Floer homology and covering spaces

Ciprian Manolescu

(joint work with Tye Lidman)

June 6, 2016

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 1 / 19

Floer homologies

Y 3 closed oriented, s ∈ Spinc(Y )

(Ozsvath-Szabo, 2001) Heegaard Floer homology:

HF+(Y , s), HF (Y , s), HF red(Y , s)

(Kronheimer-Mrowka, 2007) Monopole Floer homology:HM(Y , s), HM(Y , s) = Cone(U : HM → HM) (cf. Bloom)

(M., 2001) S1-equivariant Seiberg-Witten Floer spectrumSWF(Y , s), for b1(Y ) = 0; gives HS1

∗ (SWF(Y , s)), H∗(SWF(Y , s))

Suspension spectrum: ΣnX where n ∈ Q and X is a topological space.

H∗(ΣnX ) = H∗+n(X ).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 2 / 19

Floer homologies

Y 3 closed oriented, s ∈ Spinc(Y )

(Ozsvath-Szabo, 2001) Heegaard Floer homology:

HF+(Y , s), HF (Y , s), HF red(Y , s)

(Kronheimer-Mrowka, 2007) Monopole Floer homology:HM(Y , s), HM(Y , s) = Cone(U : HM → HM) (cf. Bloom)

(M., 2001) S1-equivariant Seiberg-Witten Floer spectrumSWF(Y , s), for b1(Y ) = 0; gives HS1

∗ (SWF(Y , s)), H∗(SWF(Y , s))

Suspension spectrum: ΣnX where n ∈ Q and X is a topological space.

H∗(ΣnX ) = H∗+n(X ).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 2 / 19

Floer homologies

Y 3 closed oriented, s ∈ Spinc(Y )

(Ozsvath-Szabo, 2001) Heegaard Floer homology:

HF+(Y , s), HF (Y , s), HF red(Y , s)

(Kronheimer-Mrowka, 2007) Monopole Floer homology:HM(Y , s), HM(Y , s) = Cone(U : HM → HM) (cf. Bloom)

(M., 2001) S1-equivariant Seiberg-Witten Floer spectrumSWF(Y , s), for b1(Y ) = 0; gives HS1

∗ (SWF(Y , s)), H∗(SWF(Y , s))

Suspension spectrum: ΣnX where n ∈ Q and X is a topological space.

H∗(ΣnX ) = H∗+n(X ).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 2 / 19

Equivalences

Heegaard Floer / Seiberg-Witten equivalence:

Theorem (Kutluhan-Lee-Taubes, Colin-Ghiggini-Honda, 2012)

If Y is any three-manifold and s ∈ Spinc(Y ), we have

HF+(Y , s) ∼= HM(Y , s), HF (Y , s) ∼= HM(Y , s)

Equivalence between the two Seiberg-Witten Floer homologies:

Theorem (Lidman-M., 2016)

If Y is a rational homology sphere and s ∈ Spinc(Y ), then

HS1

∗ (SWF(Y , s)) ∼= HM∗(Y , s), H∗(SWF(Y , s)) ∼= HM∗(Y , s)

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 3 / 19

Equivalences

Heegaard Floer / Seiberg-Witten equivalence:

Theorem (Kutluhan-Lee-Taubes, Colin-Ghiggini-Honda, 2012)

If Y is any three-manifold and s ∈ Spinc(Y ), we have

HF+(Y , s) ∼= HM(Y , s), HF (Y , s) ∼= HM(Y , s)

Equivalence between the two Seiberg-Witten Floer homologies:

Theorem (Lidman-M., 2016)

If Y is a rational homology sphere and s ∈ Spinc(Y ), then

HS1

∗ (SWF(Y , s)) ∼= HM∗(Y , s), H∗(SWF(Y , s)) ∼= HM∗(Y , s)

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 3 / 19

Outline

Typical applications of Floer theory: surgery questions, e.g.

