Floer homology, orderable groups, and taut foliations of hyperbolic 3-manifolds: An experimental study Nathan M. Dunfield (University of Illinois) These slides already posted at: http://dunfield.info/slides/Newt17.pdf
Y 3 : closed oriented irreducible with Floer homology, H ∗ ( Y ; Q ) ∼ = H ∗ ( S 3 ; Q ) . orderable groups, and taut foliations of hyperbolic 3-manifolds: Conj: For an irreducible Q HS Y , TFAE: (a) � HF ( Y ) is non-minimal. An experimental study (b) π 1 ( Y ) is left-orderable. (c) Y has a co-orient. taut foliation. Nathan M. Dunfield (University of Illinois) These slides already posted at: http://dunfield.info/slides/Newt17.pdf
Y 3 : closed oriented irreducible with Heegaard Floer Homology: An F 2 -vector space � H ∗ ( Y ; Q ) ∼ = H ∗ ( S 3 ; Q ) . HF ( Y ) , part of a 3 + 1 dimensional (almost) TQFT. Conj: For an irreducible Q HS Y , TFAE: [Kronheimer, Mrowka, Ozsváth, (a) � HF ( Y ) is non-minimal. Szabó 2003] No Dehn surgery on a nontrivial knot in S 3 yields R P 3 . (b) π 1 ( Y ) is left-orderable. � � (c) Y has a co-orient. taut foliation. � H 1 ( Y ; Z ) � . Basic fact: dim � HF ( Y ) ≥ When equal, Y is an L-space . L-spaces: Spherical manifolds, e.g. L ( p , q ) . Non-L-spaces: 1 / n -Dehn surgery on a knot in S 3 other than the unknot or the trefoil.
Left-order: A total order on a group Heegaard Floer Homology: An F 2 -vector space � G where g < h implies f · g < f · h for HF ( Y ) , part of a 3 + 1 all f , g , h ∈ G . dimensional (almost) TQFT. Orderable: ( R , + ) , ( Z , + ) , F n , B n . [Kronheimer, Mrowka, Ozsváth, Szabó 2003] No Dehn surgery on a Non-orderable: finite groups, SL n Z nontrivial knot in S 3 yields R P 3 . for n ≥ 3. � � � H 1 ( Y ; Z ) � . Basic fact: dim � HF ( Y ) ≥ For countable G , equivalent to When equal, Y is an L-space . → Homeo + ( R ) . G � L-spaces: Spherical manifolds, Y 3 is called orderable if π 1 ( Y ) is e.g. L ( p , q ) . left-orderable. Non-L-spaces: 1 / n -Dehn surgery on a knot in S 3 other than the unknot or the trefoil.
Left-order: A total order on a group G where g < h implies f · g < f · h for all f , g , h ∈ G . Orderable: ( R , + ) , ( Z , + ) , F n , B n . Taut foliation: A decomposition F Non-orderable: finite groups, SL n Z of Y into 2-dim’l leaves where: for n ≥ 3. (a) Smoothness: C 1 , 0 For countable G , equivalent to (b) Co-orientable. → Homeo + ( R ) . G � (c) There exists a loop transverse to Y 3 is called orderable if π 1 ( Y ) is F meeting every leaf. left-orderable. Example: Y fibers over S 1 . Better example: T 3 foliated by irrational planes.
Non-examples: While every closed 3-manifold has a foliation F satisfying ( a ) and ( b ) , if Y is R 3 or S 2 × R F is taut then � and so π 1 ( Y ) is infinite. Taut foliation: A decomposition F The hyperbolic 3-manifold of least of Y into 2-dim’l leaves where: volume, the Weeks manifold, is a (a) Smoothness: C 1 , 0 Q HS which has no taut foliations. (b) Co-orientable. (c) There exists a loop transverse to F meeting every leaf. Example: Y fibers over S 1 . Better example: T 3 foliated by irrational planes.
