Heegaard diagrams and optimal Morse flows on non-orientable 3-manifolds of genus 1 and genus 2

The present paper investigates Heegaard diagrams of nonorientable closed 3-manifolds, i.e. a non-orienable closed surface together with two sets of meridian disks of both handlebodies. It is found all possible non-orientable genus 2 Heegaard diagrams of complexity less than 6. Topological properties of Morse flows on closed smooth non-orientable 3-manifolds are described. Normalized Heegaard diagrams are furhter used for classification Morse flows with a minimal number of singular points and singular trajectories. Анотація. В роботі досліджуються діаграми Хегора неорієнтовних 3многовидів. Кожна така діаграма є неорієновною замкненою поверхнею з двома наборами меридіанів на ній. Також розглядаються так звані нормалізовані діаграми, які не містять криволінійних двокутників, утворених меридіанами. Описано алгоритм, що дозволяє визначити, чи є нормалізована діаграма діаграмою Хегора деякого 3-многовиду. Діаграми Хегора роду два розбито на три типи та отримано топологічну класифікацію всіх неорієнтовних тривимірних многовидів роду два і складності не вище п’яти. Кожен такий многовид гомеоморфний зв’язній сумі лінзового простору і косого добутку 2-сфери на коло. Більш того, знайдено 3многовид складності шість, який не гомеоморфний такій сумі. Нормалізовані діаграми Хегора також використано для класифікації оптимальних потоків Морса на многовидах роду два складності не вище п’яти.


INTRODUCTION
A Heegaard diagram is a common and convenient method for representing a closed connected 3-manifold. Heegaard diagrams and Heegaard splittings are closely related. Handlebodies, Heegaard splittings and Heegaard diagrams have been studied by Y. Matsumoto in [8]. The structure of genus 2 Heegaard diagrams of orientable closed 3-manifolds has been largely discussed by S. Matveev in [9]. A simple algorithm for enumeration of orientable closed three-dimensional manifolds whose genus do not exceed two has been proposed by S. Matveev and A. Fomenko in [1]. A classification of decomposition of non-orientable 3-manifolds into 3 handlebodies with disjoint boundaries has been presented by V. Nunez, J. C. Gomezlarranaga, W. Heil in [3]. A classification of Heegaard diagrams of genus 3 has been conducted by F. Korablev in [5]. The existence of a unique non-orientable manifold of genus 1 is a consequence of W. B. R. Lickorish's result in [6]. However, in this paper, we present a certain reformulation of Lickorish's result in terms of Heegaard diagrams. Our main purpose is to investigate if there exists any non-orientable 3-manifold distinct from manifolds of type L p,q #S 1 r S 2 . Given a Heegaard diagram H of M there exists a polar Morse flow X (only one source and only one sink) whose associated diagram is H. Morse flows (Morse-Smale flows without closed orbit) on manifolds with boundary have been studied by A. Prishlyak in [12,13] and A. Prishlyak and A. Prus in [14], A topological classification of Morse-Smale flows on closed surfaces has been presented by M. Peiksoto in [11], V. Sharko and A. Oshemkov in [10]. Furthermore, Morse flows on three-dimensional manifolds have been studied by Ya. Umanskii in [17], A. Prishlyak in [7], Clark Robinson in [15].