S3r (K ) = S3

r (U)⇒ K = U (cf. Kronheimer-Mrowka-Ozsvath-Szabo)

Goals of this talk:

Relate Floer homology to covering spaces

Relate L-spaces to covering spaces

Relate covering spaces to surgery (topological applications)

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 4 / 19

Outline

Typical applications of Floer theory: surgery questions, e.g.

S3r (K ) = S3

r (U)⇒ K = U (cf. Kronheimer-Mrowka-Ozsvath-Szabo)

Goals of this talk:

Relate Floer homology to covering spaces

Relate L-spaces to covering spaces

Relate covering spaces to surgery (topological applications)

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 4 / 19

Our main results

Theorem (Lidman-M., 2016)

Suppose that Y is a rational homology sphere, Y is orientable, andπ : Y → Y is a pn-sheeted regular covering, for p prime. Let s be a Spinc

structure on Y . Then, the following inequality holds:∑i

dim Hi (SWF(Y , s);Z/p) ≤∑i

dim Hi (SWF(Y ), π∗s;Z/p).

Corollary: dim HF (Y , s;Z/p) ≤ dim HF (Y , π∗s;Z/p).

Theorem (Lidman-M., 2016)

Under the same assumptions,

dim HF red(Y , s;Z/p) ≤ dim HF red(Y , π∗s;Z/p).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 5 / 19

Our main results

Theorem (Lidman-M., 2016)

Suppose that Y is a rational homology sphere, Y is orientable, andπ : Y → Y is a pn-sheeted regular covering, for p prime. Let s be a Spinc

structure on Y . Then, the following inequality holds:∑i

dim Hi (SWF(Y , s);Z/p) ≤∑i

dim Hi (SWF(Y ), π∗s;Z/p).

Corollary: dim HF (Y , s;Z/p) ≤ dim HF (Y , π∗s;Z/p).

Theorem (Lidman-M., 2016)

Under the same assumptions,

dim HF red(Y , s;Z/p) ≤ dim HF red(Y , π∗s;Z/p).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 5 / 19

Related work

Hendricks (2011):

dim HFK (S3,K ;Z/2) ≤ dim HFK (D(K ), K ;Z/2)

where D(K )→ K is the double cover branched over the knot K .

Hendricks (2012): similar inequality of dim HFK for 2-periodic knots

Lipshitz-Treumann (2011): If π : Y → Y is a 2 : 1 cover coming from amap f : Y → S1 such that at least one fiber f −1(p) has genus ≤ 2, then

rk HF (Y , s) ≤ 2 rk HF (Y , π∗s).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 6 / 19

Related work

Hendricks (2011):

dim HFK (S3,K ;Z/2) ≤ dim HFK (D(K ), K ;Z/2)

where D(K )→ K is the double cover branched over the knot K .

Hendricks (2012): similar inequality of dim HFK for 2-periodic knots

Lipshitz-Treumann (2011): If π : Y → Y is a 2 : 1 cover coming from amap f : Y → S1 such that at least one fiber f −1(p) has genus ≤ 2, then

rk HF (Y , s) ≤ 2 rk HF (Y , π∗s).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 6 / 19

Related work

Hendricks (2011):

dim HFK (S3,K ;Z/2) ≤ dim HFK (D(K ), K ;Z/2)

where D(K )→ K is the double cover branched over the knot K .

Hendricks (2012): similar inequality of dim HFK for 2-periodic knots

Lipshitz-Treumann (2011): If π : Y → Y is a 2 : 1 cover coming from amap f : Y → S1 such that at least one fiber f −1(p) has genus ≤ 2, then

rk HF (Y , s) ≤ 2 rk HF (Y , π∗s).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 6 / 19

Origin of these results

Theorem (P. Smith, 1938)

Suppose that a group G of order pn (where p is prime) acts on a compacttopological space X , and let XG denote the fixed point set. The mod pBetti numbers of X and XG are then related by:∑

i

dimHi (XG ;Z/pZ) ≤

∑i

dimHi (X ;Z/pZ). (1)

Seidel and I. Smith (2010) proved an analogue of this in LagrangianFloer homology, for Z/2 actions such that the Lagrangians have a stablenormal trivialization.

