Research Statement. Gabriel Islambouli

Transcription

1 Research Statement Gabriel Islambouli Introduction My research is focused on low dimensional topology. In particular, I am interested in the interaction between smooth 4-manifolds, mapping class groups of surfaces, and the curve complex. My main tool thus far has been trisections of 4-manifolds. A trisection is a decomposition of a smooth 4-manifold which allows one to encode its smooth topology as three sets of curves on a surface. This allows us to translate questions about 4-manifolds to questions about configurations of curves satisfying certain conditions, a purely 2-dimensional problem with many applicable tools. Trisections also serve as a bridge between three and four dimensions, as they resemble Heegaard splittings, which are classical decompositions of 3-manifolds. This statement is divided into four sections. In Section 1, I will give some background on the theory of trisections, and describe work of mine which leads to the first examples of different trisections on a fixed 4-manifold. Section 2 will describe some work connecting smooth 4-manifolds to simplicial complexes classically used to study surfaces. The work described in Sections 1 and 2 draws inspiration from the theory of Heegaard splittings, and is indicative of the deep parallels between the theories. Section 3 describes joint work on the Khovanov homology of infinite braids which includes a nondetection result for mapping classes of the punctured disk. In Section 4, I will describe ongoing projects which seek to further the connections developed in my earlier work. 1 Diffeomorphism classes of trisections In 1898, Poul Heegaard [7] introduced a decomposition of 3-manifolds which we now call a Heegaard splitting. Later, deep work of Moise [16] showed that every orientable 3-manifold admits a Heegaard splitting. It wasn t until 1970 that Engmann [1] showed that a fixed 3-manifold can admit nonequivalent Heegaard structures of the same genus. Moving to dimension four, in 2012 Gay and Kirby [3] showed that an arbitrary smooth, orientable, compact 4-manifold admits a (g, k)-trisection. A schematic of a trisection of a smooth, orientable, closed 4-manifolds can be seen in Figure 1, and should be used as a reference while absorbing the formal definition. Definition 1.1. A (g, k)-trisection of a 4-manifold, X, is a decomposition M = X 1 X 2 X 3 such that: X i is a genus k handlebody of dimension 4. X i X j = H ij is a genus g handlebody of dimension 3. X i = H ij H ki is a genus g Heegaard splitting for X i = # k S 1 S 2. The triple intersection X 1 X 2 X 3 is a genus g surface, which we will call the trisection surface Recall that a cut system for a genus g surface is a collection of g disjoint simple closed curves which cut the surface into a 2g punctured sphere. An elementary application of a theorem of Laudenbach and Poenaru [13] shows that all of the structure of a (g, k)-trisection can be encoded by 3 cut systems on the trisection surface (see the top of Figure 2 for some examples). We call the trisection surface, together with the cut systems, a trisection diagram. This parallels the well known situation of Heegaard splittings which can be described by 2 cut systems on a surface. S 4 admits a (3, 1)-trisection, and its trisection diagram is straightforward to derive. The connected sum operation is defined on trisection diagrams in the obvious way: take the connected sum of the surfaces and leave the curves on both surfaces unchanged. This produces a trisection diagram of the 1

