L-spaces and left-orderability
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1 1 L-spaces and left-orderability Cameron McA. Gordon (joint with Steve Boyer and Liam Watson) Special Session on Low-Dimensional Topology Tampa, March 2012
2 2 Left Orderability A group G 1 is left orderable (LO) if strict total order < on G such that g < h = fg < fh f G
3 3 Left Orderability A group G 1 is left orderable (LO) if strict total order < on G such that g < h = fg < fh f G R is LO
4 4 Left Orderability A group G 1 is left orderable (LO) if strict total order < on G such that g < h = fg < fh f G R is LO G LO, 1 H < G = H LO
5 5 Left Orderability A group G 1 is left orderable (LO) if strict total order < on G such that g < h = fg < fh f G R is LO G LO, 1 H < G = H LO G LO = G torsion-free
6 6 Left Orderability A group G 1 is left orderable (LO) if strict total order < on G such that g < h = fg < fh f G R is LO G LO, 1 H < G = H LO G LO = G torsion-free G, H LO G H LO (Vinogradov, 1949)
7 7 Left Orderability A group G 1 is left orderable (LO) if strict total order < on G such that g < h = fg < fh f G R is LO G LO, 1 H < G = H LO G LO = G torsion-free G, H LO G H LO (Vinogradov, 1949) G (countable) LO embedding G Homeo + (R)
8 8 Theorem (Boyer-Rolfsen-Wiest, 2005) M a compact, orientable, prime 3-manifold (poss. with boundary). Then π 1 (M) is LO π 1 (M) has an LO quotient.
9 9 Theorem (Boyer-Rolfsen-Wiest, 2005) M a compact, orientable, prime 3-manifold (poss. with boundary). Then π 1 (M) is LO π 1 (M) has an LO quotient. Hence β 1 (M) > 0 π 1 (M) LO
10 10 Theorem (Boyer-Rolfsen-Wiest, 2005) M a compact, orientable, prime 3-manifold (poss. with boundary). Then π 1 (M) is LO π 1 (M) has an LO quotient. Hence β 1 (M) > 0 π 1 (M) LO So interesting case is when H (M; Q) = H (S 3 ; Q) M is a Q-homology 3-sphere (QHS)
11 11 Foliations Suppose M has a (codim 1) co-orientable taut foliation (CTF) F π 1 (M) acts on leaf space L of universal covering of M
12 12 Foliations Suppose M has a (codim 1) co-orientable taut foliation (CTF) F π 1 (M) acts on leaf space L of universal covering of M E.g. if L = R (F is R-covered) then get non-trivial homomorphism π 1 (M) Homeo + (R) π 1 (M) is LO
13 13 Foliations Suppose M has a (codim 1) co-orientable taut foliation (CTF) F π 1 (M) acts on leaf space L of universal covering of M E.g. if L = R (F is R-covered) then get non-trivial homomorphism π 1 (M) Homeo + (R) π 1 (M) is LO Theorem (Calegari-Dunfield, 2003) M a prime, atoroidal QHS with a CTF, M the universal abelian cover of M. Then π 1 ( M) is LO.
14 14 Foliations Suppose M has a (codim 1) co-orientable taut foliation (CTF) F π 1 (M) acts on leaf space L of universal covering of M E.g. if L = R (F is R-covered) then get non-trivial homomorphism π 1 (M) Homeo + (R) π 1 (M) is LO Theorem (Calegari-Dunfield, 2003) M a prime, atoroidal QHS with a CTF, M the universal abelian cover of M. Then π 1 ( M) is LO. Thurston s universal circle construction gives ρ : π 1 (M) Homeo + (S 1 )
15 15 Foliations Suppose M has a (codim 1) co-orientable taut foliation (CTF) F π 1 (M) acts on leaf space L of universal covering of M E.g. if L = R (F is R-covered) then get non-trivial homomorphism π 1 (M) Homeo + (R) π 1 (M) is LO Theorem (Calegari-Dunfield, 2003) M a prime, atoroidal QHS with a CTF, M the universal abelian cover of M. Then π 1 ( M) is LO. Thurston s universal circle construction gives ρ : π 1 (M) Homeo + (S 1 ) Restriction of ρ to π 1 ( M) lifts to Homeo + (S 1 ) Homeo + (R)
16 16 Heegaard Floer Homology (Ozsváth-Szabó) M a QHS. ĤF(M) : finite dimensional Z 2 -vector space
17 17 Heegaard Floer Homology (Ozsváth-Szabó) M a QHS. ĤF(M) : finite dimensional Z 2 -vector space dim ĤF(M) H 1 (M)
18 18 Heegaard Floer Homology (Ozsváth-Szabó) M a QHS. ĤF(M) : finite dimensional Z 2 -vector space dim ĤF(M) H 1 (M) M is an L-space if equality holds E.g. lens spaces are L-spaces
19 19 Heegaard Floer Homology (Ozsváth-Szabó) M a QHS. ĤF(M) : finite dimensional Z 2 -vector space dim ĤF(M) H 1 (M) M is an L-space if equality holds E.g. lens spaces are L-spaces Is there a non-heegaard Floer characterization of L-spaces?
