Maps on 3-manifolds

Maps on -manifolds given by surgery

Boldizsár Kalmár Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Reáltanoda u. 13-15, 1053 Budapest, Hungary
bkalmar@renyi.hu
 and  András I. Stipsicz Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Reáltanoda u. 13-15, 1053 Budapest, Hungary and
Institute for Advanced Study, Princeton, NJ, 08540
stipsicz@renyi.hu
Abstract.

Suppose that the -manifold is given by integral surgery along a link . In the following we construct a stable map from to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of stable maps from to the plane in terms of properties of .

Key words and phrases:
Stable map, -manifold, surgery, negative knot, Thurston-Bennequin number.
2010 Mathematics Subject Classification. Primary 57R45; Secondary 57M27.

1. Introduction

It is well-known that a continuous map between smooth manifolds can be approximated by a smooth map and any smooth map on a 3-manifold can be approximated by a generic stable map. This line of argument, however, gives no concrete map on a given 3-manifold even if it is given by some explicit construction. Recall that by [Li62, Wa60] a closed oriented 3-manifold can be given by integral surgery along some link in . In the present work we construct an explicit stable map based on such a surgery presentation of .

Results in [Gr09, Gr10] give lower bounds on the topological complexity of the set of critical values of generic smooth maps and on the complexity of the fibers in terms of the topology of the source and target manifolds. In a slightly different direction, [CT08] gives a lower bound for the number of crossing singularities of stable maps from a -manifold to in terms of the Gromov norm of the -manifold. Recently [Ba08, Ba09] and [GK07] studied the topology of -manifolds through the singularities of their maps into surfaces.

In the present paper we give upper bounds on the minimal numbers of the crossings and non-simple singularities and of the connected components of the fibers of stable maps on the -manifold in terms of properties of diagrams of (e.g. the number of crossings or the number of critical points when projected to ). As an additional result, these constructions lead to upper bounds on a version of the Thurston-Bennequin number of negative Legendrian knots.

Before stating our main results, we need a little preparation. First of all, a stable map of a -manifold into the plane can be easily described by its Stein factorization.

Definition 1.1.

Let be a map of the -manifold into . Let us call two points equivalent if and only if and lie on the same component of an -fiber. Let denote the quotient space of with respect to this equivalence relation and the quotient map. Then there exists a unique continuous map such that . The space or the factorization of the map into the composition of and is called the Stein factorization of the map . (Sometimes the map is also called the Stein factorization of .)

In other words, the Stein factorization is the space of connected components of fibers of . Its structure is strongly related to the topology of the -manifold . For example, an immediate observation is that the quotient map induces an epimorphism between the fundamental groups since every loop in can be lifted to . If is a stable map, then its Stein factorization is a -dimensional CW complex. The local forms of Stein factorizations of proper stable maps of orientable -manifolds into surfaces are described in [KLP84, Le85], see Figure 1. Indeed, let be a stable map of the closed orientable -manifold into . We say that a singular point of is of type (a), …, (e), respectively, if the Stein factorization at looks locally like (a), …, (e) of Figure 1, respectively. We will call a point a singular point of type (a), …, (e), respectively, if for a singular point of type (a), …, (e), respectively. According to [KLP84, Le85] we give the following characterization of the singularities of : The singular point is a cusp point if and only if it is of type (c), the singular point is a definite fold point if and only if it is of type (a) and is an indefinite fold point if and only if it is of type (b), (d) or (e). Singular points of types (d) and (e) are called non-simple, while the others are called simple. A double point in of two crossing images of singular curves which is not an image of a non-simple singularity is called a simple singularity crossing. A simple singularity crossing or an image in of a non-simple singularity is called a crossing singularity. A stable map is called a fold map if it has no cusp singularities.

Figure 1. The local forms of Stein factorizations of stable maps from orientable -manifolds to surfaces. The map (symbolized by an arrow) maps from the CW complex to .

Let be a given link, and let denote a generic projection of it to the plane. Let and denote the number of components of and the number of crossings of , respectively.

Choose a direction in , which we represent by a vector . We can assume that satisfies the condition that the projection of the diagram to along yields only non-degenerate critical points. Let denote the number of times is tangent to . Suppose at each -tangency the half line emanating from in the direction of avoids the crossings of and intersects transversally (at the points different from ). Denote the number of transversal intersections by . Let denote the maximum of the values , where runs over the -tangencies. With these definitions in place now we can state the main result of the paper.