This is used in Hendricks’ results.

Stable normal trivialization: genus (Heegaard diagram) = 0 or 1. This isOK for knots in S3.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 7 / 19

Origin of these results

Theorem (P. Smith, 1938)

Suppose that a group G of order pn (where p is prime) acts on a compacttopological space X , and let XG denote the fixed point set. The mod pBetti numbers of X and XG are then related by:∑

i

dimHi (XG ;Z/pZ) ≤

∑i

dimHi (X ;Z/pZ). (1)

Seidel and I. Smith (2010) proved an analogue of this in LagrangianFloer homology, for Z/2 actions such that the Lagrangians have a stablenormal trivialization.

This is used in Hendricks’ results.

Stable normal trivialization: genus (Heegaard diagram) = 0 or 1. This isOK for knots in S3.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 7 / 19

Avoiding transversality

One hopes to prove the inequality dim HF (Y , s) ≤ dim HF (Y , s) for 2-foldbranched covers using the same ideas.

However, there is no stable normal trivialization. Similar problems withequivariant transversality appear with HM.

The construction of SWF(Y , s) does not require perturbing theSeiberg-Witten equations to achieve transversality. Instead, one considersthe configuration space

V = i ker d∗ ⊕ Γ(W ) ⊂ iΩ1(Y )⊕ Γ(W )

We can write the Seiberg-Witten map as ∇CSD = l + c : V → V , where

l(a, φ) = (∗da, /Dφ)

c(a, φ) = (τ(φ, φ), a · φ)

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 8 / 19

Avoiding transversality

One hopes to prove the inequality dim HF (Y , s) ≤ dim HF (Y , s) for 2-foldbranched covers using the same ideas.

However, there is no stable normal trivialization. Similar problems withequivariant transversality appear with HM.

The construction of SWF(Y , s) does not require perturbing theSeiberg-Witten equations to achieve transversality. Instead, one considersthe configuration space

V = i ker d∗ ⊕ Γ(W ) ⊂ iΩ1(Y )⊕ Γ(W )

We can write the Seiberg-Witten map as ∇CSD = l + c : V → V , where

l(a, φ) = (∗da, /Dφ)

c(a, φ) = (τ(φ, φ), a · φ)

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 8 / 19

Avoiding transversality

One hopes to prove the inequality dim HF (Y , s) ≤ dim HF (Y , s) for 2-foldbranched covers using the same ideas.

However, there is no stable normal trivialization. Similar problems withequivariant transversality appear with HM.

The construction of SWF(Y , s) does not require perturbing theSeiberg-Witten equations to achieve transversality. Instead, one considersthe configuration space

V = i ker d∗ ⊕ Γ(W ) ⊂ iΩ1(Y )⊕ Γ(W )

We can write the Seiberg-Witten map as ∇CSD = l + c : V → V , where

l(a, φ) = (∗da, /Dφ)

c(a, φ) = (τ(φ, φ), a · φ)

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 8 / 19

Finite dimensional approximation (M.)

We approximate V by finite dimensional vector spaces

V µλ = ⊕

(eigenspaces of l between λ and µ

)for µ 0 λ. Let pµλ = projection to V µ

λ .

Instead of the Seiberg-Witten flow on V , we consider the flow of l + pµλcon V µ

λ .

This has an associated Conley index Iµλ = N/L, where N is a large compactneighborhood of all finite trajectories and L ⊂ ∂N is the exit set of theflow from N. (If the flow of l + pµλc is Morse-Smale, then the homology ofIµλ is Morse homology.) However, we do not require transversality.

The Floer spectrum is defined as

SWF(Y , s) = Σ− dimV 0λ Iµλ .

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 9 / 19

Finite dimensional approximation (M.)

We approximate V by finite dimensional vector spaces

V µλ = ⊕

(eigenspaces of l between λ and µ

)for µ 0 λ. Let pµλ = projection to V µ

λ .