20 20 Heegaard Floer Homology (Ozsváth-Szabó) M a QHS. ĤF(M) : finite dimensional Z 2 -vector space dim ĤF(M) H 1 (M) M is an L-space if equality holds E.g. lens spaces are L-spaces Is there a non-heegaard Floer characterization of L-spaces? Conjecture M a prime QHS. Then M is an L-space π 1 (M) is not LO
21 21 M not an L-space (Ozsváth-Szabó, 2004) M has an R-covered CTF (Fenley-Roberts, 2012) M has a CTF (if M atoroidal) π 1 (M) LO π 1 (M) virtually LO
22 22 (A) Seifert manifolds Theorem The Conjecture is true if M is Seifert fibered.
23 23 (A) Seifert manifolds Theorem The Conjecture is true if M is Seifert fibered. Base orbifold is either S 2 (a 1,..., a n ) :
24 24 (A) Seifert manifolds Theorem The Conjecture is true if M is Seifert fibered. Base orbifold is either S 2 (a 1,..., a n ) : M does not admit a horizontal foliation (Lisca-Stipsicz, 2007) (BRW, 2005) M an L-space π 1 (M) not LO
25 25 P 2 (a 1,..., a n ) : π 1 (M) not LO (BRW, 2005)
26 26 P 2 (a 1,..., a n ) : π 1 (M) not LO (BRW, 2005) Show M an L-space by induction on n; using (OS, 2005): X compact, orientable 3-manifold, X a torus; α,β X, α β = 1, and ( ) ( ) ( ) H 1 X(α + β) = H1 X(α) + H1 X(β) Then X(α), X(β) L-spaces X(α + β) L-space ( )
27 27 (B) Graph manifolds = union of Seifert fibered spaces along tori
28 28 (B) Graph manifolds = union of Seifert fibered spaces along tori Theorem (Clay-Lidman-Watson, 2011) M a ZHS graph manifold. Then π 1 (M) is LO.
29 29 (B) Graph manifolds = union of Seifert fibered spaces along tori Theorem (Clay-Lidman-Watson, 2011) M a ZHS graph manifold. Then π 1 (M) is LO. Theorem (Boileau-Boyer, 2011) M a ZHS graph manifold. Then M admits a CTF, horizontal in every Seifert piece. Hence M is not a L-space.
30 30 (B) Graph manifolds = union of Seifert fibered spaces along tori Theorem (Clay-Lidman-Watson, 2011) M a ZHS graph manifold. Then π 1 (M) is LO. Theorem (Boileau-Boyer, 2011) M a ZHS graph manifold. Then M admits a CTF, horizontal in every Seifert piece. Hence M is not a L-space. So Conjecture true for ZHS graph manifolds
31 31 (C) Sol manifolds N = twisted I-bundle/Klein bottle N has two Seifert structures: base Möbius band; fiber ϕ 0 base D 2 (2, 2); fiber ϕ 1
32 32 (C) Sol manifolds N = twisted I-bundle/Klein bottle N has two Seifert structures: base Möbius band; fiber ϕ 0 base D 2 (2, 2); fiber ϕ 1 ϕ 0 ϕ 1 = 1 on N f : N N homeomorphism, M(f) = N f N [ ] a b f = ; M(f) QHS = c 0 c d
33 33 (C) Sol manifolds N = twisted I-bundle/Klein bottle N has two Seifert structures: base Möbius band; fiber ϕ 0 base D 2 (2, 2); fiber ϕ 1 ϕ 0 ϕ 1 = 1 on N f : N N homeomorphism, M(f) = N f N [ ] a b f = ; M(f) QHS = c 0 c d M(f) Seifert f(ϕ i ) = ±ϕ j (some i, j {0, 1}) Otherwise, M(f) a Sol manifold