Conjecture of [BGW] Y is orderable Y is not ⇐ ⇒ an L -space π 1 ( Y ) acts on R [OS] and [B, KR] All actions are nontrivial Y has a taut and orientation preserving. foliation F
Conjecture of [BGW] Y is orderable Y is not ⇐ ⇒ an L -space π 1 ( Y ) acts on R π 1 ( Y ) acts on a simply [OS] and connected 1-manifold [B, KR] (possibly non-Hausdorff) Leaf space of � F in � Y π 1 ( Y ) acts Y has a taut on S 1 foliation F Thurston’s universal circle [CD]
Evidence for the conjecture: [Hanselman-Rasmussen 2 -Watson + Boyer-Clay 2015] True for all graph manifolds. [Culler-D. 2016 + Roberts 2001] Suppose K ⊂ S 3 where ∆ K ( t ) has a simple root on the unit circle and which is lean. Then there exists ǫ > 0 so that the conjecture holds for the r Dehn surgery on K whenever r ∈ ( − ǫ , ǫ ) . [Gordon-Lidman, Tran, . . . ]
Sample: 307,301 hyperbolic Q HSs. Conjecture holds for ≥ 65%! non-L-spaces (53%) L-spaces (47%) not orderable ( ≥ 37%) taut fol ( ≥ 47%) orderable ( ≥ 30%)
Starting point: Evidence for the conjecture: [Hanselman-Rasmussen 2 -Watson + hyp Q -homology solid tori C = triang by ≤ 9 ideal tets Boyer-Clay 2015] True for all graph [Burton 2014] manifolds. � � hyp Q HS fillings on C ∈ C [Culler-D. 2016 + Roberts 2001] Y = with systole ≥ 0 . 2 Suppose K ⊂ S 3 where ∆ K ( t ) has a # C = 59 , 068 # Y = 307 , 301 simple root on the unit circle and which is lean. Then there exists ǫ > 0 Mean vol ( Y ∈ Y ) is 6 . 9 with σ = 0 . 9. so that the conjecture holds for the r 59% of Y ∈ Y have a unique Dehn Dehn surgery on K whenever filling description involving C ; the r ∈ ( − ǫ , ǫ ) . remaining 41% average 3.4. [Gordon-Lidman, Tran, . . . ]
Determining L-spaces Starting point: Alg. decidable [Sarkar-Wang 2006] hyp Q -homology solid tori Bordered Floer [LOT, L-Zhan] C = triang by ≤ 9 ideal tets [Burton 2014] � � A Q -homology solid torus M is Floer hyp Q HS fillings on C ∈ C Y = simple if it has at least two L-space with systole ≥ 0 . 2 Dehn fillings. # C = 59 , 068 # Y = 307 , 301 [Rasmussen 2 2015] If you know two L-space fillings on M , then the Mean vol ( Y ∈ Y ) is 6 . 9 with σ = 0 . 9. precise set of L-space fillings can be 59% of Y ∈ Y have a unique Dehn read off from the Turaev torsion of M . filling description involving C ; the remaining 41% average 3.4. [Berge; D 2015] There are at least 54,790 finite fillings on C ∈ C .
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR]
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite 0 32% 68% 20% 13% 67% 2 finite fillings
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite 0 32% 68% 20% 13% 67% 2 finite fillings 8% 33% 59% 20% 13% 67% Y ⇐ = C via [RR]
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite 0 32% 68% 20% 13% 67% 2 finite fillings 8% 33% 59% 20% 13% 67% Y ⇐ = C via [RR] 8% 33% 59% 45% 13% 42% Y = ⇒ C via def
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite 0 32% 68% 20% 13% 67% 2 finite fillings 8% 33% 59% 20% 13% 67% Y ⇐ = C via [RR] 8% 33% 59% 45% 13% 42% Y = ⇒ C via def 40% 46% 14% 45% 13% 42% Y ⇐ = C via [RR]
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite 0 32% 68% 20% 13% 67% 2 finite fillings 8% 33% 59% 20% 13% 67% Y ⇐ = C via [RR] 8% 33% 59% 45% 13% 42% Y = ⇒ C via def 40% 46% 14% 45% 13% 42% Y ⇐ = C via [RR] 40% 46% 14% 51% 13% 36% ⇒ C via def Y =
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite 0 32% 68% 20% 13% 67% 2 finite fillings 8% 33% 59% 20% 13% 67% Y ⇐ = C via [RR] 8% 33% 59% 45% 13% 42% Y = ⇒ C via def 40% 46% 14% 45% 13% 42% Y ⇐ = C via [RR] 40% 46% 14% 51% 13% 36% ⇒ C via def Y = 47% 51% 2% 51% 13% 36% Y ⇐ = C via [RR]
Y = 307,301 Q HSs C = 59,068 Q HSTs L-sp non-L L-sp? F-simp non-F simp? 0 0 100% 0 0 100% init state 0 0 100% 0 13% 87% Turaev obstr [RR] 0 32% 68% 0 13% 87% Y ⇐ = C via finite 0 32% 68% 20% 13% 67% 2 finite fillings 8% 33% 59% 20% 13% 67% Y ⇐ = C via [RR] 8% 33% 59% 45% 13% 42% Y = ⇒ C via def 40% 46% 14% 45% 13% 42% Y ⇐ = C via [RR] 40% 46% 14% 51% 13% 36% ⇒ C via def Y = 47% 51% 2% 51% 13% 36% Y ⇐ = C via [RR] 47% 51% 2% 51% 13% 36% final fixed point
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