Instead of the Seiberg-Witten flow on V , we consider the flow of l + pµλcon V µ

λ .

This has an associated Conley index Iµλ = N/L, where N is a large compactneighborhood of all finite trajectories and L ⊂ ∂N is the exit set of theflow from N. (If the flow of l + pµλc is Morse-Smale, then the homology ofIµλ is Morse homology.) However, we do not require transversality.

The Floer spectrum is defined as

SWF(Y , s) = Σ− dimV 0λ Iµλ .

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 9 / 19

Finite dimensional approximation (M.)

We approximate V by finite dimensional vector spaces

V µλ = ⊕

(eigenspaces of l between λ and µ

)for µ 0 λ. Let pµλ = projection to V µ

λ .

Instead of the Seiberg-Witten flow on V , we consider the flow of l + pµλcon V µ

λ .

This has an associated Conley index Iµλ = N/L, where N is a large compactneighborhood of all finite trajectories and L ⊂ ∂N is the exit set of theflow from N. (If the flow of l + pµλc is Morse-Smale, then the homology ofIµλ is Morse homology.) However, we do not require transversality.

The Floer spectrum is defined as

SWF(Y , s) = Σ− dimV 0λ Iµλ .

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 9 / 19

Proof of the theorem

Let π : Y → Y be a regular pn-fold cover with automorphism group G .

A metric g on Y gives a pulled-back metric g on Y . We have:(iΩ1(Y )⊕ Γ(W )

)G= iΩ1(Y )⊕ Γ(W )

with ‖π∗x‖ = |G | · ‖x‖.

From here we get (V )G = V , then (V µλ )G = V µ

λ and (Iµλ )G = Iµλ .

Applying the finite dimensional Smith inequality to Iµλ and Iµλ , we obtain∑i

dim Hi (SWF(Y , s);Z/p) ≤∑i

dim Hi (SWF(Y , π∗s);Z/p).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 10 / 19

Proof of the theorem

Let π : Y → Y be a regular pn-fold cover with automorphism group G .

A metric g on Y gives a pulled-back metric g on Y . We have:(iΩ1(Y )⊕ Γ(W )

)G= iΩ1(Y )⊕ Γ(W )

with ‖π∗x‖ = |G | · ‖x‖.

From here we get (V )G = V , then (V µλ )G = V µ

λ and (Iµλ )G = Iµλ .

Applying the finite dimensional Smith inequality to Iµλ and Iµλ , we obtain∑i

dim Hi (SWF(Y , s);Z/p) ≤∑i

dim Hi (SWF(Y , π∗s);Z/p).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 10 / 19

Proof of the theorem

Let π : Y → Y be a regular pn-fold cover with automorphism group G .

A metric g on Y gives a pulled-back metric g on Y . We have:(iΩ1(Y )⊕ Γ(W )

)G= iΩ1(Y )⊕ Γ(W )

with ‖π∗x‖ = |G | · ‖x‖.

From here we get (V )G = V , then (V µλ )G = V µ

λ and (Iµλ )G = Iµλ .

Applying the finite dimensional Smith inequality to Iµλ and Iµλ , we obtain∑i

dim Hi (SWF(Y , s);Z/p) ≤∑i

dim Hi (SWF(Y , π∗s);Z/p).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 10 / 19

L-spaces

Y is called an L-space if HF (Y , s) ∼= Z for all s. (This implies b1(Y ) = 0.)

Examples: lens spaces, elliptic manifolds, double branched covers overquasi-alternating links, large surgeries on L-space knots.

Properties of L-spaces

If Y is an L-space, then:

Y has no taut foliations,

No contact structure on Y has a symplectic filling with b+2 > 0,

If a 4-manifold X can be written as X1 ∪Y X2 withb+2 (Xi ) > 0, i = 1, 2, then X has trivial Seiberg-Witten invariants.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 11 / 19

L-spaces

Y is called an L-space if HF (Y , s) ∼= Z for all s. (This implies b1(Y ) = 0.)

Examples: lens spaces, elliptic manifolds, double branched covers overquasi-alternating links, large surgeries on L-space knots.

