RESEARCH STATEMENT MARGARET NICHOLS 1. Introduction My research lies in geometry and topology, particularly in the study of 3-manifolds. A major theme in 3-manifold topology since the work of Haken in the 60s is to study 3-manifolds inductively by cutting them repeatedly along embedded surfaces. For a good theory, the surfaces must be as simple as possible; in a homological version of this theory (due to Thurston), this means they should be of minimal complexity in their homology class. This approach requires analyzing 3-manifolds with boundary, and properly embedded surfaces therein. This leads to the study of sutured 3-manifolds, whose boundaries are decomposed along a system of embedded loops (the sutures) into an upper and a lower surface. The sutured manifold is taut if these two surfaces are of minimal complexity in their relative homology class. The property of tautness is of great interest for a number of applications, but it can be difficult in practice to certify that a sutured manifold is taut. Recently Friedl and Kim [FK13] observed that tautness follows from a purely homological condition, that the sutured manifold is a (relative) homology product. This condition is sufficient, albeit unnecessary. However, using the work of Agol on virtual fibration [Ago08], they show that a sutured manifold is taut if and only if it is a twisted homology product for some representation of the fundamental group. We call any such representation product-like. The existence of a product-like representation is not effective, and it is a challenge to relate the property of being product-like directly to the topology of the manifold. Thus we are motivated to study the following: Problem 1. Given a taut sutured manifold, relate the minimal complexity of a product-like representation to topological features of the manifold. For example, we might want an effective bound on the dimension of some product-like representation. Conversely, given such a bound, we might want to understand what constraints this puts on the manifold. My main results to date give the following progress on this problem: Theorem A. I describe a class of 3-manifolds for which no twisting is required, that is, for which the trivial representation is product-like; Theorem B. I give an example of a manifold which is, topologically, only slightly more complicated than those in the class above, but which does require twisting by a representation of dimension 1; Theorem C. I describe an infinite family of 3-manifolds which require twisting by a representation of dimension 2; Theorem D. I describe a sequence of 3-manifolds with arbitrarily high complexity when restricting to solvable representations: any product-like solvable representation associated to the nth manifold must have image with derived length at least n. As a corollary, these examples do not admit solvable product-like representations of small dimension (dependent on n). Date: October 30, 2018. 1
2 MARGARET NICHOLS 2. Background My main objects of study are sutured 3-manifolds. Precisely: Definition 2. A sutured 3-manifold is a 4-tuple (M, R +, R, γ) consisting of a 3-manifold M, a collection of embedded loops γ M, and subsurfaces R +, R of M, such that M decomposes as M = R + γ R. We say the sutured manifold M is taut if R + (equivalently, R ) is of minimal complexity in [R +, γ] H 2 (M, N(γ)). In this setting, complexity refers to a modified notion of Euler characteristic, and a surface has minimal complexity exactly when it achieves the Thurston norm of the homology class. When M is a product manifold, that is, when M is homotopy equivalent (relative to γ) to R ±, it is necessarily taut. Friedl and Kim make the observation that this can be weakened to a homological condition, and that tautness follows from M having the homology of a product manifold. Crucially, this is true when considering homology with twisted coefficients. Definition 3. A sutured manifold (M, R ± ) is a twisted homology product if, for some representation α : π 1 (M) GL n (C), the inclusion ι : R ± M induces isomorphisms on the homology groups with coefficients E α twisted by α: We say such a representation is product-like. ι : H (R ± ; E α ) = H (M; E α ). (1) With the knowledge in hand that such a representation must exist ([FK13]), we are prompted to ask the nature of a product-like representation. We might hope it arises from a geometric structure on the manifold; in [DFJ12], Dunfield, Friedl, and Jackson conjecture exactly this in the case that M is hyperbolic (within the somewhat different but related setting of twisted Alexander polynomials), or somewhat weaker, Agol and Dunfield conjecture the following. Conjecture 4 ([AD15]). Any taut sutured manifold M is a twisted homology product for some α : π 1 (M) SL 2 (C). More broadly, we ask if we can at least bound the minimal dimension of product-like representations, or describe their complexity in some other terms, such as the minimal nilpotency class or derived length required of a nilpotent or solvable representation. Previous work. Other work on this problem has primarily focused on the conjecture of Dunfield, Friedl, and Jackson, exploring the relationship between the Thurston norm of [R ± ] and the twisted Alexander polynomial arising from the hyperbolic structure on M. In particular, Morifuji and Tran show the conjecture holds when M is a 2-bridge knot complement ([Mor12],[MT14]), and Agol and Dunfield expand this to the class of libroid knots ([AD15]). 3. My results In [Nic18], I restrict attention to the case of a sutured handlebody. As the fundamental group is free, the representations π 1 (M) GL n (C) form a very simple space, and the complexity of finding product-like representations reduces to understanding the sutures. We see that even when the genus of the handlebody is small, a variety of behaviors arise.
