Monodromy Substitutions and Rational Blowdowns
We introduce several new families of relations in the mapping class groups of planar surfaces, each equating two products of right-handed Dehn twists. The interest of these relations lies in their geometric interpretation in terms of rational blowdowns of 4-manifolds, specifically via monodromy substitution in Lefschetz fibrations. The simplest example is the lantern relation, already shown by the first author and Gurtas  to correspond to rational blowdown along a sphere; here we give relations that extend that result to realize the “generalized” rational blowdowns of Fintushel-Stern  and Park  by monodromy subsitution, as well as several of the families of rational blowdowns discovered by Stipsicz-Szabó-Wahl .
57R65, 20F38, 57R17 \extralineThe first author was partially supported by Grand-in-Aid for Scientific Research (C), No. 21540079, Japan Society for the Promotion of Science. The second author was partially supported by NSF grant DMS-0905380.
It has been known for some time that a rational blowdown of a smooth 4-manifold can be performed symplectically. Moreover, symplectic 4-manifolds are well-known to correspond to Lefschetz pencils or, after suitable blowup, to Lefschetz fibrations. Our aim here is to combine these two ideas to show that under certain circumstances one can perform a rational blowdown on a 4-manifold equipped with a Lefschetz fibration, and preserve the fibration structure.
Recall that a Lefschetz fibration on a closed smooth 4-manifold is a smooth map that is a fiber bundle projection away from finitely many singular points. Near the singular points is required to appear in appropriate oriented local complex coordinates as . The Lefschetz fibration is said to have genus if the typical fiber of is a smooth surface of genus . A theorem of Gompf (, Theorem 10.2.18) shows that under mild hypotheses the total space of a Lefschetz fibration admits a symplectic structure.
The monodromy of around a critical value is well-known to be isotopic to a (right-handed) Dehn twist around a simple closed curve in , the vanishing cycle associated to the critical point. Arranging the critical values in a cyclic order, the Lefschetz fibration is determined up to isomorphism by the global monodromy given by the sequence of corresponding Dehn twists. Observe that the composition of all these Dehn twists is isotopic to the identity (it is the sequence itself that determines , not the composition). Hoping it does not lead to confusion, we often refer to this global monodromy as an (unreduced) word in right-handed Dehn twists, though the monodromy is determined only up to cyclic permutation of the twists and simultaneous conjugation by some diffeomorphism of the fiber . We write for the Lefschetz fibration determined by the word .
For a surface let (or ) denote the mapping class group of , the group of isotopy classes of orientation-preserving diffeomorphisms of (or of , fixing the boundary pointwise). Suppose that and are two words in right-handed Dehn twists with the property that in , and suppose is a Lefschetz fibration with fiber and global monodromy given by for some word That is to say, and are distinct factorizations of a given element of into products of Dehn twists, and one of these factorizations appears in the monodromy word associated to . By a monodromy substitution, we mean the operation
of replacing by the Lefschetz fibration having global monodromy . Observe that the products and are equal in , so that the boundaries of and are naturally diffeomorphic. In particular, if is isotopic to the identity so that is a closed Lefschetz fibration, the same is true of . Generally and will be different smooth 4-manifolds: indeed, if and involve a different number of Dehn twists, then and have differing Euler characteristics. However, does still possess the structure of a Lefschetz fibration (with the same fiber) and hence is symplectic.
Examples: 1) If is a curve that bounds a disk on , then the corresponding Dehn twist is isotopic to the identity. Writing also for the right-handed Dehn twist around , suppose for some word . The result of monodromy substitution based on the relation in gives rise to the fibration with monodromy . As a smooth -manifold, is obtained from by contracting a sphere of self-intersection , i.e., an ordinary blowdown operation.
2) The lantern relation states that , where the letters indicate (twists around) the curves shown in Figure 3 below—thinking of the 3-holed disk in that figure as a subsurface of . It was shown in  that in the corresponding monodromy substitution , the resulting manifold is obtained from by cutting out a neighborhood of a sphere with normal bundle of degree and replacing it by a rational homology ball: a rational blowdown.
