Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

AN INTRINSIC CHARACTERIZATION OF p-SYMMETRIC HEEGAARD SPLITTINGS

MICHELE MULAZZANI

Abstract. We show that every p-fold strictly-cyclic branched covering of a b-bridge link in S^{3} admits a p-symmetric Heegaard splitting of genus g = (b − 1)(p − 1). This gives a complete converse to a result of Birman and Hilden, and gives an intrinsic characterization of p-symmetric Heegaard splittings as p-fold strictly-cyclic branched coverings of links.

1. Introduction

The concept of p-symmetric Heegard splittings has been introduced by Birman and Hilden (see [2]) in an extrinsic way, depending on a particular embedding of the handlebodies of the splitting in the ambient space E^{3}. The definition of such particular splittings was motivated by the aim to prove that every closed, orientable 3-manifold of Heegaard genus g ≤ 2 is a 2-fold covering of S^{3} branched over a link of bridge number g + 1 and that, conversely, the 2-fold covering of S^{3} branched over a link of bridge number b ≤ 3 is a closed, orientable 3-manifold of Heegaard genus b − 1 (compare also [7]).

A genus g Heegaard splitting M = Y_{g} ∪_{ϕ} Y′_{g} is called p-symmetric, with p > 1, if there exist a disjoint embedding of Y_{g} and Y′_{g} into E^{3} such that Y′_{g} = τ(Y_{g}), for a translation τ of E^{3}, and an orientation-preserving homeomorphism ρ : E^{3} → E^{3} of period p, such that ρ(Y_{g}) = Y_{g} and, if G denotes the cyclic group of order p generated by ρ and Φ : ∂Y_{g} → ∂Y_{g} is the

2000 Mathematics Subject Classification. Primary 57M12, 57R65; Secondary 20F05, 57M05, 57M25.

Key words and phrases. 3-manifolds, Heegaard splittings, cyclic branched coverings, links, plats, bridge number, braid number.

This contribution is extracted from [6] M. Mulazzani, On p-symmetric Heegaard splittings, J. Knot Theory Ramifications 9 (2000), no. 8, 1059–1067. Reprinted with permission from World Scientific Publishing Co.

Michele Mulazzani, An intrinsic characterization of p-symmetric Heegaard splittings, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 217–222, Topology Atlas, Toronto, 2002; arXiv:math.GT/0112231.

orientation-preserving homeomorphism Φ = τ_{|∂Y′_{g}}^{−1}ϕ, the following conditions are fulfilled:

i) Y_{g}/G is homeomorphic to a 3-ball; ii) Fix(ρ_{|Y_{g}}^{h}) = Fix(ρ_{|Y_{g}}), for each 1 ≤ h ≤ p − 1; iii) Fix(ρ_{|Y_{g}})/G is an unknotted set of arcs^{1} in the ball Y_{g}/G; iv) there exists an integer p_{0} such that Φρ_{|∂Y_{g}}Φ^{−1} = (ρ_{|∂Y_{g}})^{p_{0}}.

Remark 1. By the positive solution of the Smith Conjecture [4] it is easy to see that necessarily p_{0} ≡ ±1 mod p.

The map ρ′ = τρτ^{−1} is obviously an orientation-preserving homeomorphism of period p of E^{3} with the same properties as ρ, with respect to Y′_{g}, and the relation ϕρ_{|∂Y_{g}}ϕ^{−1} = (ρ′_{|∂Y′_{g}})^{p_{0}} easily holds.

The p-symmetric Heegaard genus g_{p}(M) of a 3-manifold M is the smallest integer g such that M admits a p-symmetric Heegaard splitting of genus g.

The following results have been established in [2]:

(1) Every closed, orientable 3-manifold of p-symmetric Heegaard genus g admits a representation as a p-fold cyclic covering of S^{3}, branched over a link which admits a b-bridge presentation, where g = (b − 1)(p − 1).

(2) The p-fold cyclic covering of S^{3} branched over a knot of braid number b is a closed, orientable 3-manifold M which admits a p-symmetric Heegaard splitting of genus g = (b − 1)(p − 1).

Note that statement 2 is not a complete converse of 1, since it only concerns knots and, moreover, b denotes the braid number, which is greater than or equal to (often greater than) the bridge number. In this paper we fill this gap, giving a complete converse to statement 1. Since the coverings involved in 1 are strictly-cyclic (see next section for details on strictly-cyclic branched coverings of links), our statement will concern this kind of coverings. More precisely, we shall prove in Theorem 4 that a p-fold strictlycyclic covering of S^{3}, branched over a link of bridge number b, is a closed, orientable 3-manifold M which admits a p-symmetric Heegaard splitting of genus g = (b − 1)(p − 1), and therefore has p-symmetric Heegaard genus g_{p}(M) ≤ (b − 1)(p − 1). This result gives an intrinsic interpretation of psymmetric Heegaard splittings as p-fold strictly-cyclic branched coverings of links.