2 H 23 X 2 H 12 X 3 X 1 H 31 Figure 1: A schematic of a trisection. Each X i is diffeomorphic to a 4-dimensional handlebody and each H ij is diffeomorphic to a 3-dimensional handlebody. The three H ij meet in a closed surface indicated by a dot in the center of the trisection. Any two of the H ij form a Heegaard splitting. connected sum of the 4-manifolds. In particular, taking a connected sum with the (3, 1)-trisection of S 4 does not change the underlying 4 manifold, but increases the genus of the trisection surface by 3. This process of increasing the genus without changing the underlying manifold is called called stabilization. Two trisections X = X 1 X 2 X 3 and X = X 1 X 2 X 3 are said to be diffeomorphic if there exists a diffeomorphism, f : X X, such that f(x i ) = X i. Gay and Kirby showed that any two trisections of a fixed 4-manifold become diffeomorphic after some number of stabilizations. This theorem brings up a natural question: can a manifold admit non-diffeomorphic trisections of the same genus? My work in [9] affirmatively answers this question in a strong sense. Theorem 1.1. For any k > 1, there exist infinitely many manifolds admitting 2 k 1 non-diffeomorphic (3k, k)-trisections. The proof of this theorem connects trisections to geometric group theory by showing that a classical property, Nielsen equivalence in groups, can provide an obstruction to a diffeomorphism of trisections. It also parallels a method of distinguishing Heegaard splittings, giving evidence that successful techniques in 3-manifold theory can be imported to say something about smooth 4-manifolds. 2 Trisections and the Pants Complex Among the deepest and most subtle relations in 3-manifold theory is the connection between the topology of a 3-manifold and the Hempel distances of its Heegaard splittings. This distance controls the genus of incompressible surfaces, the geometric structures supported by its topology, and the structure of Heegaard splittings under stabilization. The Hempel distance is defined by a distance in the curve complex of the Heegaard splitting surface. In 2005, Jesse Johnson [10] showed that the distances in other complexes typically used to study mapping class groups also contain interesting information about 3-manifolds via a construction using Heegaard splittings. My work in [8] extends the work of Johnson to four dimensions. We first describe a natural number valued invariant of two trisections which comes from a distance in the pants complex. Recall that a trisection can be described by 3 sets of cut systems drawn on a surface. These cut systems can be extended to a pants decomposition, which is a set of 3g 3 curves on the surface which cut it into 2g 2 pairs of pants. As a consequence of a classical theorem of Waldhausen [20], any two of these sets of pants decompositions can be put into a standard position, so that all of the complexity lies in the third set of curves. See Figure 2 for an example of the standard curves as well as pants decompositions of a surface. To obtain the distance between two (g, k)-trisections, first standardize two sets of curves in each trisection, which makes them identical. Now any difference between these two manifolds is contained in the difference between the final two sets of curves. We extend these sets of curves to pants decompositions and compute the distance between these pants decompositions in the pants complex. There were a few choices suppressed in this description (for example the extension of a cut system to a pants decomposition is not unique), but minimizing over all such choices gives a well defined distance, D(T 1, T 2 ), between two trisections. 2

3 Figure 2: Top: Trisection diagrams for S 2 S 2 and S 2 S 2. Bottom: Pants decomposition extensions of the non-standard set of curves for S 2 S 2 and S 2 S 2 Given two (g, k) trisections, we can stabilize both of them to obtain (g + 3, k + 1)-trisections, which we can again compare in the pants complex. One may of course repeat this action, and compare the twice stabilized trisections to each other. Given a trisection T, let T n be the trisection obtained from T by stabilizing n times. The main result of [8] is the following theorem. Theorem 2.1. Let T 1 and T 2 be (g, k)-trisections of M 1 and M 2, respectively. The natural number lim n D(T n 1, T n 2 ) exists, and depends only on the underlying manifolds M 1 and M 2. In light of this theorem, we define the distance between two 4-manifolds, D(M 1, M 2 ), to be lim n D(T n 1, T n 2 ) where T 1 is any trisection of M 1, and T 2 is any trisection of M 2. Computing this distance directly amounts to solving numerous difficult problems and is quite impractical. Nevertheless, low complexity cases are still amenable to study. For example, there is a close relation between adjacent manifolds and framings of 2-handles in Kirby diagrams. Theorem 2.2. If D(M 1, M 2 ) = 1, then there exist Kirby diagrams for M 1 and M 2 which are identical, except for the framing of some 2-handle. Moreover, if M 1 and M 2 have Kirby diagrams which are identical, except for a framing difference of 1 on some 2-handle, then D(M 1, M 2 ) = 1. As another alternative to computing the distance directly, one can instead bound the distance using other invariants which are easier to compute. In this direction, if σ(m) is the signature of the 4-manifold, then we obtain the following proposition. Proposition 2.1. D(M 1, M 2 ) 1 2 σ(m 1) σ(m 2 ) Since it is easy to construct manifolds with arbitrarily large difference in signature (for example # k CP 2 and # k CP 2 will do) this gives us examples of manifolds which are arbitrarily far apart in the pants complex. 3 Khovanov Homology of Infinite Braids Khovanov homology has become an indispensable tool in knot theory and quantum topology. Applications have ranged from concordance invariants [17] to unknot detection [12]. A common goal among knot homologists is to categorify classical objects associated to knots. Along these lines, in 2014, Rozansky [18] showed that the limiting Khovanov chain complex of the infinite twist on n-strands (i.e. the braid obtained by iterating the full twist on n strands), categorifies the n th Jones-Wenzl projector. In [21], Michael Willis and I prove a vast generalization of Rozansky s result. Before stating the theorem, we first fix some notation and definitions. If B is a braid, then we denote by B k the braid obtained by multiplying B by itself k times in the braid group, and we let 3