It is natural, particularly in light of the second example, to ask whether there are other relations in appropriate mapping class groups that correspond to other rational blowdowns under monodromy substitution. By a rational blowdown, we will mean the operation of replacing the neighborhood of a configuration of spheres in a smooth 4-manifold, intersecting according to some connected plumbing graph, by a rational ball having the same oriented boundary. There are many examples of plumbed 4-manifolds whose boundary also bounds a rational ball, the first having been studied in this context by Fintushel and Stern . A generalization of their examples to a family of linear plumbing graphs was considered by Park , and several more families were discovered by Stipsicz-Szabó-Wahl . Our main result shows that many of these rational blowdowns can be realized by monodromy substitution.
Let be any plumbing graph among the families , , or (see below). Then there exists a planar surface and a relation of the form in , where and are words in right-handed Dehn twists, with the following property. Suppose that is a Lefschetz fibration with fiber contaning as a subsurface, and global monodromy of the form . Then contains an embedded copy of the plumbing of disk bundles over spheres determined by , and furthermore the Lefschetz fibration given by monodromy substitution is diffeomorphic to the rational blowdown of along this plumbing.
The families and consist of graphs and , respectively, for arbitrary nonnegative integers ; they are shown in Figure 1. The family contains linear plumbing graphs indexed by a pair of relatively prime integers , with vertices weighted by the integers . The are the continued fraction coefficients of :
A simple example of a relation arising in the theorem is the one corresponding to Fintushel-Stern’s original construction: a linear plumbing graph (, corresponding to ) whose vertices are weighted by the (negatives of the) coefficients of the continued fraction expansion of (that is, by where occurs times). The corresponding relation is in the mapping class group of a sphere with holes; the case is just the lantern relation. The relation for general is shown in Figure 2, and is called a “daisy relation” for obvious reasons. We remark that this relation has appeared also in the work of Plamenevskaya and the third author .
We close this introduction with a couple of remarks. First, Theorem 1.1 is entirely local: it proceeds by finding a Lefschetz fibration on the neighborhood of a plumbing in the listed families, having a planar surface as fiber. Then we apply a relation to write the monodromy of the corresponding open book on as a different product of Dehn twists, and the corresponding different Lefschetz fibration is a rational ball with diffeomorphic boundary. In particular, there is no need to assume that is a closed Lefschetz fibration to begin with. On the other hand, the hypotheses of Theorem 1.1 appear to be somewhat restrictive, in the sense that if one encounters a (symplectic) manifold containing a plumbing among the listed families, it is not obvious whether a global Lefschetz fibration structure exists having a monodromy of the required form. However it can be shown that the local Lefschetz fibration on can be extended to as a broken Lefschetz fibration (c.f. , , , ). It is possible, moreover, that the purely local structures introduced here can be used to show that rational blowdowns (or more general operations) obtained by monodromy substitution may be performed symplectically. We leave this as a future project.
Finally, there are several families of rational blowdowns (notably the family of ) not covered by Theorem 1.1. It is an interesting question to decide whether such blowdowns can also be recovered by substitution techniques. On the other hand, the methods of this paper yield many more relations in planar mapping class groups (or in mapping class groups of surfaces of higher genus). So far, only those relations in Theorem 1.1 have been given an “interesting” geometric interpretation, but it is natural to hope for new examples of operations on 4-manifolds arising from these ideas.
Organization: The relations needed for our monodromy substitutions are derived in the following section. On some level, the proof of Theorem 1.1 is entirely elementary once the relations are in hand, and requires only some basic checking. However, in section 3 we produce Kirby diagrams for the rational balls appearing in our rational blowdowns, and verify that they are diffeomorphic to those found in , , and . In the final section we construct a family of closed Lefschetz fibrations over to which a monodromy substitution corresponding to a daisy relation may be applied, yielding minimal symplectic manifolds homeomorphic but not diffeomorphic to for each .
Acknowledgements: The first author is grateful to Kouichi Yasui for helpful discussions; we also thank András Stipsicz for his interest and encouragement.