Properties of L-spaces

If Y is an L-space, then:

Y has no taut foliations,

No contact structure on Y has a symplectic filling with b+2 > 0,

If a 4-manifold X can be written as X1 ∪Y X2 withb+2 (Xi ) > 0, i = 1, 2, then X has trivial Seiberg-Witten invariants.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 11 / 19

L-spaces

Y is called an L-space if HF (Y , s) ∼= Z for all s. (This implies b1(Y ) = 0.)

Examples: lens spaces, elliptic manifolds, double branched covers overquasi-alternating links, large surgeries on L-space knots.

Properties of L-spaces

If Y is an L-space, then:

Y has no taut foliations,

No contact structure on Y has a symplectic filling with b+2 > 0,

If a 4-manifold X can be written as X1 ∪Y X2 withb+2 (Xi ) > 0, i = 1, 2, then X has trivial Seiberg-Witten invariants.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 11 / 19

Open problem

Question (folklore, or Boyer-Gordon-Watson + Juhasz)

For closed, orientable, irreducible 3-manifolds, is it true that:

Y is an L-space ⇐⇒ Y has no taut foliations⇐⇒ π1(Y ) is not left-orderable ?

This is true for Seifert and Sol manifolds, by work of Lisca-Stipsicz andBoyer-Gordon-Watson; and in fact for all graph manifolds, by work ofBoyer-Clay and Hanselman-J.Rasmussen-S.Rasmussen-Watson.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 12 / 19

Open problem

Question (folklore, or Boyer-Gordon-Watson + Juhasz)

For closed, orientable, irreducible 3-manifolds, is it true that:

Y is an L-space ⇐⇒ Y has no taut foliations⇐⇒ π1(Y ) is not left-orderable ?

This is true for Seifert and Sol manifolds, by work of Lisca-Stipsicz andBoyer-Gordon-Watson; and in fact for all graph manifolds, by work ofBoyer-Clay and Hanselman-J.Rasmussen-S.Rasmussen-Watson.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 12 / 19

Behavior under covers

Suppose that Y → Y is a covering between compact 3-manifolds. Then:

If Y has a co-orientable taut foliation, then so does Y ;

If π1(Y ) admits a left-ordering, then so does it subgroup π1(Y ).

Question

If π : Y → Y is a covering map, Y is orientable, and Y is an L-space,does Y have to be an L-space?

Remark: There are covers of L-spaces that are not L-spaces, for example

M(−2; 2/3, 2/3, 1/2, 1/2)→ M(−1; 2/3, 1/4, 1/4).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 13 / 19

Behavior under covers

Suppose that Y → Y is a covering between compact 3-manifolds. Then:

If Y has a co-orientable taut foliation, then so does Y ;

If π1(Y ) admits a left-ordering, then so does it subgroup π1(Y ).

Question

If π : Y → Y is a covering map, Y is orientable, and Y is an L-space,does Y have to be an L-space?

Remark: There are covers of L-spaces that are not L-spaces, for example

M(−2; 2/3, 2/3, 1/2, 1/2)→ M(−1; 2/3, 1/4, 1/4).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 13 / 19

Behavior under covers

Suppose that Y → Y is a covering between compact 3-manifolds. Then:

If Y has a co-orientable taut foliation, then so does Y ;

If π1(Y ) admits a left-ordering, then so does it subgroup π1(Y ).

Question

If π : Y → Y is a covering map, Y is orientable, and Y is an L-space,does Y have to be an L-space?

Remark: There are covers of L-spaces that are not L-spaces, for example

M(−2; 2/3, 2/3, 1/2, 1/2)→ M(−1; 2/3, 1/4, 1/4).

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 13 / 19

Results

Our inequality for the Floer homology of covers implies:

Corollary (Lidman-M.)

Suppose that Y is a rational homology sphere, Y is orientable, andπ : Y → Y is a pn-sheeted regular covering, for p prime. If Y is aZ/p-L-space (i.e. HF (Y , s;Z/p) ∼= Z/p, ∀s), then so is Y .