Theorem 1.2.

Suppose that the 3-manifold is obtained by integral surgery on the link . Then there is a stable map such that

  1. the Stein factorization is homotopy equivalent to the bouquet ,

  2. the number of cusps of is equal to ,

  3. all the non-simple singularities of are of type (d), and their number is equal to ,

  4. the number of non-simple singularities which are not connected by any singular arc of type (b) to any cusp is equal to ,

  5. the number of simple singularity crossings of in is no more than

  6. the number of connected components of the singular set of is no more than , and

  7. the maximal number of the connected components of any fiber of is no more than .

  8. Suppose we got by cutting out and gluing back the regular neighborhood of from . Then the indefinite fold singular set of contains a link in , which is isotopic to in and whose -image coincides with .

Remark 1.3.
  1. Let be a closed orientable -manifold, a given smooth map of into and a link disjoint from the singular set of . Suppose furthermore that is an immersion. Let denote the -manifold obtained by some integral surgery along . Then the method developed in the proof of Theorem 1.2 provides a stable map of into (relative to ).

  2. In constructing the map , the proof of Theorem 1.2 provides a sequence of stable maps of into , where each is obtained from by some deformation, . Finally, the map is obtained from . Suppose that is a compact -manifold which admits a handle decomposition with only - and -handles, i.e. can be given by attaching 4-dimensional 2-handles to along . Using our method we can construct a stable map of into .

Recall that according to [BR74] a closed orientable -manifold has a stable map into without singularities of types (b), (c), (d) and (e) if and only if is a connected sum of finitely many copies of . According to [Sa96] a closed orientable -manifold has a stable map into without singular points of types (c), (d) and (e) if and only if is a graph manifold. By [Le65] a -manifold always has a stable map into without singular points of type (c). Our arguments imply a constructive proof for

Theorem 1.4.

Every closed orientable -manifold has a stable map into without singular points of types (c) and (e).

Remark 1.5.
  1. One cannot expect to eliminate the singular points of types (a), (b) or (d) of stable maps from arbitrary closed orientable -manifolds to . In this sense our Theorem 1.4 gives the best possible elimination on -manifolds.

  2. By taking an embedding we get for every closed orientable -manifold a stable map into as well without singular points of types (c) and (e). Then by using the method of [Sa06], for example, for eliminating the singular points of type (a), we get a stable map, which is a direct analogue of the indefinite generic maps appearing in [Ba08, Ba09, GK07].

The construction also implies certain relations between quantities one can naturally associate to stable maps and to surgery diagrams.

Definition 1.6.

Suppose that is a fixed closed, oriented -manifold and is a stable map with singular set .

  • Let denote the number of simple singularity crossings of .

  • Let denote the number of non-simple singularities of .

  • Let denote the number of crossing singularities of . Clearly .

  • Let denote the number of non-simple singularities of which are not connected by any singular arc of type (b) to any cusp.

  • Let denote the number of cusps of . Clearly .

  • Let denote the number of connected components of . Clearly it is no more than the number of connected components of .

  • Let denote the maximum number of connected components of the fibers of .

The inequality

has been shown to hold in [Gr09, Section 2.1].111The paper [MPS95] is also closely related. In addition, by [CT08, Theorem 3.38] we have , where is the Gromov norm of , cf. also [Gr09, Section 3].

Theorem 1.2 provides several estimates for upper bounds on the topological complexity of smooth maps of a -manifold given by surgery. For example, by summing quantities in Definiton 1.6 and their estimates in Theorem 1.2, we immediately obtain

Corollary 1.7.

Suppose that the 3-manifold is obtained by integral surgery on the link . Let be any diagram of and a general position vector in . Then

  • ,

  • ,

  • ,

where the minima are taken for all the stable maps of into . Evidently, we can estimate other properties in Definiton 1.6 of stable maps on as well.

These expressions can be simplified by estimating as

(1.1)

cf. Lemma 3.7.

The number of tangencies of a projection of a knot in a fixed direction is reminiscent to the number of cusp singularities of a front projection of a Legendrian knot in the standard contact 3-space. Based on this analogy, our previous results imply an estimate on a quantity attached to a Legendrian knot in the following way.