4 KC(B) be the Khovanov chain complex of B (in the categorified Temperley-Lieb algebra). We say that an n-strand braid B is complete if, when B expressed as the usual generators σ 1...σ n, each σ i appears at least once, after free reductions. We say B is positive if the generators all appear with a positive exponent. Theorem 3.1. If B be a positive, complete, n-stranded braid, then the limiting complex lim k KC(B k ) categorifies the Jones-Wenzl projector. This result naturally leads to a non-detection result for Khovanov homology. Recall that the braid group on n strands is isomorphic to the mapping class group of the n-punctured disk. A braid is called reducible, periodic, or pseudo-anosov if its image in the mapping class group is reducible, periodic, or pseudo-anosov, respectively. As a corollary of Theorem 3.1, the limiting complexes of any of these iterated braids are chain homotopy equivalent. Since there are positive, complete braids of all three Nielsen-Thurston types, we see that the the limiting Khovanov complex, and so in particular the limiting Jones polynomial, fail to detect the Nielsen-Thurston type of a braid. Nevertheless, it remains possible that other aspects of this limiting process, such as the rate of convergence or the leading terms, do detect the Nielsen-Thurston type. 4 Future Work 4.1 Trisections and Mapping Class Groups My work on connecting 4-manifolds to the pants complex opens many directions for future work. The previously described examples of trisections realizing the arbitrarily large distance in the pants complex have unbounded trisection genus. In contrast, for 3 manifolds, a standard construction based on iterating a generic pseudo-anosov map shows that the distances in the pants complex can grow arbitrarily large in any fixed genus. In the 4-manifold case, there are a finite number of genus 1 and 2 trisections, so the exact analogue can not hold. Nevertheless, it seems likely that the following question has a positive answer. Question 4.1. Given an arbitrary natural number n, do there exist genus 3 trisections, T 1 and T 2, such that D(T 1, T 2 ) > n? While the answer to this question is interesting in its own right, perhaps more interesting is that a solution will likely involve interactions between trisections and the mapping class group. To elaborate, after fixing some auxiliary identifications, a trisection can be described by two positive integers, g and k, together with a mapping class of a genus g surface. This mapping class element is quite restricted, as it must, in some sense, be a gluing map for a genus g Heegaard splitting for # k S 1 S 2. A solution to Question 4.1 could come from a pseudo-anosov mapping class whose powers both satisfy the gluing map condition and are generic with respect to some of the handlebodies in the trisection. Since this map contains essentially all of the information of the 4-manifold, one can, quite generally, ask which properties of a mapping class correspond to interesting properties of a 4-manifold. Ambitiously, one can ask if it is possible, in practice, to identify which mapping classes lead to a 4-manifold admitting a geometric structure. As a stepping stone towards a solution, we offer the following intermediary question. Question 4.2. Under what conditions does a mapping class give rise to a 4-manifold with a hyperbolic fundamental group? In a similar vein, given a trisection of a 4-manifold, M, with trisection surface Σ, one may ask which mapping classes of Σ extend to all of M. Phrased differently, this is the group of diffeomorphisms of M which fix Σ setwise, up to isotopies also fixing Σ setwise. We call this the mapping class group of a trisection and denote the group by Mod(M, Σ). If φ is an element Mod(M, Σ) then we let the Nielsen-Thurston type of φ be the Nielsen-Thurston type of φ Σ. The analogous group for a Heegaard splitting is the well known Goeritz group. The challenging problem of determining generators of the Goeritz group for splittings of S 3 has sustained nearly a century of inquiry [5] [2]. The problem in dimension four is expected to be more challenging, but there are still many tractable problems which are still of interest. For example, drawing inspiration from [11], which addressed the related problem for 3-manifolds, one may ask the following question. 4