2. Main results

Let

β = {(p_{k}(t), t) | 1 ≤ k ≤ 2n , t ∈ [0, 1]} ⊂ E^{2} × [0, 1]

1A set of mutually disjoint arcs {t_{1}, …, t_{n}} properly embedded in a handlebody Y is unknotted if there is a set of mutually disjoint discs D = {D_{1}, …, D_{n}} properly embedded in Y such that t_{i} ∩ D_{i} = t_{i} ∩ ∂D_{i} = t_{i}, t_{i} ∩ D_{j} = ∅ and ∂D_{i} − t_{i} ⊂ ∂Y for 1 ≤ i, j ≤ n and i ≠ j.

CHARACTERIZATION OF p-SYMMETRIC HEEGAARD SPLITTINGS 219

be a geometric 2n-string braid of E^{3} [1], where p_{1}, …, p_{2n} : [0, 1] → E^{2} are continuous maps such that p_{k}(t) ≠ p_{k′}(t), for every k ≠ k′ and t ∈ [0, 1], and such that {p_{1}(0), …, p_{2n}(0)} = {p_{1}(1), …, p_{2n}(1)}. We set P_{k} = p_{k}(0), for each k = 1, …, 2n, and

A_{i} = (P_{2i−1}, 0), B_{i} = (P_{2i}, 0), A′_{i} = (P_{2i−1}, 1), B′_{i} = (P_{2i}, 1),

for each i = 1, …, n (see Figure 1). Moreover, we set F = {P_{1}, …, P_{2n}}, F_{1} = {P_{1}, P_{3} …, P_{2n−1}} and F_{2} = {P_{2}, P_{4}, …, P_{2n}}.

The braid β is realized through an ambient isotopy

β̂: E^{2} × [0, 1] → E^{2} × [0, 1], β̂(x, t) = (β_{t}(x), t),

where β_{t} is an homeomorphism of E^{2} such that β_{0} = Id_{E^{2}} and β_{t}(P_{i}) = p_{i}(t), for every t ∈ [0, 1]. Therefore, the braid β naturally defines an orientationpreserving homeomorphism β̃ = β_{1} : E^{2} → E^{2}, which fixes the set F. Note that β uniquely defines β̃, up to isotopy of E^{2} mod F.

Connecting the point A_{i} with B_{i} by a circular arc α_{i} (called top arc) and the point A′_{i} with B′_{i} by a circular arc α′_{i} (called bottom arc), as in Figure 1, for each i = 1, …, n, we obtain a 2n-plat presentation of a link L in E^{3}, or equivalently in S^{3}. As is well known, every link admits plat presentations and, moreover, a 2n-plat presentation corresponds to an nbridge presentation of the link. So, the bridge number b(L) of a link L is the smallest positive integer n such that L admits a representation by a 2n-plat. For further details on braid, plat and bridge presentations of links we refer to [1].

Figure 1. A 2n-plat presentation of a link.

Remark 2. A 2n-plat presentation of a link L ⊂ E^{3} ⊂ S^{3} = E^{3} ∪ {∞} furnishes a (0, n)-decomposition [5] (S^{3}, L) = (D, A_{n}) ∪_{ϕ′} (D′, A′_{n}) of the link, where D and D′ are the 3-balls

D = (E^{2}×] − ∞, 0]) ∪ {∞} and D′ = (E^{2} × [1, +∞[) ∪ {∞},
A_{n} = α_{1} ∪ ⋯ ∪ α_{n}, A′_{n} = α′_{1} ∪ ⋯ ∪ α′_{n}

and ϕ′ : ∂D → ∂D′ is defined by ϕ′(∞) = ∞ and ϕ′(x, 0) = (β̃(x), 1), for each x ∈ E^{2}.

If a 2n-plat presentation of a μ-component link L = ⋃_{j=1}^{μ} L_{j} is given, each component L_{j} of L contains n_{j} top arcs and n_{j} bottom arcs. Obviously, ∑_{j=1}^{μ} n_{j} = n. A 2n-plat presentation of a link L will be called special if:

(1) the top arcs and the bottom arcs belonging to L_{1} are α_{1}, …, α_{n_{1}} and α′_{1}, …, α′_{n_{1}} respectively, the top arcs and the bottom arcs belonging to L_{2} are α_{n_{1}+1}, …, α_{n_{1}+n_{2}} and α′_{n_{1}+1}, …, α′_{n_{1}+n_{2}} respectively, …, the top arcs and the bottom arcs belonging on L_{μ} are

α_{n_{1}+···+n_{μ−1}+1}, …, α_{n_{1}+···+n_{μ}} = α_{n}

and

α′_{n_{1}+···+n_{μ−1}+1}, …, α′_{n_{1}+···+n_{μ}} = α′_{n}

respectively;

(2) p_{2i−1}(1) ∈ F_{1} and p_{2i}(1) ∈ F_{2}, for each i = 1, …, n.

It is clear that, because of (2), the homeomorphism β̃, associated to a 2nstring braid β defining a special plat presentation, keeps fixed both the sets F_{1} and F_{2}. Although a special plat presentation of a link is a very particular case, we shall prove that every link admits such kind of presentation.