Recall first that the standard contact structure on is the 2-plane field given by the kernel of the 1-form . A knot is Legendrian if the tangent vectors of are in . (To indicate the Legendrian structure on the knot, we will denote it by and reserve the notation for smooth knots and links.) If is chosen generically within its Legendrian isotopy class, its projection to the plane will have no vertical tangencies, and at any crossing the strand with smaller slope will be over the one with higher slope. Consider now a Legendrian knot and let denote such a projection (called a front projection) of . The Thurston-Bennequin number of is given by the formula , where stands for the writhe (i.e. the signed sum of the double points) of the projection. Although the definition of tb uses a projection of the Legendrian knot , it is not hard to show that the resulting number is an invariant of the Legendrian isotopy class of .

In case the projection has only negative crossings, we have that , hence the resulting Thurston-Bennequin number can be identified with after choosing appropriately, cf. [Ge08, OS04]. (In this case the generic projection used in the definitions of and is derived from the front projection by rounding the cusps.)

As it is customary, we define as the maximum of all Thurston-Bennequin numbers of Legendrian knots smoothly isotopic to . (It is a nontrivial fact, and follows from the tightness of that this maximum exists.) A modification of this definition for negative knots (i.e. for knots admitting projections with only negative crossings) provides

Definition 1.8.

For a negative knot let denote the value where runs over those Legendrian knots smoothly isotopic to which admit front diagrams with only negative crossings.

It is rather easy to see that if the knot admits a projection with only negative crossings, then it also has a front projection with the same property. Clearly .

Theorem 1.9.

For a negative knot and any -manifold obtained by an integral surgery along we have

  • ,

  • ,

  • ,

where the minima are taken for all the stable maps of into .

By Theorem 1.9 and [CT08, Theorem 3.38] we obtain

Corollary 1.10.

For a negative knot and any -manifold obtained by an integral surgery along , we have .


Acknowledgements: The authors were supported by OTKA NK81203 and by the Lendület program of the Hungarian Academy of Sciences. The first author was partially supported by Magyary Zoltán Postdoctoral Fellowship. The authors thank the anonymous referee for the comments which improved the paper.

2. Preliminaries

In this section, we recall and summarize some technical tools. First, we show that a cusp can be pushed through an indefinite fold arc as in Figure 2:

Lemma 2.1 (Moving cusps).

Suppose that in a neighborhood of a point the Stein factorization of a map is given by Figure 2(a). Then can be deformed in this neighborhood to a map so that the Stein factorization of is as the diagram of Figure 2(b).

Proof.

Suppose is the cusp singular point and is the indefinite fold arc at hand. Let be a point on the other side of in . Connect and by an embedded arc . Then there is an arc such that , starts at and and do not intersect. By using the technique of [Le65] we can now deform in a small tubular neighborhood of to achieve the claimed map . Note that during this move one singular point of type (d) appears. ∎

An analogous statement holds if we move a cusp from a -sheeted region to a -sheeted region.

Figure 2. Moving cusps. A map can be deformed so that the image of a cusp point goes to the other side of the image of an indefinite fold arc.

According to the next result, two cusps can be eliminated as in Figure 3:

Lemma 2.2 (Eliminating cusps).

Suppose that in a neighborhood of a point the Stein factorization of a map is given by Figure 3(a). Then can be deformed in this neighborhood to a map so that the Stein factorization of is as the diagram of Figure 3(b).

Proof.

This statement is the elimination in [Le65, pages 285–295] for -dimensional source manifolds. ∎

Figure 3. Eliminating cusps.

Recall that if is a stable map and denotes its singular set, then is a generic immersion with cusps, i.e. if denotes the set of cusp points, then is a generic immersion with finitely many double points and is disjoint from .

The following result will be the key ingredient in our subsequent arguments for proving Theorem 1.2.

Lemma 2.3 (Making wrinkles).