HEEGAARD DIAGRAMS
Let M 1 and M 2 be compact three-dimensional manifolds with homeomorphic boundaries and let h : BM 1 Ñ BM 2 be any homeomorphism. Gluing the manifolds M 1 and M 2 together along this homeomorphism gives a topological space  Let H Y H 1 be a Heegaard splitting of a 3-manifold M , F = BH = BH 1 the common boundary of genus g of the two handlebodies, u = tu 1 , . . . , u g u the set of meridians of the handlebody H, and v = tv 1 , . . . , v g u the set of meridians of the handlebody H 1 .
Conversely, let F be a closed non-orientable surface of genus g (Heegaard genus g) and u = (u 1 , . . . , u g ), v = (v 1 , . . . , v g ) be two sets of simple closed curves on F such that u i Xu j = ∅ and v i Xv j = ∅ for i ‰ j. Then the triple (F, u, v) is a Heegaard diagram of a certain 3-manifod iff the complements F zu and F zv are homeomorphic to 2-sphere with holes. Definition 2.6. Two Heegaard diagrams (F, u, v) and (F, u 1 , v 1 ) are called isotopic if there exists an isotopy φ i : F Ñ F such that φ 0 = 1, φ 1 (u) = u 1 and φ 1 (v) = v 1 . Definition 2.7. Two Heegaard diagrams (F, u, v) and (F, Both operations of isotopy and semi-isotopy of Heegaard diagrams does not change the underlying 3-dimensional manifold M as well as its Heegaard splitting of M into handlebodies H and H 1 . Such operations only replace in H and H 1 the meridional disks by isotopic ones. Let (F, u, v) be a Heegaard diagram and let β be a simple curve joining the meridians u 1 and u 2 of the diagram and having no other common points with the curves of u. Let C be a closed neighborhood of the union u 1 Yu 2 Yβ homeomorphic to a disc with two holes and intersecting no other curves of u. The boundary component of this neighborhood which is not isotopic to the curve u 1 or to the curve u 2 , will be denoted by u 1 # u 2 . The set tu 1 # u 2 , u 2 , . . . , u g u will be denoted byũ. 36 C. Hatamian, A. Prishlyak Definition 2.8. We will say that the diagram (F,ũ, v) is obtained from the diagram (F, u, v) by adding u 2 to u 1 along β.
The operation of adding curves from the set v is defined similarly. Notice that addition of curves from the set u to curves of the set v and of curves from the set v to curves of the set u is not allowed. The addition of one curve of a set to another curve of the same set fails not only to affect the corresponding manifold, but also its Heegaard splitting. The handlebodies of the splitting remain unchanged, and it is only their sets of meridional discs that are altered. Proof. The proof is similar to the proof of [1, Proposition 5.3]. As well as in the orientable case, the equivalence of Heegaard diagrams implies the existence of a homeomorphism between the corresponding Heegaard surfaces, which can further be extended to a homeomorphism between meridional disks, and then to a homeomorphism between 3-manifolds. Suppose that a certain part is not homeomorphic to a disk. Then in this part (and, therefore, in the surface F ) there exists a non-trivial simple closed curve l intersecting neither the meridians v of the handlebody H 1 nor the meridians u of the handlebody H. Hence, this curve bounds discs in each of the handlebodies H and H 1 . The union of those disks a 2-sphere S which either splits the manifold M into a connected sum or gives rise to a summand S 2ˆS1 . In both cases, the examination of the diagram reduces to the examination of one or two diagrams of lower genus.
Such a situation is therefore of no interest for us. In what follows we will assume that the curves u and v split the surface into disks. This happens if and only if the union u Y v is a connected graph. Such diagrams will be called connected.
So, let (F, u, v) be a connected Heegaard diagram. The regions homeomorphic to disks, into which the graph u Y v cuts the surface F will be interpreted as curvilinear polygons whose vertices are intersection points of meridians from different sets. Each polygon has, of course, an even number of sides. Definition 2.11. The Heegaard diagram (F, u, v) is called normalized if among the regions into which the meridians split the surface there are no lunes.

AN ALGORITHM TO DETERMINE HEEGAARD DIAGRAMS
Let (F, u, v) be a normalized diagram. Curring the surface F along the meridians u = tu 1 , . . . , u g u, we will obtain a sphere with 2g holes D 1 , . . . , D g . The holes can be naturally divided into pairs so that each pair D 2i´1 , D 2i consists of the two holes corresponding to the meridian u i . The meridians v will then be cut into arcs joining the holes in various ways.
To recover uniquely the Heegaard diagram from such a picture, we should know how the boundary of each hole D 2i is glued with the boundary of its companion hole D 2i´1 . It is convenient to enumerate the points at which the meridian u i intersects the meridians v in the order they are met when traversing the meridian u i , and to retain these numbers in the cutting. The total number of those crossing points is called the Heegaard complexity.
We take the boundary orientations of glued holes into account by assigning a plus sign to the orientedly glued holes and a minus sign to the ones that are glued non-orientedly. In order to define the gluing maps φ i : BD 2i Ñ BD 2i´1 , i = 1, 2, . . . , g, we introduce topological symmetries s i : BD 2i Ñ BD 2i´1 and topological rotations r i : BD 2i´1 Ñ BD 2i´1 , for i = 1, 2, . . . , g by the following rules: v i so that the endpoint of each arc joining D 2i to D 2i´1 is taken to the other endpoint of the same arc; Then an algorithm to determine Heegaard diagrams of closed 3-manifolds can be suggested as follows.
(1) An equal number of arcs should join D 2i and D 2i´1 .
(2) To determine how the boundaries of the holes are identified we need to enumerte the endpoints of the arcs on each of them in the cyclic order.
In the case of orientedly glued holes, D 2i and D 2i´1 have opposite orientation, whereas in the case of non-orientedly glued holes, D 2i and D 2i´1 have the same orientation. ( 3) The surface F should be orientable. To verify that, we construct a dual graph for u curves in F zv. The vertices of the graph represent disjoint regions in the plane divided by the arcs of the diagram. The edges of We assign a plus sign to each edge representing the intersection of orientedly glued holes and a minus sign to the other edges representing intersection of the ones that are glued non-orientedly. If the graph contains no cycles with odd number of minus signs, then the surface will be orientable.
(4) Every Heegaard diagram has exactly g closed curves or cycles. These curves split the surface and give a sphere with 2g holes which implies the surface F is connected.