In principle, being a Z/p-L-space is weaker than being a L-space.

However, in practice no torsion has ever been observed in Floer homologyfor rational homology spheres.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 14 / 19

Results

Our inequality for the Floer homology of covers implies:

Corollary (Lidman-M.)

Suppose that Y is a rational homology sphere, Y is orientable, andπ : Y → Y is a pn-sheeted regular covering, for p prime. If Y is aZ/p-L-space (i.e. HF (Y , s;Z/p) ∼= Z/p, ∀s), then so is Y .

In principle, being a Z/p-L-space is weaker than being a L-space.

However, in practice no torsion has ever been observed in Floer homologyfor rational homology spheres.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 14 / 19

More on covers

If we assume no torsion in HF for b1 = 0, then:

Suppose π : Y → Y is a regular covering of orientable 3-manifolds withAut(π) solvable. Then:

Y is an L-space ⇒ Y is an L-space

Question

If π : Y → Y is a cover and s ∈ Spinc(Y ), is it true that

rk HF (Y , s) ≤ rk HF (Y , π∗s) ?

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 15 / 19

More on covers

If we assume no torsion in HF for b1 = 0, then:

Suppose π : Y → Y is a regular covering of orientable 3-manifolds withAut(π) solvable. Then:

Y is an L-space ⇒ Y is an L-space

Question

If π : Y → Y is a cover and s ∈ Spinc(Y ), is it true that

rk HF (Y , s) ≤ rk HF (Y , π∗s) ?

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 15 / 19

More on covers

If we assume no torsion in HF for b1 = 0, then:

Suppose π : Y → Y is a regular covering of orientable 3-manifolds withAut(π) solvable. Then:

Y is an L-space ⇒ Y is an L-space

Question

If π : Y → Y is a cover and s ∈ Spinc(Y ), is it true that

rk HF (Y , s) ≤ rk HF (Y , π∗s) ?

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 15 / 19

Covers vs. surgery

Let S3r (K ) = r -surgery on K ⊂ S3, for r ∈ Q

Ozsvath-Szabo (2004): HF (S3r (K )) can be expressed in terms of

CFK∞(K ) and r ; cf. also Jabuka.

This, combined with our inequalities for covers, gives constraints on whichS3r (K ) can cover S3

r ′(K′).

Here are some examples...

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 16 / 19

Covers vs. surgery

Let S3r (K ) = r -surgery on K ⊂ S3, for r ∈ Q

Ozsvath-Szabo (2004): HF (S3r (K )) can be expressed in terms of

CFK∞(K ) and r ; cf. also Jabuka.

This, combined with our inequalities for covers, gives constraints on whichS3r (K ) can cover S3

r ′(K′).

Here are some examples...

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 16 / 19

Covers vs. surgery

Let S3r (K ) = r -surgery on K ⊂ S3, for r ∈ Q

Ozsvath-Szabo (2004): HF (S3r (K )) can be expressed in terms of

CFK∞(K ) and r ; cf. also Jabuka.

This, combined with our inequalities for covers, gives constraints on whichS3r (K ) can cover S3

r ′(K′).

Here are some examples...

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 16 / 19

Constraints on surgeries and covers

Theorem (Lidman-M.)

Let K and K ′ be non-trivial Z/rZ-L-space knots and p, p′, q, q′ positiveintegers satisfying

(2g(K )− 1)dq/pe < (2g(K ′)− 1)bq′/p′c,

then S3p/q(K ) is not an rn-sheeted regular covering of S3

p′/q′(K′) for any

prime r .

Theorem (Lidman-M.)

Let K be a hyperbolic alternating knot and let L be any alternating link.Then, the double branched cover Σ(L) is not an rn-sheeted regular coverof S3

p/q(K ) for any prime r .

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 17 / 19

Constraints on surgeries and covers

Theorem (Lidman-M.)