Suppose that is a stable map and let denote an embedded closed -dimensional manifold such that is disjoint from the singular set , is a generic immersion and is a generic immersion with cusps. Let be a small tubular neighborhood of disjoint from and fix an identification of with the normal bundle of . Let be a non-zero section such that for any . Then is homotopic to a smooth stable map such that

  1. outside ,

  2. the singular set of is ,

  3. has indefinite fold singularities along ,

  4. has definite fold singularities along ,

  5. ,

  6. is an immersion parallel to and

  7. if for a double point of the two points in lie in the same connected component of the fiber , then the double point of correspond to a singularity of type (d).

Proof.

We perform the homotopy inside fiberwise as shown by Figure 4. Since is the trivial bundle, the homotopy of the fibers yields a homotopy of the entire . ∎

Figure 4. The deformation of to in a fiber of .
Remark 2.4.

If the submanifold has boundary, we can still get something similar. In this case the section should be zero at the boundary points of , and the homotopy yields a stable map with cusps at .

3. Proof of the results

3.1. Construction of the stable map on

Proof of Theorem 1.2.

We will prove the theorem by presenting an algorithm which produces the map on with the desired properties. This algorithm will be given in seven steps; the first six of these steps are concerned with maps on . Let us start with a fold map with one unknotted circle as singular set such that is an embedding and is a circle for each regular point . Then the Stein factorization of is a disk together with its embedding into . By cutting out the interior of a sufficiently small tubular neighborhood of from , we get a solid torus whose boundary is mapped into by as a circle fibration over a circle parallel to , and is a trivial circle bundle . Suppose the link is disjoint from . Then by identifying with and with the projection onto , we get a link diagram . Now we start modifying this map . In Steps 1 through 6 we will deal with maps on , and the goal will be to obtain a map which is suitable with respect to the fixed surgery link . In particular, we aim to find a map on with the property that its restriction to any component of is an embedding into . We suppose that the modifications through Step 1, …, Step 6 happen so that all the images of the maps , …, lie completely inside the disk determined by the (unchanged) circle , . This can be reached easily by choosing to bound an area “large enough” in and supposing that the diameter of is small.

Step 1

Our first goal is to deform so that the resulting map has fold singularities along . Apply Lemma 2.3 to the map and the embedded -dimensional manifold , and denote the resulting stable map by . It is a fold map, its indefinite fold singular set is and its definite fold singular set is , where is isotopic to ; for an example see Figure 5.

Figure 5. The image of the singular set of the map , where is the trefoil knot. The outer circle is , the inner solid curve is and the dashed curve is . The double points of correspond to singularities of type (d).

Since is isotopic to , the integral surgery along giving can be equally performed along . Recall that doing surgery along simply means that we cut out a tubular neighborhood of the definite fold curve (which is diffeomorphic to ), and glue it back by a diffeomorphism of its boundary . If the image was an embedding of circles, then it would be easy to construct the claimed map on the -manifold given by the integral surgery. Since this is not the case in general, we need to further deform the map .

Let denote the interior of the bands (one for each component of ) bounded by and in the Stein factorization . Then is immersed into by . The Stein factorizations of the maps in the next steps will be built on . Let denote the surface .

Step 2

Now, our goal is to deform so that the Stein factorization of the resulting map has small “flappers” near at the points where is tangent to the general position vector . These “flappers” will help us to move the image of so that it will become an embedding into .

First, we use Lemma 2.3 together with Remark 2.4 as follows. Let be the set of points in such that for each the direction is tangent to at . For each take a small embedded arc in a small neighborhood of in such that is an embedding parallel to . For each arc there exists an embedded arc in such that is an embedding onto . See, for example, the upper picture of Figure 6, where the small dashed arcs having cusp endpoints represent the arcs for all .

Figure 6. We obtain the upper picture by applying Lemma 2.3 and Remark 2.4 to the small arcs in which are mapped by to the dashed arcs near the points of the diagram where it is tangent to . We obtain the lower picture by applying Lemma 2.3 and Remark 2.4 to the new arcs added to the upper picture. The solid arcs correspond to singularities of type (a) and the black double points of the dashed arcs correspond to singularities of type (d).

Apply Lemma 2.3 and Remark 2.4 to the map and the arcs to obtain a map . The section in Lemma 2.3 is chosen so that if we project the -images of the arising new definite fold curves in to , then for each curve there is only one critical point, which is a maximum. An example for the resulting map can be seen in the upper picture of Figure 6. Note that the deformation yielded small “flappers” in attached to along the arcs . Next, for each take small arcs in which intersect generically the previous arcs , lie in and on the “flappers” and are mapped into almost parallel to . See the new small dashed arcs in the lower picture of Figure 6. Once again, there are small arcs embedded in mapped by onto , respectively.