RESEARCH STATEMENT 3 Handlebodies of genus two with a single suture. In this simplest of situations topologically, the corresponding algebraic picture is also as simple as we could hope, with M always being a rational homology product. Here, every sutured handlebody is taut except when the suture actually bounds a disk in M. Theorem A ([Nic18]). Let M be a taut sutured handlebody of genus 2 with a single suture. Then ι : H (R ± ; Q) = H (M; Q) is an isomorphism. In order to establish this, I must understand the delicate relationship between the algebraic property of being a commutator in the fundamental group of the handlebody and the geometric property of being realizable as a simple closed curve on the boundary of the handlebody. I then analyze this relationship using the theory of commutator calculus. Handlebodies of genus two with multiple sutures. In contrast with Theorem A, I show the situation of a genus-2 handlebody with multiple sutures is inherently more complicated. Theorem B ([Nic18]). There exists a genus-2 taut sutured handlebody which is not a rational homology product. Figure 1 shows such a handlebody. The key observation is that, without the geometric constraint of a single suture, we have significantly more flexibility in the arrangement of the sutures, while still giving a taut manifold. This allows us to choose γ so that there is an injection π 1 (R + ) π 1 (M), but the generators of H 1 (R + ) have the same image in H 1 (M), and the induced map on homology fails to be injective. Figure 1. The three curves shown divide M into two pairs of pants. While this example fails to be a rational homology product, we can find plenty of product-like 1- dimensional representations. Any choice of representation α : π 1 (M) GL 1 (C) such that α(xy) 1 realizes M as a twisted homology product. Examples in higher genus. There are many examples of handlebodies of genus at least 3 that are not twisted homology products for any representation with dimension less than 2. Remember that, according to the conjecture of Agol-Dunfield, every taut sutured manifold should be a twisted homology product for some 2-dimensional representation. Theorem C ([Nic18]). For all g 3, there exists a taut sutured handlebody M of genus g which fails to be a twisted homology product for every representation α : π 1 (M) GL 1 (C). For all even g 4, we may make the additional assumption that the suture set consists of a single component.
4 MARGARET NICHOLS These examples are all products for some choice of representation to GL 2 (C). To construct these examples, I find an algebraic obstruction to the sutured manifold admitting a product-like abelian representation, which is characterized by how π 1 (R ± ) sit as subgroups within π 1 (M). I then realize a simple closed curve satisfying this obstruction on the boundary of a sutured handlebody. We define R + to be a closed neighborhood of this curve and the two additional curves shown in Figure 2. Figure 2. The green curve obstructs the existence of a product-like representation to GL 1 (C). Solvable representations. Consistent with Agol-Dunfield s conjecture (Conjecture 4), I have not found examples of taut sutured manifolds that admit no 2-dimensional product-like representations. However, if we restrict the class of representations we consider, we can make finer distinctions between the minimal complexity of product-like representations. For instance, among the class of solvable representations, the derived length of the representation gives such a measure of complexity. Even within the setting of handlebodies, I show that a taut sutured manifold may require a solvable representation to have arbitrarily large derived length in order to be product-like. In the setting of representations to GL n (C), the derived length of a solvable representation places restrictions on its dimension. Theorem D ([Nic18]). There exists a family of taut sutured handlebodies M k, so that for each k, the manifold M k fails to be a twisted homology product for any solvable representation of degree less than or equal to k. Moreover, the manifold M k is not a twisted homology product for any solvable representation α : π 1 (M k ) GL ϕ(k) (C), where ϕ(k) with k.
RESEARCH STATEMENT 5 The construction of this family exploits a generalization of the obstruction to the existence of product-like abelian representations from Theorem C. 4. Future work My overarching program is to concretely describe the complexity of representation required to certify tautness purely in terms of the geometry and topology of the sutured manifold. Some steps in this direction include the following. (1) In the case of a genus-two sutured handlebody, we have shown a one-dimensional representation is sometimes required. Does this always suffice in genus two, or are there examples which require higher dimensional representations? (2) In the setting of one-dimensional representations of sutured handlebodies M g, product-like representations form a (possibly empty) subvariety of (GL 1 (C)) g, which we are able to compute explicitly from the Fox derivatives. To what extent can this perspective be applied general sutured manifolds? To higher dimensional representations? Can algebro-geometric techniques be utilized to better understand when these varieties are nonempty? (3) Theorem D demonstrates that, within the class of solvable representations, there is no uniform bound on dimension of product-like representations. Can this be extended to broader classes of representations? Is there a nice description of a similar obstruction to admitting nilpotent product-like representations? (4) Does Conjecture 4 hold: does there always exist a two-dimensional product-like representation? If so, do any or all such representations have geometric meaning? If the Conjecture is false, is there any universal bound on dimension?
6 MARGARET NICHOLS References [AD15] I. Agol and N. M. Dunfield. Certifying the Thurston norm via SL(2, C)-twisted homology. Pre-print, http: //arxiv.org/abs/1501.02136, 2015. [Ago08] I. Agol. Criteria for virtual fibering. J. Topol., 1(2):269 284, 2008. [DFJ12] N. M. Dunfield, S. Friedl, and N. Jackson. Twisted Alexander polynomials of hyperbolic knots. Exp. Math., 21(4):329 352, 2012. [FK13] S. Friedl and T. Kim. Taut sutured manifolds and twisted homology. Math. Res. Lett., 20(2):289 303, 2013. [Mor12] T. Morifuji. On a conjecture of Dunfield, Friedl and Jackson. C. R. Math. Acad. Sci. Paris, 350(19-20):921 924, 2012. [MT14] T. Morifuji and A. T. Tran. Twisted Alexander polynomials of 2-bridge knots for parabolic representations. Pacific J. Math., 269(2):433 451, 2014. [Nic18] M. Nichols. Taut sutured handlebodies as twisted homology products. Pre-print, https://arxiv.org/abs/ 1810.07353, 2018.