34 34 (C) Sol manifolds N = twisted I-bundle/Klein bottle N has two Seifert structures: base Möbius band; fiber ϕ 0 base D 2 (2, 2); fiber ϕ 1 ϕ 0 ϕ 1 = 1 on N f : N N homeomorphism, M(f) = N f N [ ] a b f = ; M(f) QHS = c 0 c d M(f) Seifert f(ϕ i ) = ±ϕ j (some i, j {0, 1}) Otherwise, M(f) a Sol manifold π 1 (M(f)) not LO (BRW, 2005)
35 35 Theorem M is an L-space
36 36 Theorem M is an L-space (1) True if f = [ ] a : f(ϕ 1 ) = ϕ 0, so M(f) Seifert
37 37 Theorem M is an L-space [ ] a 1 (1) True if f = : f(ϕ ) = ϕ 0, so M(f) Seifert [ ] [ ][ ] [ ] a b a 1 1 d a 1 (2) True if f = = = (t 1 d ) d where t 0 : N N is Dehn twist along ϕ 0 Bordered ĤF calculation shows ĤF(M(f)) = ĤF(M(f t 0)) So reduced to case (1)
38 38 Theorem M is an L-space [ ] a 1 (1) True if f = : f(ϕ ) = ϕ 0, so M(f) Seifert [ ] [ ][ ] [ ] a b a 1 1 d a 1 (2) True if f = = = (t 1 d ) d where t 0 : N N is Dehn twist along ϕ 0 Bordered ĤF calculation shows ĤF(M(f)) = ĤF(M(f t 0)) So reduced to case (1) (3) In general, induct on c : do surgery on suitable simple closed curves N and use ( )
39 39 Theorem M is an L-space [ ] a 1 (1) True if f = : f(ϕ ) = ϕ 0, so M(f) Seifert [ ] [ ][ ] [ ] a b a 1 1 d a 1 (2) True if f = = = (t 1 d ) d where t 0 : N N is Dehn twist along ϕ 0 Bordered ĤF calculation shows ĤF(M(f)) = ĤF(M(f t 0)) So reduced to case (1) (3) In general, induct on c : do surgery on suitable simple closed curves N and use ( ) Hence: Conjecture is true for all non-hyperbolic geometric manifolds.
40 40 (D) Dehn surgery
41 41 (D) Dehn surgery Theorem (OS, 2005) K a hyperbolic alternating knot. Then K(r) is not an L-space r Q
42 42 (D) Dehn surgery Theorem (OS, 2005) K a hyperbolic alternating knot. Then K(r) is not an L-space r Q So Conjecture = π 1 (K(r)) LO
43 43 (D) Dehn surgery Theorem (OS, 2005) K a hyperbolic alternating knot. Then K(r) is not an L-space r Q So Conjecture = π 1 (K(r)) LO Theorem (Roberts, 1995) K an alternating knot. Then K(r) has a CTF r Q if K is not special, and either r > 0 or r < 0 if K is special.
44 44 (D) Dehn surgery Theorem (OS, 2005) K a hyperbolic alternating knot. Then K(r) is not an L-space r Q So Conjecture = π 1 (K(r)) LO Theorem (Roberts, 1995) K an alternating knot. Then K(r) has a CTF r Q if K is not special, and either r > 0 or r < 0 if K is special. Corollary The Conjecture is true for K(1/q), K alternating, either q Q, or q > 0 or q < 0.
45 45 Theorem Let K be the figure eight knot. Then π 1 (K(r)) is LO for 4 < r < 4.