NON-ORIENTABLE HEEGAARD DIAGRAMS OF GENUS 1 AND GENUS 2
In the case of orientable closed 3-manifolds, there are infinitely many orientable closed 3-manifold of genus 1, called lens spaces L p,q , whereas in the case of non-orientable ones, the following theorem being a direct consequence of results by W.B.R. Lickorish [6] holds. We will present a proof based on Heegaard diagrams. In the case of Heegaard complexity 1, there is only one possible diagram. Although this diagram has exactly one cycle. As shown in Figure 4.2 the resulting graph has a loop with a minus sign and thus the surface is not orientable. Hence there is no genus 1 Heegaard diagram of complexity 1.
In the case of Heegaard complexity 2, there are two possible diagrams as shown in Figure 4  (1, D 1 ). This gives a cycle. Similarly, if we move along the arc starting from (2, D 2 ), then we will return to the starting point as well, and thus we will obtain another cycle.
Thus we get a diagram with two cycles, and according to the algorithm, it is not a genus 1 Heegaard diagram.
Diagram (b): If we move along the arc starting from (1, D 2 ), then we pass to (2, D 1 ) which means we have arrived at (2, D 2 ). Then we get to (1, D 1 ) = (1, D 2 ) and once again for the same reason we return to the starting point. Thus, we ontain one cycle. As shown in Figure 4.4, although the diagram has exactly one cycle, it is not a Heegaard diagram since the resulting graph has a loop and an edge with minus signs meaning the surface F is not orientable.

Diagram (a)
Diagram (b) The total number of cycles in a non-orientable genus 1 diagram can be calculated using the following formula: where P is the complexity of the diagram and L is the number of single-arc cycles in the diagram. If the complexity of a diagram is even, then it has  In Lickorish [6] there was obtained a classification of isotopy classes of simple closed curves on the Klein bottle K and proved that the mapping class group of K is isomorphic to Z 2ˆZ2 . One can easily deduce that any homeomorphism of a Klein bottle extends to a homeomorphism of a solid Klein bottle, which directly implies uniqueness of non-orientable closed 3-manifold of genus 1, i.e. Theorem 4.1.

Definition 4.3.
We will say that a Heegaard diagram (F, u, v) of genus 2 has type I, II or III as it is depicted in Figure 4.6 a, b or c, respectively. a) type I b) type II, b, c ą 0 c) type III d ą 0   1, 1, 1, 1, 0, 0, 3) which represents a type II non-orientable Heegaard diagram of complexity 6. First, we  1,1,1,1,0,0,3) need to make sure that the diagram shown in Figure 4.7 has exactly two closed curves.
Similarly, if we move along the curve starting from (1, D 2 ), we eventually get to the starting point. Therefore this curve is also closed.
Having checked that the diagram has exactly two curves, we verify whether or not the surface F is orientable. To do so, we construct the required graph   The resulting graph in Figure 4.9 contains no cycles with an odd number of minus signs and hence F is orientable.
Finally, we need to verify whether or not the two curves split the surface. In order to do that, we construct the diagram illustrated in Figure 4.10 and make sure that we have exactly four cycles. If we move along the arc starting from the end of (b, D 2 ), we get to the beginning of (a, D 1 ) = (a, D 2 ). Since we have returned to the starting point we've got a cycle.
According to [2] one can often show that two spaces are not homeomorphic by showing that their fundamental groups are not isomorphic. The fundamental group of the Heegaard diagram represented by the 8-tuple (2, 1, 1, 1, 1, 0, 0, 3) is . This example shows that there exists a non-orientable Heegaard diagram of complexity 6 which is not homeomorphic to L p,q #S 1 r S 2 . Since every genus 2 Heegaard diagram of any type has an even number of a-and d-arcs, the following formula holds true: It can be shown that 933 non-orientable genus 2 diagrams of complexity less than 6, most of which are homeomorphic or symmetric, satisfy the equation (4.2). According to the algorithm, only 34 of them meet the criteria and represent distinct non-orientable genus 2 Heegaard diagrams. These diagrams are listed in Table 4.1.

OPTIMAL MORSE FLOWS ON 3-MANIFOLDS
A vector field (and the corresponding flow) on a closed manifolds is called a Morse if the following conditions hold true: (1) it has finitely many singular points which are non-degenerate; (2) stable and unstable manifolds of singular points intersects transversely; (3) α-and ω-limit sets of each trajectory are singular points.