Floer homology, orderable groups, and taut foliations of - PowerPoint PPT Presentation

Floer homology, orderable groups, and taut foliations of hyperbolic 3-manifolds: An experimental study Nathan M. Dunfield (University of Illinois and IAS) These slides already posted at: http://dunfield.info/slides/IAS.pdf

Floer homology, Y 3 : closed oriented irreducible with orderable groups, H ∗ ( Y ; Q ) ∼ = H ∗ ( S 3 ; Q ) . and taut foliations of hyperbolic 3-manifolds: Conj: For an irreducible Q HS Y , TFAE: An experimental study (a) � HF ( Y ) is non-minimal. (b) π 1 ( Y ) is left-orderable. (c) Y has a co-orient. taut foliation. Nathan M. Dunfield (University of Illinois and IAS) These slides already posted at: http://dunfield.info/slides/IAS.pdf

Y 3 : closed oriented irreducible with H ∗ ( Y ; Q ) ∼ = H ∗ ( S 3 ; Q ) . Heegaard Floer: An F 2 -vector space � HF ( Y ) where Conj: For an irreducible Q HS Y , TFAE: (a) � HF ( Y ) is non-minimal. � � dim � � H 1 ( Y ; Z ) � HF ( Y ) ≥ (b) π 1 ( Y ) is left-orderable. (c) Y has a co-orient. taut foliation. 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.

Heegaard Floer: An F 2 -vector Left-order: A total order on a group space � HF ( Y ) where G where g < h implies f · g < f · h for all � � dim � � H 1 ( Y ; Z ) � HF ( Y ) ≥ f , g , h ∈ G . When equal, Y is an L-space . For countable G , equivalent to → Homeo + ( R ) . G � L-spaces: Spherical manifolds, e.g. L ( p , q ) . Orderable: ( R , + ) , ( Z , + ) , F n . Non-L-spaces: 1 / n -Dehn surgery on a knot in S 3 other than the unknot Non-orderable: finite groups, SL n Z or the trefoil. for n ≥ 2. Y 3 is called orderable if π 1 ( Y ) is left-orderable.

Left-order: A total order on a group G where g < h implies f · g < f · h for all f , g , h ∈ G . For countable G , equivalent to → Homeo + ( R ) . G � Taut foliation: A decomposition F Orderable: ( R , + ) , ( Z , + ) , F n . of Y into 2-dim’l leaves where: Non-orderable: finite groups, SL n Z for n ≥ 2. (a) Smoothness: C 1 , 0 Y 3 is called orderable if π 1 ( Y ) is (b) Co-orientable left-orderable. (c) There exists a loop transverse to F meeting every leaf. Y ∼ If Y has a taut foliation then � = R 3 and so π 1 ( Y ) is infinite.

Evidence for the conjecture: [Hanselman-Rasmussen 2 -Watson, Boyer-Clay 2015] True for all graph manifolds. [Li-Roberts 2012, Culler-D. 2015] Suppose K ⊂ S 3 and ∆ K ( t ) has a simple root on the unit circle whose complement 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, . . . ]

A few rat’l homology 3-spheres: Evidence for the conjecture: 265,503 hyperbolic Q HSs which are [Hanselman-Rasmussen 2 -Watson, 2-fold branched covers over non-alt Boyer-Clay 2015] True for all graph links in S 3 with ≤ 15 crossings. manifolds. 0.25 µ =7 . 4 Volume µ = 7 . 4 [Li-Roberts 2012, Culler-D. 2015] 0.20 median =7 . 5 σ = 2 . 0 Suppose K ⊂ S 3 and ∆ K ( t ) has a 0.15 σ =2 . 0 simple root on the unit circle whose 0.10 complement is lean. Then there 0.05 exists ǫ > 0 so that the conjecture 0.00 holds for the r Dehn surgery on K 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Volume whenever r ∈ ( − ǫ , ǫ ) . 2.5 Injectivity radius µ =0 . 3 µ = 0 . 3 2.0 median =0 . 3 [Gordon-Lidman, . . . ] σ = 0 . 2 1.5 σ =0 . 2 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 Injectivity radius H-W census has 10,903 Q HSs.

Sample: 265,503 hyperbolic Q HSs. Conjecture holds so far! non-L-sps (27%) L-spaces (73%) order ≥ 3% taut ≥ 24% nonorderable ≥ 44%

Finding 63,977 taut folations. T a 1-vertex triangulation of Y . Def. A laminar orientation of T is: (a) An orientation of the edges where every face is acyclic. (b) Every edge is adjacent to a tet in which it is not very long. (c) The relation on faces has one equiv class. [D. 2015] If Y has a tri with a laminar orient, then Y has a taut foliation.

Finding 63,977 taut folations. T a 1-vertex triangulation of Y . Def. A laminar orientation of T is: (a) An orientation of the edges where every face is acyclic. (b) Every edge is adjacent to a tet in which it is not very long. (c) The relation on faces has one equiv class. [D. 2015] If Y has a tri with a laminar orient, then Y has a taut foliation.

� � � H 1 ( Y ) � increases the odds that Y is an L-space. The pattern: Large � HF ( Y ) µ = 33 . 0 400 400 median = 31 . 3 350 350 σ = 21 . 3 300 300 250 250 HF ( M ) 200 200 0 20 40 60 80 100 120 140 | H 1 ( Y ) | / vol ( Y ) 150 150 L -space density 1 . 0 100 100 0 . 8 50 50 0 . 6 0 . 4 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 | H 1 ( M ) | 0 . 2 | H 1 ( Y ) | 0 . 0 0 10 20 30 40 50 60 | H 1 ( Y ) | / vol ( Y )

Computing � HF : Used [Zhan] which implements the bordered Heegaard Floer homology of [LOT]. Nonordering π 1 ( Y ) : Try to order the ball in the Cayley graph of radius 3-5 in a presentation with many generators. Solved word problem using matrix multiplication. Ordering π 1 ( Y ) : Find reps to � PSL 2 R . Reps to PSL 2 R are plentiful (mean 8 per manifold) but the the Euler class in H 2 ( Y ; Z ) must vanish to lift, so only get 7,382 orderable manifolds from 2.13 million PSL 2 R reps.