AN IRREDUCIBLE HEEGAARD DIAGRAM OF THE REAL PROJECTIVE 3-SPACE P

We give a genus 3 Heegaard diagram H of the real projective space P3, which has no waves and pairs of complementary handles. So Negami’s result that every genus 2 Heegaard diagram of P3 is reducible cannot be extended to Heegaard diagrams of P3 with genus 3.


Introduction.
In the study of 3-manifolds, the construction of an algorithm for recognizing the 3-sphere S 3 among all 3-manifolds is a very important problem.The first work in this direction was done by Whitehead [12], and later Volodin, Kuznetsov, and Fomenko [11] conjectured that Heegaard diagrams for S 3 are reducible, except for the canonical one.
Homma, Ochiai, and Takahashi [4] proved that the conjecture is true for the case of genus 2. But for the case of genera greater than two it is not true anymore.Morikawa [5] gave a counterexample for the case of genus 3, and Ochiai [8,9] gave counterexamples for the case of genera 3 and 4. Negami [6,7] proved that every 3-bridge projection of a link can be transformed into a minimum crossing one by a finite sequence of wave moves if and only if the link is equivalent to one of a trivial knot, a splittable link, and the Hopf link.Consequently, any genus 2 Heegaard diagrams of S 3 , S 2 × S 1 #L(p, q) and P 3 are reducible.
In this paper, we give a genus 3 Heegaard diagram H of the real projective space P 3 , which has no waves and pairs of complementary handles.Moreover, we construct a crystallization Γ corresponding to the Heegaard diagram H and show that at least one among the Heegaard diagrams associated with Γ is transformed into a Heegaard diagram with some pairs of complementary handles by a finite sequence of wave moves, and so it is reducible to the canonical diagram of P 3 .

Preliminaries.
Let M be a closed orientable 3-manifold and let T n , Tn be solid tori of genera n and h : ∂T n → ∂ Tn a homeomorphism of the boundary surface.Then the triad (T n , Tn ; M) is called a Heegaard splitting of genus n for M when M = T n ∪ h Tn .
A collection of mutually disjoint n meridian disks m 1 ,...,m n in a solid torus T of genus n is called a complete system of meridian disks of T if Cl(T −∪ n i=1 N(m i ,T )) is a 3-ball, where N(m i ,T ) is a regular neighborhood of m i in T .We call a collection of mutually disjoint (n+1) meridian disks in T an extended complete system of meridian disks of T provided that any n subcollection is a complete system of meridian disks of T .
Next, we give the concept of wave of Heegaard diagrams.Let H = (F ; u, v) be a Heegaard diagram for M, and w an arc on F such that for a meridian or a longitude of H, say u 1 , and both ends of w attach to the same side of u 1 .Then one of two circles in u 1 ∪ w, different from u 1 , bounds a meridian disk of H, say u 1 , and We call w a wave for H, and the replacement of u 1 with u 1 a wave move with w if C(H ) < C(H), where C(H) is the complexity of H which is defined as the cardinality of u ∩ v.
Let H be a Heegaard diagram of the real projective space P 3 other than the canonical one H associated with Figure 5. Then H is said to be reducible if there is a finite sequence of (normal) Heegaard diagrams, H n ,...,H 0 , with H n = H and H 0 = H, such that H i−1 is a wave move of H i (i = 1, 2,...,n).
Wave moves are also defined for n-bridge decompositions of links; the relations between two wave theories are investigated in [7].In particular, for 3-bridge decomposition of links, we have the following theorem.

Theorem 2.1 [6]. Every 3-bridge projection of a link can be transformed into a minimum crossing one by a finite sequence of wave moves if and only if the link is equivalent to one of a trivial knots, a splittable link, and the Hopf link.
By a 4-colored graph G = (Γ ,γ), we mean a regular graph Γ (with possibly multiple edges, but no loops) of degree 4, endowed with a proper edge coloration; a coloration γ : E(Γ ) → ∆ 3 = {0, 1, 2, 3}, where E(Γ ) is the set of edges of Γ , such that γ(e 1 ) = γ(e 2 ) for any two adjacent edges e 1 , e 2 .
A 4-colored graph G representing a PL manifold M is called a crystallization if, for each colour c ∈ Γ 3 , the subgraph obtained by deleting all coloured edges c is connected.Crystallizations exist for all PL manifolds (see [10]).

3.
A Heegaard diagram of P 3 .As mentioned in Section 2, genus 2 Heegaard splittings of closed orientable 3-manifolds are closely related to 3-bridge decompositions of links.In fact, Birman and Hilden [1] proved that there is a bijective correspondence between the equivalence classes of 3-bridge projections and those of genus 2 Heegaard diagrams.By Theorem 2.1, every Heegaard diagram of genus 2 of P 3 , other than the canonical one, contain at least one wave.
In this section, we give a Heegaard diagram of genus 3 of P 3 which has no waves and pairs of complementary handles.
In Figure 1, it is easily checked that this Heegaard diagram H has no waves and pairs of complementary handles.Proof.Construct a crystallization Γ associated with the above Heegaard diagram via Gagliardi's method [3].
Since the dotted lines in Figure 2 are axes for an involution, this crystallization represents a 2-fold branched covering of S 3 branched over the following link (Figure 3) by Ferris' construction of 2-fold branched coverings of S 3 [2].In Figure 4, dotted lines are eliminated overbridges by jump moves [7].By a couple of jump moves about underbridge, it is not hard to see that this link is equivalent to the standard Hopf link (Figure 5).Therefore, M 3 is the same as P 3 .Remark.In Figure 3, this link represents a 4-bridge projection which has no waves.Now, we construct an extended Heegaard diagram H associated with H = (F ; u, v).The extended Heegaard diagram H contains 16 Heegaard diagrams for P 3 .At least one of them can be transformed into a Heegaard diagram H with a pair of complementary handles by a finite sequence of wave moves (Figure 6).
By Singer moves on H , we have a Heegaard diagram of genus 2 of P 3 and so it is transformed into the canonical one [6].In Figure 6, a pair of complementary handles occurs at black dot.

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