2 Monodromy relations
All of our relations may be derived from the basic lantern relation (indeed, the planar mapping class group may be given a presentation in which the lantern is the only relation, aside from commutation relations: see ). Recall that if is a sphere with four holes and are the curves indicated in Figure 3, then the lantern relation states that
Here we are sloppy and use the same symbol for (the isotopy class of) a curve and (the isotopy class of) the right-handed Dehn twist about that curve, and in the above relation we use group multiplication: that is, in the product , the twist around precedes the twist around . Note that the surface of Figure 3 may also be a subsurface of a more general surface.
Our generalizations of this relation are all based on the following simple construction. Consider a planar surface containing as a subsurface the pair-of-pants , and let and be the boundary-parallel curves marked as in Figure 4(a).
We construct a new surface with one additional boundary component by removing a small disk from (and ), the closure of which we take to lie in the interior of the region between the curve and the hole of it encloses. If the component of corresponding to coincides with a component of , then we can think of as obtained from by gluing a disk with two holes into the hole enclosed by . (Intuitively, we think of this as “splitting” the hole enclosed by in two.) Extending by the identity across this new 2-holed disk induces a homomorphism that we use implicitly in the following lemma. Note that in this situation, the twist around commutes with the image of .
Key Lemma 2.1
Given the setup of the previous paragraph (in particular, encloses only one hole of ), suppose that in the planar mapping class group the relation
holds, for some . Assume that commutes with either and , or with and . Then in we have the relation
where are the curves in marked in Figure 4(b).
Since commutes with and (either on or ) and with , the given relation shows
in . Multiply both sides by on the left (if commutes with , ) or right (in the other case), and observe that both and are central in . The lantern relation completes the proof.
Observe that the commutativity assumption on is trivially satisfied if the hole of corresponding to coincides with a hole of : in this case is boundary parallel and therefore central.
Our plan in this section is to inductively construct several families of relations on planar surfaces using the Key Lemma. For reasons that will become clear subsequently, we are looking for relations of the form satisfying the following properties:
Both and are words in right-handed Dehn twists.
consists of twists around a collection of pairwise disjoint curves.
If is an -holed disk, then is a product of twists around curves that span the rational first homology of .
The prototypical example of such a relation is the lantern relation itself. Observe that if relation is obtained from relation by successive applications of the Key Lemma 2.1, and if satisfies (1), (2), and (3) above, then will satisfy (1) and (3). By careless choices of subsurface , however, it is easy to disrupt property (2).
2.1 Relations corresponding to family
Theorem 2.2 ((-family of relations))
For three integers , let be a disk with holes, with corresponding boundary-parallel curves , , (arranged on a circle and numbered clockwise for convenience). Let , , be disjoint simple closed curves respectively enclosing only the , or (see Figure 5), and let be a circle parallel to the outer boundary of . For , let be a “convex” curve enclosing exactly the holes corresponding to and . Similarly let enclose all ’s and a single , and enclose all ’s and a single . Then in we have
Begin with the lantern relation of Figure 3, and apply the Key Lemma to the pair of holes enclosed by and to split . Since encloses just one hole of the surface, the commutation requirement of the Key Lemma is satisfied. We obtain a relation on a disk with four holes, of the form .
Applying the same procedure to the holes and gives on a disk with five holes, and inductively we get the relation
on a disk with holes as in Figure 6 (this is, incidentally, the daisy relation of the introduction).
Repeat this procedure using the holes and to split into holes and replace in the relation above by a product around curves each enclosing and one of the holes . We find
where is a circle enclosing all the .
Finally we use the curve to split into holes. Observe that in this case one of the holes of the subsurface from the Key Lemma encloses more than one hole of our surface; however the commutation requirement is still trivial here since all the curves on the left hand side of the relation above are disjoint so that the corresponding twists pairwise commute, while the word is trivial in this case. The desired relation follows.
Clearly, the relation just obtained satisifies conditions (1), (2), and (3) mentioned previously.
2.2 Relations for family
To derive this family of relations, take in Figure 6 and redraw it so that the hole enclosed by becomes the outer boundary: we have the relation in the labeling conventions of Figure 7. For , apply the Key Lemma consecutively times, using the highest-numbered curve to split the corresponding hole into two (the lemma applies since here). The result is the relation , where is a curve enclosing the -holes.
