered Floercalculation of this particular splice.
![Page 8: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/8.jpg)
Application: The L-space gluing theorem
DefinitionA rational homology sphere Y is an L-space whenever
dim HF (Y ) = |H1(Y ;Z)|
QuestionWhen is M0 ∪h M1 an L-space?
When one (or both) of the Mi is a solid torus, the answer is“sometimes”. Define
LM = {α | the Dehn filling M(α) is an L-space} ⊂ SM
![Page 9: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/9.jpg)
Application: The L-space gluing theorem
LM = {α | the Dehn filling M(α) is an L-space} ⊂ SM
Theorem (Hansleman-Rasmussen-W.)
Suppose Mi is irreducible and boundary irreducible (in particular,Mi � D2 × S1). Then M0 ∪h M1 is an L-space if and only if
L◦M0∪ h(L◦M1
) = SM0
Special cases of this were known: Hedden-Levine, Hanselman,Hanselman-W., Hanselman-Rasmussen-Rasmussen-W.
![Page 10: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/10.jpg)
Consequences of the L-space gluing theorem
Corollary
Set N = |H1(Y ;Z)| for Y = M0 ∪h M1 with Mi 6= D2 × S1. IfN = 1, 2, 3, 6 then Y is not and L-space.
In particular, there do not exist toroidal integer homology sphereL-spaces (see also Eftekhary).
Conjecture (Ozsvath-Szabo)
The only prime integer homology sphere L-spaces are thethree-sphere and the Poincare homology sphere.
![Page 11: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/11.jpg)
Consequences of the L-space gluing theorem
A knot in the three-sphere admitting non-trivial L-space surgeriesis called an L-space knot. That is, such K are characterized by theproperty |LS3\ν(K)| > 1.
Corollary
Suppose K is a satellite L-space knot. Then both the pattern knotand the companion knot must be L-space knots.
This was conjectured by Hom-Lidman-Vafaee. More can be saidabout the companion knot; see Baker-Motegi.
![Page 12: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/12.jpg)
Application: Degree one maps
QuestionGiven a degree one map Y → Y0, what is the relationship betweenHF (Y ) and HF (Y0)?
Given an integer homology sphere M0 ∪h M1, consider the slope
αh = h(λ1) ∈ SM0
Note that this is a meridian for M0, that is, ∆(λ0, αh) = 1.
Theorem (Hanselman-Rasmussen-W.)
Let Y = M0 ∪h M1 and Y0 = M0(αh). Then there is a degree onemap Y → Y0 and
dim HF (Y ) ≥ dim HF (Y0)
![Page 13: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/13.jpg)
A sketch of the proof
µ0
λ0
Step 0: make any local systemsappearing trivial
![Page 14: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/14.jpg)
A sketch of the proof
µ0
λ0
Step 0: make any local systemsappearing trivialStep 1: remove any closedcomponents
![Page 15: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/15.jpg)
A sketch of the proof
µ0
λ0
Step 0: make any local systemsappearing trivialStep 1: remove any closedcomponentsStep 2: pull the remaining curvetight
![Page 16: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/16.jpg)
A sketch of the proof
µ0
λ0
Step 0: make any local systemsappearing trivialStep 1: remove any closedcomponentsStep 2: pull the remaining curvetightStep 3: check that none of thesesteps created new intersection points
![Page 17: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/17.jpg)
Where does HF (M) come from?
The curve-set HF (M) is a geometric interpretation of the bordered
Floer homology CFD(M, α, β), which was defined byLipshitz-Ozsvath-Thurston.
The invariant CFD(M, α, β) is a type D structure, which is alinear-algebraic object defined over an algebra A associated withT = ∂M \ ?.
The interpretation HF (M) comes from describing type Dstructures as geometric objects in T , and then providing astructure theorem that simplifies them.
![Page 18: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/18.jpg)
Type D structures associated with a point
Consider a 0-handle with a markedpoint ? near the boundary.
Consider points (stations) on theboundary collected into groups(towns).
A type D structure is a train trackthat
(1) only travels to a *next* town;
(2) doesn’t pass the basepoint; and
(3) has an even number of possibleconnections.
![Page 19: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/19.jpg)
Type D structures associated with a point
Desired property: Extendability
An extension of a type D structure isa rail system upgrade.
The additional tracks can pass ? atmost once.
![Page 20: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/20.jpg)
Type D structures associated with a point
LemmaAny extension of a type D structureis equivalent to one in standardform: a collection of properlyembedded arcs, together withcrossover arrows running clockwisealong the boundary.
Convention:
![Page 21: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/21.jpg)
Type D structures associated with a point
Crossover arrows moving betweengroups can be removed.
Structure TheoremEvery extendable type D structureassociated with a zero handle can beput in the (simplified) standard formillustrated on the left.
![Page 22: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/22.jpg)
Adding 1-handles
![Page 23: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/23.jpg)
Adding 1-handles
![Page 24: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/24.jpg)
Adding 1-handles
![Page 25: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/25.jpg)
Adding 1-handles
![Page 26: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/26.jpg)
Adding 1-handles
![Page 27: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/27.jpg)
Adding 1-handles
![Page 28: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/28.jpg)
Adding 1-handles
![Page 29: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/29.jpg)
Adding 1-handles
![Page 30: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/30.jpg)
Adding 1-handles
![Page 31: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/31.jpg)
Adding 1-handles
![Page 32: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/32.jpg)
Adding 1-handles
![Page 33: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/33.jpg)
Adding 1-handles
![Page 34: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/34.jpg)
Adding 1-handles
![Page 35: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/35.jpg)
Adding 1-handles
![Page 36: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/36.jpg)
Adding 1-handles
![Page 37: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/37.jpg)
Adding 1-handles
![Page 38: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/38.jpg)
Adding 1-handles
![Page 39: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/39.jpg)
Adding 1-handles
![Page 40: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/40.jpg)
Adding 1-handles
( 1 11 0 )
![Page 41: Liam Watson Universit e de Sherbrooke July 26, 2017liam/files/talk-MCA-V2.pdf · Liam Watson Universit e de Sherbrooke MCA Montr eal July 26, 2017. Heegaard Floer homology ... The](https://reader036.vdocuments.site/reader036/viewer/2022070803/5f032b147e708231d407e0a7/html5/thumbnails/41.jpg)
Three-manifold invariants
Structure TheoremExtendable Type D structures in a surface with a fixed 0- and1-handle decomposition are immersed curves with local systems.
Extension TheoremThe type D structure CFD(M, α, β) is extendable.