The application of Lemma 2.3 and Remark 2.4 for these arcs provides us a map, which we denote by . This map will have one additional flapper for every flapper of . We choose the section in Lemma 2.3 so that the -images of the arising new definite fold curves lie inward222At a point of let us call the direction which is perpendicular to and points toward the direction where locally lies “inward”. from the arcs , respectively, in the -image of and the previous flappers. For an enlightening example, see the lower picture of Figure 6. Note that after this step new singular points of type (d) appeared. Also note that for each we have four cusp singular points in , three of which are mapped by into . We denote the set of these three cusps by . For each the -images of two of these three cusps in point to the direction . We denote the set of these two cusps by . Note that the definite fold curves in the images of the two cusps in are on opposite sides.

Step 3

Now our goal is to obtain definite fold arcs connecting points of where had cusps. Moreover these definite fold arcs will be mapped into parallel to the diagram . (These curves will be translated in the next step so that later they will lead to an embedding of into .)

In order to reach this goal, we deform the map further by eliminating half of the cusps as follows. We proceed for each component of separately and in the same way, thus in the following we can suppose that is connected. Take a cusp which is in for an such that the entire lies to the right hand side of its tangent at . By going along the band in in the direction to which the -image of this cusp points, we reach another cusp in for some at the next -tangency of . If this cusp does not belong to , then it is possible to apply Lemma 2.2 and eliminate these two cusps, since they are in the position of Figure 3. Then we continue by taking the cusp in whose Stein factorization is folded inward. If the cusp does belong to , then we choose that cusp from which can be used to eliminate (it is easy to see that this is exactly the cusp in whose Stein factorization is folded inward), we eliminate them, then we continue by taking the cusp belonging to . This procedure goes all along the band , meets all and eliminates half of the cusps. After finishing this process, we obtain a stable map, which we denote by ; cf. Figure 7 for an example.

Figure 7. Eliminating half of the cusps in the lower picture of Figure 6. The black double points correspond to singularities of type (d).

The cusp elimination results new definite fold curves whose -image is an immersion, and which have double points near the crossings of the diagram . In the next step we will deform so that the double points of these new curves will be localized near the images of the remaining cusps.

Step 4

Now our goal is to deform to a map such that the definite fold arcs obtained in the previous step will be mapped into far from the diagram . (Informally, we will “lift” some of the arcs in the direction of .) Moreover, the immersion of these definite fold arcs into will have double points only near some cusps of . This brings us closer to the original goal to have a map which embeds a link isotopic to into the plane.

The cusp eliminations above affect only small tubular neighborhoods of curves connecting cusps in . Denote by the new definite fold arcs which appear in these tubular neighborhoods after the eliminations. Note that by the algorithm above, the arcs are mapped into so that by an elementary deformation they can be moved “upward” in the direction of , see Figure 7.

So we further deform to get a stable map denoted by as indicated in Figure 8: as it is shown by the picture, the arcs are “lifted”. In fact, we deform : we move the top of the “flappers” corresponding to the -curves of Step 2 and the -image of the curves in the direction of and far from . We proceed for each component of separately and in the same way, thus in the following we can suppose that is connected. First we choose a point such that the entire lies to the right hand side from its tangent at . Then, by walking along the band starting from , we deform the flappers and the curves to be mapped into the plane as a “zigzag” far away from the diagram . More precisely, consider the coordinate system in with origin and coordinate axes and , respectively, where denotes the vector obtained by rotating clockwise by degrees. By extending the -image of the flappers in the direction of deform the -image of the curves so that by going along between the points , where and , the corresponding component of the curve is mapped into a small tubular neighborhood of a line with slope for . Finally, arrange the last component of starting with slope and ending at the first (extended) flapper belonging to , see Figure 8.

Note that as a result the double points of the immersion of the deformed curves are in a small neighborhood of the cusps mapped close to the tops of the flappers.

Figure 8. The Stein factorization of , i.e. the deformation of of Figure 7. (The straight line represents the line used to cut in Step 5.) The upper part of from the bold -complex is denoted by . (As usual, the circle is omitted.)