46 46 Theorem Let K be the figure eight knot. Then π 1 (K(r)) is LO for 4 < r < 4. Uses special representations ρ : π 1 (S 3 \ K) PSL 2 (R)
47 47 Theorem Let K be the figure eight knot. Then π 1 (K(r)) is LO for 4 < r < 4. Uses special representations ρ : π 1 (S 3 \ K) PSL 2 (R) Also true for r Z (Fenley-Roberts)
48 48 Theorem Let K be the figure eight knot. Then π 1 (K(r)) is LO for 4 < r < 4. Uses special representations ρ : π 1 (S 3 \ K) PSL 2 (R) Also true for r Z (Fenley-Roberts) ( ) implies K a knot in S 3, if K(s) an L-space for some s Q, s > 0, then K(r) an L-space for all r 2g(K) 1 (K is an L-space knot)
49 49 Theorem Let K be the figure eight knot. Then π 1 (K(r)) is LO for 4 < r < 4. Uses special representations ρ : π 1 (S 3 \ K) PSL 2 (R) Also true for r Z (Fenley-Roberts) ( ) implies K a knot in S 3, if K(s) an L-space for some s Q, s > 0, then K(r) an L-space for all r 2g(K) 1 (K is an L-space knot) So Conjecture = π 1 (K(r)) not LO, r 2g(K) 1
50 50 Theorem Let K be the figure eight knot. Then π 1 (K(r)) is LO for 4 < r < 4. Uses special representations ρ : π 1 (S 3 \ K) PSL 2 (R) Also true for r Z (Fenley-Roberts) ( ) implies K a knot in S 3, if K(s) an L-space for some s Q, s > 0, then K(r) an L-space for all r 2g(K) 1 (K is an L-space knot) So Conjecture = π 1 (K(r)) not LO, r 2g(K) 1 some results in this direction (Clay-Teragaito; Clay-Watson)
51 51 (E) 2-fold branched covers L a link in S 3 Σ(L) =2-fold branched cover of L
52 52 (E) 2-fold branched covers L a link in S 3 Σ(L) =2-fold branched cover of L Theorem (OS, 2005) If L is a non-split alternating link then Σ(L) is an L-space.
53 53 8L L0 L (E) 2-fold branched covers L a link in S 3 Σ(L) =2-fold branched cover of L Theorem (OS, 2005) If L is a non-split alternating link then Σ(L) is an L-space. (uses ( ) ; = Σ(L), Σ(L 0 ), Σ(L ) a surgery triad with det L = det L 0 + det L )
54 54 8L L0 L (E) 2-fold branched covers L a link in S 3 Σ(L) =2-fold branched cover of L Theorem (OS, 2005) If L is a non-split alternating link then Σ(L) is an L-space. (uses ( ) ; = Σ(L), Σ(L 0 ), Σ(L ) a surgery triad Theorem with det L = det L 0 + det L ) If L is a non-split alternating link then π 1 (Σ(L)) is not LO.
55 55 8L L0 L (E) 2-fold branched covers L a link in S 3 Σ(L) =2-fold branched cover of L Theorem (OS, 2005) If L is a non-split alternating link then Σ(L) is an L-space. (uses ( ) ; = Σ(L), Σ(L 0 ), Σ(L ) a surgery triad Theorem with det L = det L 0 + det L ) If L is a non-split alternating link then π 1 (Σ(L)) is not LO. (Also proofs by Greene, Ito)
56 56 (F) Questions Question 1 If M is a QHS with a CTF, is π 1 (M) LO?
57 57 (F) Questions Question 1 If M is a QHS with a CTF, is π 1 (M) LO? Question 2 If K is a hyperbolic alternating knot, is π 1 (K(r)) LO r Q?
58 58 (F) Questions Question 1 If M is a QHS with a CTF, is π 1 (M) LO? Question 2 If K is a hyperbolic alternating knot, is π 1 (K(r)) LO r Q? L quasi-alternating = Σ(L) an L-space
59 59 (F) Questions Question 1 If M is a QHS with a CTF, is π 1 (M) LO? Question 2 If K is a hyperbolic alternating knot, is π 1 (K(r)) LO r Q? L quasi-alternating = Σ(L) an L-space Question 3 Does L quasi-alternating = π 1 (Σ(L)) not LO?
60 60 (F) Questions Question 1 If M is a QHS with a CTF, is π 1 (M) LO? Question 2 If K is a hyperbolic alternating knot, is π 1 (K(r)) LO r Q? L quasi-alternating = Σ(L) an L-space Question 3 Does L quasi-alternating = π 1 (Σ(L)) not LO? Conjecture = Q s 1, 2 and 3 have answer yes
61 61 Question 4 If M is a QHS with π 1 (M) LO does M admit a CTF? (Maybe no?)
62 62 Question 4 If M is a QHS with π 1 (M) LO does M admit a CTF? (Maybe no?) Question 5 Does the Conjecture hold for graph manifolds?
63 63 Question 4 If M is a QHS with π 1 (M) LO does M admit a CTF? (Maybe no?) Question 5 Does the Conjecture hold for graph manifolds? Only known prime ZHS L-spaces are S 3 and Poincaré HS
64 64 Question 4 If M is a QHS with π 1 (M) LO does M admit a CTF? (Maybe no?) Question 5 Does the Conjecture hold for graph manifolds? Question 6 Only known prime ZHS L-spaces are S 3 and Poincaré HS M a hyperbolic ZHS. Is π 1 (M) LO?
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