Let K and K ′ be non-trivial Z/rZ-L-space knots and p, p′, q, q′ positiveintegers satisfying

(2g(K )− 1)dq/pe < (2g(K ′)− 1)bq′/p′c,

then S3p/q(K ) is not an rn-sheeted regular covering of S3

p′/q′(K′) for any

prime r .

Theorem (Lidman-M.)

Let K be a hyperbolic alternating knot and let L be any alternating link.Then, the double branched cover Σ(L) is not an rn-sheeted regular coverof S3

p/q(K ) for any prime r .

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 17 / 19

Virtually cosmetic surgeries

Question

For what knots K ⊂ S3 do there exist pairs of surgery coefficients pq 6=

p′

q′

such that S3p/q(K ) is a cover of S3

p′/q′(K )?

Compare: cosmetic surgeries are those such that S3p/q(K ) = S3

p′/q′(K ).

Theorem (Lidman-M.)

Let K be a non-trivial knot in S3 and let p, q, p′, q′ be positive integers. Ifpq ≤ 1 and dq/pe < bq′/p′c, then S3

p/q(K ) cannot be an rn-sheeted regular

cover of S3p′/q′(K ) for any prime r .

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 18 / 19

Virtually cosmetic surgeries

Question

For what knots K ⊂ S3 do there exist pairs of surgery coefficients pq 6=

p′

q′

such that S3p/q(K ) is a cover of S3

p′/q′(K )?

Compare: cosmetic surgeries are those such that S3p/q(K ) = S3

p′/q′(K ).

Theorem (Lidman-M.)

Let K be a non-trivial knot in S3 and let p, q, p′, q′ be positive integers. Ifpq ≤ 1 and dq/pe < bq′/p′c, then S3

p/q(K ) cannot be an rn-sheeted regular

cover of S3p′/q′(K ) for any prime r .

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 18 / 19

Examples of virtually cosmetic surgeries

K = U (unknot): for all non-zero q ∈ Z and p′/q′ ∈ Q, the surgeryS31/q(U) = S3 is a p-fold regular cyclic cover of S3

p′/q′(U) = L(p′, q′).

K = the right-handed trefoil. Let rn be a prime power of the form6k + 1 for a positive integer k. Then S3

(6q±1)/q(K ) = L(6q ± 1, 4q) is

a regular rn-cover of S3(6q′±1)/q′(K ) = L(6q′ ± 1, 4q′) for

q′ = q + k(6q ± 1). Note that these examples have surgerycoefficients greater than 1, unlike in our theorem.

Similar examples can be found for other torus knots.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 19 / 19

Examples of virtually cosmetic surgeries

K = U (unknot): for all non-zero q ∈ Z and p′/q′ ∈ Q, the surgeryS31/q(U) = S3 is a p-fold regular cyclic cover of S3

p′/q′(U) = L(p′, q′).

K = the right-handed trefoil. Let rn be a prime power of the form6k + 1 for a positive integer k. Then S3

(6q±1)/q(K ) = L(6q ± 1, 4q) is

a regular rn-cover of S3(6q′±1)/q′(K ) = L(6q′ ± 1, 4q′) for

q′ = q + k(6q ± 1). Note that these examples have surgerycoefficients greater than 1, unlike in our theorem.

Similar examples can be found for other torus knots.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 19 / 19

Examples of virtually cosmetic surgeries

K = U (unknot): for all non-zero q ∈ Z and p′/q′ ∈ Q, the surgeryS31/q(U) = S3 is a p-fold regular cyclic cover of S3

p′/q′(U) = L(p′, q′).

K = the right-handed trefoil. Let rn be a prime power of the form6k + 1 for a positive integer k. Then S3

(6q±1)/q(K ) = L(6q ± 1, 4q) is

a regular rn-cover of S3(6q′±1)/q′(K ) = L(6q′ ± 1, 4q′) for

q′ = q + k(6q ± 1). Note that these examples have surgerycoefficients greater than 1, unlike in our theorem.

Similar examples can be found for other torus knots.

Ciprian Manolescu (UCLA) Floer homology and covering spaces June 6, 2016 19 / 19