Step 5

In this step, we modify the stable map so that the cusps of the resulting map will be easy to eliminate in the next step. Let be a line perpendicular to located near , separating it from the other parts moved to the direction of in Step 4, as indicated in Figure 8.

Now, we cut the -complex (recall that denotes , see Step 1) along the -preimage of the line , thus we obtain the decomposition

where denotes the -dimensional CW complex containing and denotes the closure of . Then is a -manifold with boundary. Let us denote the -complex by . In order to visualize in Figure 8, we suppose that the cutting of along is a little bit perturbed and thus the bold -complex in Figure 8 represents . Before proceeding further, we need a better understanding of the -preimages of the sets appearing in the above decomposition. The preimage is clearly diffeomorphic to for a link . The following statements show much more about . It is easy to see that the numbers of components of and are equal. However, we have the stronger

Lemma 3.1.

A longitudinal curve in is isotopic to .

Proof.

The -complex decomposes as a union of -cells: some of them (which we depict as “small -cells” in Figure 8) are attached at one of their endpoints to the union of the other 1-cells, we denote these small cells by for . Others are attached by both of their endpoints. Let denote the -complex . Then the PL embedding is isotopic to the subcomplex of formed by the arcs of type (b) in the open bands connecting the singular points of type (d) in . Furthermore, the subcomplex is isotopic to . Take a small closed regular neighborhood of . Then is naturally a -bundle over . The boundary of in is a -manifold isotopic to , and we will denote it by . Clearly is diffeomorphic to . Note that any section of is isotopic to .

The isotopy between and and the isotopy between and can be chosen easily so that they give a PL embedding such that and correspond to and , respectively. For , let denote small regular neighborhoods of the singular points of type (d) located near the cusp points in in , such a and the restriction can be seen in Figure 1(d). Then the intersection consists of a union of disks, which will be denoted by .

First, observe that for each there exists a disk embedded into in whose boundary is mapped by homeomorphically onto the boundary , i.e.  is a lifting of . To see this, consider the -manifold for each . By [Le85] the manifold is diffeomorphic to , where is a disk with three holes and it is mapped by into as we can see in Figure 9(a).

Figure 9. In (a) we can see the manifold and how it is mapped onto the regular neighborhood and into , cf. Figure 1(d). is mapped onto the left side of the rectangle as a proper Morse function with two indefinite critical points. The two “figure eights” in are the two singular fibers. is mapped similarly onto the right side of . The middle fiber in is mapped to the singular point of type (d). For a detailed analysis see [Le85]. In (b) we can see the boundary in and its image in represented by a bold -complex.

Each disk can be located in essentially in four ways, for example the lower picture of Figure 9(b) shows the disk for the leftmost non-simple singularity crossing of type (d) in Figure 8. We get on the picture by cutting out the two shaded areas from the -complex . It is easy to see in the upper picture of Figure 9(b) how to put the disk into . The other three possibilities for the location of a disk in and the disk in can be described in a similar way.

Now observe that can be lifted to extending because of the following. First, the regular neighborhoods of the singular points of type (c) in (see Figure 1(c)) intersect in disks which can be lifted to . Then the intersection of the small regular neighborhoods of the singular curves of type (b) and can be lifted as well since there is no constraint for the lift at the regular points of . Finally observe that the rest of intersects only in areas of non-singular points which are attached to the boundary of , so the previous lifts extend over the entire .

Hence we obtain an embedding with and corresponding to lifts of and , respectively. Thus we obtain an isotopy between a longitude of and a lift of . The fact that any lift of is isotopic to finishes the proof. ∎

Figure 10.
Lemma 3.2.

The preimage is isotopic to a regular neighborhood of .

Proof.

It is enough to show that is diffeomorphic to extending naturally the structure on its boundary since by Lemma 3.1 the union of tori contains a longitude isotopic to . Moreover it is enough to show that the -preimage of the part of homeomorphic to the CW complex in Figure 10 is diffeomorphic to , where the -preimage of the two vertical edges on the right-hand side of the -complex of Figure 10 corresponds to . Clearly the -preimage of the two vertical edges on the right-hand side is diffeomorphic to since is a circle for any lying in the two vertical edges except if is one of the two top ends. If is one of the two top ends, then is one point since it is a definite fold singularity. The -preimage of the backward sheet in Figure 10 is diffeomorphic to minus for an interval . The -preimage of the forward sheet is diffeomorphic to . ∎

Corollary 3.3.