Now for repeat the process times using the highest-numbered to split the corresponding in two. Observe that the curve playing the role of in the Key Lemma is a circle enclosing all the holes , and the twist around this circle commutes with all the . Hence the commutativity requirement of the Key Lemma holds. The result is a relation depicted in Figure 8, which is drawn on a sphere with holes rather than a disk with holes. This will be the case of our family of relations corresponding to the family of rational blowdowns.
Observe that the curve is isotopic (on the sphere with holes) to a curve enclosing the holes and . For given , apply the Key Lemma times to split hole into (and into curves ), noting that here the element from the Key Lemma is trivial: we obtain the relation of Figure 9.
Theorem 2.3 (-family of relations)
Fix three integers , and let be a sphere with holes , , and , and use the same letters to indicate boundary-parallel circles around these holes (and also Dehn twists around those curves). Let and be circles enclosing holes and holes respectively, and let and be circles enclosing all the holes labeled and all holes labeled respectively. Finally, let , , be circles such that:
For , the curve encloses all holes and hole .
For , the curve encloses all holes and hole .
For , the curve encloses holes and hole .
See Figure 9. Then if we have the following relation in :
If , then, writing for ,
2.3 Relations corresponding to linear plumbings
Recall that given relatively prime integers , there is a unique continued fraction expansion
where each . Moreover, this expansion can be obtained from the base case by repeated applications of the two operations
See, for example, , Proposition 4.1. We indicate here how to obtain relations in the mapping class groups of planar surfaces by following a parallel procedure. The base case is the lantern relation itself. Supposing the construction of the continued fraction to have begun with applications of the move (a) above, we apply the Key Lemma times to obtain the relation of Figure 6, with replaced by . If the next stage calls for applications of (b), apply the Key Lemma times to the last-constructed curve in the previous stage, , to split hole into holes.
And so on: the next stage of construction of the continued fraction calls for, say, applications of (a). The parallel is to apply the Key Lemma times using the last-constructed curve to split the hole that was not split during the previous stage. The process continues in this manner to obtain a relation in a planar mapping class group corresponding to each fraction . As an example, the relation corresponding to and , with continued fraction , is shown in Figure 10. Note that here , .
3 Kirby diagrams for rational balls
The proof that the monodromy substitutions outlined previously result in rational blowdowns amounts to analyzing Kirby pictures of Lefschetz fibrations with bordered fibers over : specifically, the Lefschetz fibrations with planar pages arising in our monodromy relations. In the following, we consider Kirby pictures for the 4-manifolds described by each side of the various relations and show that one side describes a plumbing of spheres while the other side is a rational ball. (The second of these points is essentially obvious, but we give explicit Kirby pictures in any case to relate these rational blowdowns to those in  and .)
To begin with, we observe that if is a planar surface, diffeomorphic to a disk with smaller disks removed, and if is a Lefschetz fibration over having standard fiber and described by the global monodromy word , then there is a standard Kirby diagram for obtained as follows. First note that a neighborhood of a regular fiber is diffeomorphic to the complement of standard disks in , and therefore has a Kirby diagram consisting of an -component unlink decorated by dots. In fact, drawing as a round disk in the plane with disks deleted, we can consider the components of this unlink as vertical line segments passing through the centers of the deleted disks (and closed in a standard way). Indeed, the decomposition of into corresponds to the usual decomposition of into the union of solid tori, a meridian disk of one of which we identify with the disk containing . Vertically-displaced parallel copies of apparent in the picture are then nearby fibers. The standard disks we must remove from then have boundary equal to the circles swept out by the centers of the disks to delete from , as we move around one of the solid tori.
To build from this point, we add framed 2-handles to vanishing circles on parallel copies of , in the order (from bottom to top) in which they appear in the word . In particular, if the collection of vanishing cycles has the property that their rational homology classes form a basis for , then it is easy to see that has trivial rational homology. Furthermore, the effect of a monodromy substitution is local: it amounts to replacing a piece of a global Lefschetz fibration diffeomorphic to by another piece (here we think of as Lefschetz fibrations whose fiber is a subsurface of that of —a planar subsurface in all the cases at hand). Thus we need only understand the topology of : the properties (1), (2), and (3) of the previous section ensure that is diffeomorphic to a plumbing (but see below for a direct verification of this), and that is a rational ball.