Proposition 3. Every link L admits a special 2n-plat presentation, for each n ≥ b(L).

Proof. Let L be presented by a 2n-plat. We show that this presentation is equivalent to a special one, by using a finite sequence of moves on the plat presentation which changes neither the link type nor the number of plats. The moves are of the four types I, I′, II and II′ depicted in Figure 2. First of all, it is straightforward that condition (1) can be satisfied by applying a suitable sequence of moves of type I and I′. Furthermore, condition (2) is equivalent to the following: (2′) there exists an orientation of L such that, for each i = 1, …, n, the top arc α_{i} is oriented from A_{i} to B_{i} and the bottom arc α′_{i} is oriented from B′_{i} to A′_{i}. Therefore, choose any orientation on L and apply moves of type II (resp. moves of type II′) to the top arcs (resp. bottom arcs) which are oriented from B_{i} to A_{i} (resp. from A′_{i} to B′_{i}). □

A p-fold branched cyclic covering of an oriented μ-component link L = ⋃_{j=1}^{μ} L_{j} ⊂ S^{3} is completely determined (up to equivalence) by assigning to each component L_{j} an integer c_{j} ∈ Z_{p} − {0}, such that the set {c_{1}, …, c_{μ}} generates the group Z_{p}. The monodromy associated to the covering sends each meridian of L_{j}, coherently oriented with the chosen orientations of L and S^{3}, to the permutation (1 2 ⋯ p)^{c_{j}} ∈ Σ_{p}. Multiplying each c_{j} by the same invertible element of Z_{p}, we obtain an equivalent covering.

Following [3] we shall call a branched cyclic covering:

a) strictly-cyclic if c_{j′} = c_{j′′}, for every j′, j′′ ∈ {1, …, μ},

b) almost-strictly-cyclic if c_{j′} = ±c_{j′′}, for every j′, j′′ ∈ {1, …, μ},

Figure 2. Moves on plat presentations.

c) meridian-cyclic if gcd(b, c_{j}) = 1, for every j ∈ {1, …, μ},

d) singly-cyclic if gcd(b, c_{j}) = 1, for some j ∈ {1, …, μ},

e) monodromy-cyclic if it is cyclic. The following implications are straightforward:

a) ⇒ b) ⇒ c) ⇒ d) ⇒ e).

Moreover, the five definitions are equivalent when L is a knot. Similar definitions and properties also hold for a p-fold cyclic covering of a 3-ball, branched over a set of properly embedded (oriented) arcs.

It is easy to see that, by a suitable reorientation of the link, an almoststrictly-cyclic covering becomes a strictly-cyclic one. As a consequence, it follows from Remark 1 that every branched cyclic covering of a link arising from a p-symmetric Heegaard splitting – according to Birman-Hilden construction – is strictly-cyclic.

Now we show that, conversely, every p-fold branched strictly-cyclic covering of a link admits a p-symmetric Heegaard splitting.

Theorem 4. A p-fold strictly-cyclic covering of S^{3} branched over a link L of bridge number b is a closed, orientable 3-manifold M which admits a psymmetric Heegaard splitting of genus g = (b−1)(p−1). So the p-symmetric Heegaard genus of M is

g_{p}(M) ≤ (b − 1)(p − 1).

Proof. Let L be presented by a special 2b-plat arising from a braid β, and let (S^{3}, L) = (D, A_{b}) ∪_{ϕ′} (D′, A′_{b}) be the (0, b)-decomposition described in Remark 2. Now, all arguments of the proofs of Theorem 3 of [2] entirely apply and the condition of Lemma 4 of [2] is satisfied, since the homeomorphism β̃ associated to β fixes both the sets F_{1} and F_{2}. □

As a consequence of Theorem 4 and Birman-Hilden results, there is a natural one-to-one correspondence between p-symmetric Heegaard splittings and p-fold strictly-cyclic branched coverings of links.

References

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[2] Joan S. Birman and Hugh M. Hilden, Heegaard splittings of branched coverings of S^{3}, Trans. Amer. Math. Soc. 213 (1975), 315–352. MR 52 #1662

[3] John P. Mayberry and Kunio Murasugi, Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982), no. 1, 143–173. MR 84d:57004

[4] John W. Morgan and Hyman Bass (eds.), The Smith conjecture, Academic Press Inc., Orlando, FL, 1984, Papers presented at the symposium held at Columbia University, New York, 1979. MR 86i:57002

[5] Kanji Morimoto and Makoto Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), no. 1, 143–167. MR 92e:57015

[6] Michele Mulazzani, On p-symmetric Heegaard splittings, J. Knot Theory Ramifications 9 (2000), no. 8, 1059–1067. MR 2002a:57027

[7] O. Ja. Viro, Links, two-sheeted branching coverings and braids, Mat. Sb. (N.S.) 87(129) (1972), 216–228, English translation in Math. USSR-Sb. 16 (1972), 223–236. MR 45 #7701

Department of Mathematics, University of Bologna, I-40127 Bologna, Italy, and C.I.R.A.M., Bologna, Italy

E-mail address: mulazza@dm.unibo.it