Any longitudinal curve in is isotopic to .

In order to obtain the map , we modify the map outside a neighborhood of as it is shown by Figure 11: our goal is to have the arrangement that if for a cusp singularity the point is connected in to by a -cell mapped into parallel to and corresponding to an indefinite fold curve, then a definite fold curve should connect to another cusp with the same property for . Thus we obtain a map such that is isotopic to a regular neighborhood of by the same argument as in Lemma 3.2. Also coincides with and coincides with in a neighborhood of .

We arrange the cusps of in to form pairs as follows. In sheets are attached to along arcs of type (b) (possibly containing points of type (c) at some endpoints). Walking along the bands and restricting ourselves to the intersection of the sheets and , we have that every sheet contains a pair of cusps and every second sheet contains a singular arc of type (a) connecting its pair of cusps, for example, see Figure 11.

Figure 11. The Stein factorization of . There are two -pairs of cusps.

A natural pairing is that two cusps form a pair if they are in the same sheet and they are connected by a singular arc of type (a). We refer to this pairing as -pairing. We also define another pairing : two cusps form a -pair if they are in the same sheet and they are not connected by any singular arc of type (a).

Step 6

In this step, we eliminate the cusps of contained in . These cusps are mapped by in the direction of far from and arranged into -pairs in the previous step. The restriction of the resulting map to a link isotopic to will be an embedding. (Hence after this step the construction of the claimed map on will be easy.)

We have exactly -pairs of cusps in . Observe that for each component of one -pair can be eliminated immediately: for example in Figure 11 the pair on the “highest” sheet is in the sufficient position to eliminate. In the following, we deal with the other -pairs.

More concretely, we perform the deformations and the eliminations of the pairs of cusps of in as shown in Figure 12 as follows.

Figure 12. Moving and eliminating the cusps. We move and eliminate the -pair of cusps along the arrows. The dashed arcs represent -complexes used to deform in the proof of Lemma 3.4.

First, by using Lemma 2.1 we move each pair of cusps having the position as in Figure 12(a) to the position as in Figure 12(b) thus creating a singularity of type (d). Then by using Lemma 2.2 we eliminate each pair of cusps, see Figures 12(b) and 12(c).

The resulting map will be denoted by (see Figure 13). Notice that and coincide in a neighborhood of . The deformations above yield definite fold curves , whose image under is an embedding into as indicated in Figure 13 by the bold curve.

Figure 13. The Stein factorization of the stable map . (The circle is omitted.)
Lemma 3.4.

The link is isotopic to .

Proof.

By Lemma 3.1 the link is isotopic to a longitude of the union of tori . In Step we modify only inside . The subcomplex of used in the proof of Lemma 3.1 is PL-isotopic to a -dimensional PL submanifold of such that goes through the singular curves of type (a) appearing in the -pairing at the end of Step 5 and goes through the top of , i.e. the top of the -complex in Figure 11. To be more precise, in Figure 12(a) the part of connecting the two cusp endpoints of the singular arcs of type (a) is represented by a bold dashed arc and denoted by . During the moving of the pair of cusps as depicted by the arrows in Figure 12(a), is deformed to the curve represented by a bold dashed arc in Figure 12(b). This deformation gives an isotopy between some liftings to of and . Since a part of is collinear to a singular arc of type (a) as we can see in Figure 12(a), any lifting to of is isotopic to any other lifting. Hence further deforming to represented by the bold dashed curve in Figure 12(c) yields an isotopy between some liftings of and . Finally, changing again the lifting to of if necessary, we eliminate the pair of cusps as indicated in Figure 12(b) and deform to be identical to the type (a) singular arc appearing at the elimination in Figure 12(c). All this process gives an isotopy in between and a lifting of , hence an isotopy between and . ∎

Step 7

As a final step, we perform the given surgeries along with the appropriate coefficients. Since is an embedding into on each component of , and consists of definite fold singular curves such that the local image of a small neighborhood of the definite fold curve is situated “outside” of the image of the definite fold curve, a map of