3.1 Graphs in family
We first verify that the Lefschetz fibration described by the longer of our two words in the relation for the -family of substitutions is diffeomorphic to a plumbing of disk bundles over spheres according to the graph of . The procedure outlined above gives rise to the Kirby picture in Figure 11 (note that all the twists in this word commute, so the order of the handle additions is immaterial).
Slide the outermost circle over the three 2-handles that are outermost in their respective families to obtain Figure 12.
Then successively slide each of the outermost 2-handles in the three families over the next one in to get a chain of -framed circles as shown in Figure 13.
Now slide the last remaining “large” 2-handles over each remaining circle in its family, cancelling the latter with its corresponding 1-handle at each stage. This has the effect of subtracting the number of 1-handles in that cluster from the framing on the last 2-handle; the end result is the desired plumbing.
Turning to the word on the right hand side of the relation of Theorem 2.2, note first that since the length of this word is equal to and since the vanishing cycles are easily seen to span , the fact that the corresponding Lefschetz fibration over is a rational ball is clear. For the sake of completeness, however, we derive an explicit picture of this rational ball, and show it is diffeomorphic to one constructed in . In fact, we begin with the latter construction.
Recall from  that to a negative-definite star-shaped plumbing graph one can associate a “dual graph” such that the corresponding plumbed 4-manifolds and have diffeomorphic boundaries with opposite orientation, and is diffeomorphic to a blowup of . In certain cases one can also find the dual plumbing embedded in a (different) blowup of in such a way that spans the rational homology; its complement in this embedding is then a rational ball bounded by with the correct orientation. This plan is carried out in ; in that case the dual graph to is a star-shaped graph with three legs shown in Figure 14, where the undecorated vertices have weight .
Following  we find such a plumbing in a rational surface by blowing up a configuration of four general lines in .
Here the basic dual plumbing is visible (for ); , , and additional blowups at the indicated intersections allow us to construct the dual of any desired . The complement of is given by the three -framed 2-handles together with the 3- and 4-handles in Figure 16. To obtain a standard Kirby picture for it, we must turn these handles upside down: this is complicated by the presence of 3-handles, which become 1-handles upon inverting the handlebody and should therefore be attached before the (inverted) 2-handles.
To obtain the picture, recall that the 2-handle dual to one of the framed circles is attached along a 0-framed meridian to that circle. Now, the 3-manifold described by all the 2-handles in Figure 16 is necessarily diffeomorphic to , and can therefore be simplified to a diagram of a 0-framed 3-component unlink. Replacing 0’s by dots then describes the duals to the 3-handles. Our task is to follow the three meridians through this simplification; the resulting framed link together with the dotted unknots gives the desired diagram for our rational ball, after reversing its orientation.
Our claim is that the rational ball described by the monodromy word under consideration is diffeomorphic to this one. To see this, consider the natural Kirby picture for that Lefschetz fibration, shown in Figure 21(a) (where we have taken for simplicity). Perform the indicated 1-handle slides, and then slide one 2-handle of each pair over the other as in (b). Cancelling the 1-handles that were slid with the 2-handles that were not gives Figure 21(c), where the reader will have no trouble generalizing to arbitrary .
This last figure is identical with that obtained by “pulling tight” the large circles in Figure 19, changing ’s for dots, and reversing orientation.
3.2 Graphs in family
The construction here follows the same line as for the previous case; in particular we leave the verification that the Lefschetz fibration determined by the left side of our relation for this family is diffeomorphic to the plumbing of spheres given by the graph to the reader. Note that if one draws the fiber surface as a sphere with holes rather than a disk with holes, the corresponding Kirby picture can be obtained just as before, with a dotted vertical circle for each hole, provided one encloses all of these circles by a 0-framed 2-handle (c.f. Figure 22).
It was observed in  that this graph can be found spanning the rational homology in a blowup of , by blowing up a configuration of four generic lines. Beginning with Figure 15 again, this time we blow up to obtain Figure 24 which, ignoring the circles and the and handles, describes the plumbing .
To find the complement of the plumbing, we add -framed meridians to the circles as before, and simplify the resulting diagram to the standard one for a connected sum of three copies of (with additional framed circles corresponding to the added meridians). Adding the -framed meridians and blowing back down to our original four lines gives Figure 25.