5 Question 4.3. Under what conditions does M od(m, Σ) admit a pseudo-anosov/periodic/reducible mapping class? 4.2 Pants Complexes The connection between Morse functions (i.e. generic 1-parameter functions) on a surface and the 1-skeleton of the pants complex is well understood and straightforward. On the other hand, the connection between Cerf functions (i.e. generic 2-parameter functions) on a surface and the 2-skeleton of the pants complex is more involved. The latter connection has been employed in numerous contexts, most notably to derive a presentation of the mapping class group of a surface. Trisections were originally born from the notion of a Cerf function and so this leads naturally to the following question: Question 4.4. What is the relationship between trisections and the 2-skeleton of the pants complex? Some evidence that this interaction is worth pursuing is work in progress which reduces the main theorem of [14] to a question of simple connectivity of a subset of the pants complex. The original proof employs deep results about Dehn surgery and the fact that the pants complex sees these results as well is very promising. One can also interpret the work of [4] in terms of the 2-skeleton of the pants complex, giving nice descriptions of cobordisms of a trisected 4-manifold to a connected sum of copies of CP 2 and CP 2. In another direction, Grigory Mikahlkin [15] has developed a notion of a pants decomposition of certain even dimensional manifolds. This decomposition comes from tropical geometry and there is some evidence that the homological mirror symmetry conjecture can be attacked from this angle [19]. Golla and Martelli [6] have furthered Mikhalkin s work by showing that a large class of 4-manifolds admit pants decompositions. While it is well known that pants decompositions form a basis for a variety of additional structures on a surface, it is unknown what exactly pants decompositions on 4-manifolds parameterize. The first step in discovering this is answering the following question: Question 4.5. Classify all of the ways a 4-manifold can decompose into pants. Preliminary results yield a list of moves which can change pants decompositions on 4-manifolds; whether or not these moves suffice is a subject of future work. References [1] Renate Engmann. Nicht-homöomorphe Heegaard-Zerlegungen vom Geschlecht 2 der zusammenhängenden Summe zweier Linsenräume. Abh. Math. Sem. Univ. Hamburg, 35:33 38, [2] M. Freedman and M. Scharlemann. Powell moves and the Goeritz group. ArXiv e-prints, April [3] David Gay and Robion Kirby. Trisecting 4 manifolds. Geom. Topol., 20(6): , [4] David T. Gay. Trisections of Lefschetz pencils. Algebr. Geom. Topol., 16(6): , [5] Lebrecht Goeritz. Die abbildungen der brezelfläche und der vollbrezel vom geschlecht 2. Abh. Math. Sem. Univ. Hamburg, 9(1): , [6] Marco Golla and Bruno Martelli. Pair of pants decomposition of 4-manifolds. Algebr. Geom. Topol., 17(3): , [7] P. Heegaard. Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhang. Kjöbenhavn. 104 S. 8 (1898)., [8] Gabriel Islambouli. Comparing 4-manifolds in the pants complex via trisections. Algebr. Geom. Topol., 18(3): , [9] Gabriel Islambouli. Nielsen equivalence and trisections of 4-manifolds. ArXiv e-prints, April [10] Jesse Johnson. Heegaard splittings and the pants complex. Algebr. Geom. Topol., 6: ,

6 [11] Jesse Johnson and Hyam Rubinstein. Mapping class groups of Heegaard splittings. J. Knot Theory Ramifications, 22(5): , 20, [12] P. B. Kronheimer and T. S. Mrowka. Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Études Sci., (113):97 208, [13] François Laudenbach and Valentin Poénaru. A note on 4-dimensional handlebodies. Bull. Soc. Math. France, 100: , [14] Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11): , [15] Grigory Mikhalkin. Decomposition into pairs-of-pants for complex algebraic hypersurfaces. Topology, 43(5): , [16] Edwin E. Moise. Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann. of Math. (2), 56:96 114, [17] Jacob Rasmussen. Khovanov homology and the slice genus. Invent. Math., 182(2): , [18] Lev Rozansky. An infinite torus braid yields a categorified Jones-Wenzl projector. Fund. Math., 225(1): , [19] Paul Seidel. Some speculations on pairs-of-pants decompositions and Fukaya categories. In Surveys in differential geometry. Vol. XVII, volume 17 of Surv. Differ. Geom., pages Int. Press, Boston, MA, [20] Friedhelm Waldhausen. Heegaard-Zerlegungen der 3-Sphäre. Topology, 7: , [21] Michael Willis and Gabriel Islambouli. The Khovanov homology of infinite braids. Quantum Topol., 9(3): ,