Again, we can check that Figure 27 describes the same rational ball as that given by the Lefschetz fibration corresponding to our monodromy word: the handle picture for the latter is shown in Figure 28.
3.3 Linear plumbings
As a warmup we consider the case of . Figure 30 is a Kirby diagram for , as is easily checked. For , blow up one of the clasps between the and curves times to obtain Figure 31, in which we see the configuration embedded in .
The complement of that configuration is given by the framed 2-handle, together with the 3- and 4-handles, and is clearly a rational ball. Rather than pausing to derive the (well-known) Kirby diagram for this ball, we continue to the general case.
It is well known that to construct the general one continues to blow up Figure 31 at one or other clasp of the curve to see a copy of embedded in a connected sum and spanning the rational homology. In principle, one could carry out the procedure of the previous subsections to obtain a diagram for from this picture, but complications arise when the blowups alternate between clasps: in this case the curve begins to link the curve to blow down many times, and the diagram quickly becomes very complicated. To alleviate this difficulty somewhat we introduce the following device. The blowups performed to obtain Figure 31 have been performed at the clasp marked with a star; we suppose that the next stage in the construction of calls for a blowup at the other clasp of the curve. Introduce a cancelling pair of - and -handles and slide the 2-handle over the two circles between which the next blowup is to be performed (i.e., the circle and the adjacent circle), as indicated in Figure 32.
Then continue the blowups using the clasp of the curve marked with a star in that figure. Continue this procedure: each time the recipe for calls for the sequence of blowups to switch from one clasp to the other of the curve, we introduce a new -handle pair and “save a copy” of the curve before proceeding.
The result of this plan is a Kirby picture for a connected sum of copies of in which is visible, with some number of additional 2- and 3-handles. (The number of additional 2/3 handle pairs is one fewer than the number of continued fraction coefficients of different from .) As before, we introduce 0-framed meridians on each “extra” 2-handle and blow back down. After all blowdowns are complete we will be left with a 0-framed unlink together with the (now knotted and nontrivially framed) former meridians; changing the 0-framings to dots gives the Kirby diagram for .
To determine this diagram suppose the final sequence of blowups is as described in Figure 33; here we have drawn only the part of the diagram impinging on the last stage of the construction, which we suppose to have involved blowups for some . (In framing the outermost curves by and , we are also implicitly assuming that , i.e., this is not a diagram for .) In Figure 33 we have introduced -framed meridians dual to the circles, and also a -curve that will be useful for us momentarily. Perform blowdowns to reach Figure 34(a). One additional blowdown yields Figure 34(b); sliding one circle over the other as indicated, an isotopy then reaches Figure 34(c).
The iterative step is now reasonably clear. Think of the linking - and -framed circles as lying in a genus 2 handlebody, which is a neighborhood of a -curve linking the curve and the -framed meridian (the boundary of this neighborhood is indicated by the dashed lines in Figure 34(c)). To continue our blowdowns with the -circle followed by the -circles to the right in Figure 34(c), we can return with a rotation to Figure 33, with the -curve in that figure replaced by the “-link” in Figure 34(c). Note, however, that one blowdown in this next stage has already been performed, corresponding to the transition from Figure 34(a) to (b). The result of this iteration is shown in Figure 35, and the process clearly continues with another “-link” added in the center of Figure 35(b) each time the sequence of blowdowns switches from one clasp of the curve to the other. (Note that while the framing and number of twists appearing in the first -link differs from the framing appearing in the original plumbing by one, for successive -links this number is adjusted by 2: in other words the next -link to appear in this process will have framing and twisting number both equal to .)
The final stage of the construction is shown in Figure 36, which depicts the situation after blowing down Figure 32 once and applying the construction of Figure 34, with the additional -links indicated.
To state the end result we need only determine the framings and numbers of twists each -link receives.
Let be the continued fraction expansion of (where each ), so that can be obtained from the length-one expansion by repeated application of the operations
Let be the sequence of integers different from appearing among the , written in the order in which they arise during the application of (a) and (b) above. Then a Kirby diagram for a rational homology ball with boundary is given by Figure 37.
Indeed, the numbers