I organize the "Gauge theory" seminar, Fridays 3:30--4:30pm in Science Center 507.

__Fall 2021__

**???: **Allison Miller (Rice)

*Title/A**bstract:* TBA

**October 9: **Chris Wendl (Humboldt-Berlin)

*Title/Abstract:* TBA

**???: **Semon Rezchikov (Columbia)

*Title/Abstract:* TBA

**???: **Boyu Zhang (Princeton)

*Title/Abstract:* TBA

--------------------------------------------------------------------------------------------------------------------------

__Spring 20120__

**January 31: **Vinicius Ramos (IMPA-Brazil)

*Title: *Symplectic embeddings, integrable systems and billiards

*Abstract:* Symplectic embedding problems are at the core of symplectic topology. Many results have been found involving balls, ellipsoids and polydisks. More recently, there has been progress on problems involving lagrangian products and related domains. In this talk, I explain what is known about symplectic embeddings of these domains. There are rigid and flexible phenomena and for some problems, the transition between the two happen at a surprising place. In order to get to the results, we will use the Arnold-Liouville theorem and billiard dynamics.

**February 7: **Zack Sylvan (Columbia)

*Title: *Viterbo transfer as localization

*Abstract:* I'll discuss the Viterbo transfer functor from the (partially) wrapped Fukaya category of a Liouville domain to that of a subdomain. It is a localization when everything in sight is Weinstein, and I'll explain how much of that survives if we drop the assumption that the cobordism is Weinstein. The result allows us to turn natural questions about exact Lagrangians into interesting questions in homotopical algebra.

**February 14: **no meeting

**February 21: **no meeting

**February 28: **David Gabai (Princeton)

*Title: *Knotted 3-balls in the 4-sphere

*Abstract:* We give the first examples of codimension-1 knotting in the 4-sphere, i.e. there is a 3-ball B_1 with boundary the standard linear 2-sphere, which is not isotopic rel boundary to the standard linear 3-ball B_0. Actually, there is an infinite family of distinct isotopy classes of such balls. This implies that there exist inequivalent fiberings of the unknot in 4-sphere, in contrast to the situation in dimension-3. Also, that there exists diffeomorphisms of S^1 x B^3 homotopic rel boundary to the identity, but not isotopic rel boundary to the identity. Joint work with Ryan Budney

**March 6: **Bob Gompf (UT Austin)

*Title: *A topological approach to convexity in complex surfaces

*Abstract:* We will discuss the classical notion of J-convexity of subsets of complex manifolds, and the closely related notion of Stein manifolds. The theory is particularly subtle in complex dimension 2. Surprisingly, progress can be made using topological 4-manifold theory. Every tame CW 2-complex topologically embedded in a complex surface can be perturbed so that it becomes J-convex in the sense of being a nested intersection of Stein open subsets. These Stein neighborhoods are all topologically equivalent to each other, but can be very different when viewed in the smooth category. As applications, we obtain uncountable families of distinct smoothings of R^4 admitting convex or concave holomorphic structures. We can also generalize the notion of J-convex hypersurfaces to the topological category. The resulting topological embeddings behave like their smooth counterparts, but are much more common.

**March 13: **the remaining talks are cancelled due to Coronavirus

--------------------------------------------------------------------------------------------------------------------------

__Fall 2019__

**September 13: **Dmitry Tonkonog (Harvard)

*Title: *Floer-theoretic heaviness and Lagrangian skeleta

*Abstract:* I will talk on what I have learned from Umut Varolgunes, and on our work in preparation. In 2007, Entov and Polterovich introduced the notion of heavy and superheavy subsets of symplectic manifolds; such subsets have remarkable symplectic rigidity properties. One can define a variant version of (super)heaviness using relative symplectic homology studied by Varolgunes. I will explain this story, and prove that Lagrangian skeleta of symplectic divisor complements of Calabi-Yau's are superheavy in this sense.

**September 20: **Yi Ni (Caltech)

*Title: *Thurston norm minimizing Seifert surfaces

*Abstract:* Let K be a null-homologous knot in a closed 3-manifold Y, and F be a Seifert surface. One can cap off the boundary of F with a disk in the zero surgery on K to get a closed surface F_0. If we know that F is Thurston norm minimizing, we can ask whether F_0 is also Thurston norm minimizing. A classical theorem of Gabai says that the answer is Yes when Y is the 3-sphere. Gabai's theorem can be generalized to many other 3-manifolds using Heegaard Floer homology. In this talk, we will discuss a sufficient condition for F_0 to be Thurston norm minimizing.

**September 27: **Jon Bloom (Broad Institute)

*Title: *Principal component analysis for topologists

*Abstract:* Geometrically, PCA finds the k-dimensional subspace nearest a point cloud in m-dimensional Euclidean space. Numerically, PCA also specifies a basis for this subspace, the principal directions of the data. Deep learning-ly, PCA is the shallowest of autoencoder models, that of a linear autoencoder (LAE) which learns the subspace but not the basis. I’ll relate these views by considering the gradient dynamics of learning from the perspective of a simple F_2-perfect Morse function on the real Grassmannian. I’ll then prove that L_2-regularized LAEs are symmetric at all critical points and recover the principal directions as singular vectors of the decoder, with implications for eigenvector algorithms and perhaps learning in the brain. If asked, I’ll speculate on a role for Morse homology in deep learning. Based on “Loss Landscapes of Regularized Linear Autoencoders” (ICML 2019) with Daniel Kunin, Aleksandrina Goeva, and Cotton Seed at the Broad Institute.

**October 4: **no meeting

**October 11: **Agustín Moreno (Augsburg)

*Title: *Bourgeois contact structures: tightness, fillability and applications

*Abstract:* Starting from a contact manifold and a supporting open book decomposition, an explicit construction by Bourgeois provides a contact structure in the product of the original manifold with the two-torus. In this talk, we will discuss recent results concerning rigidity and fillability properties of these contact manifolds. For instance, it turns out that Bourgeois contact structures are, in dimension 5, always tight, independent on the rigid/flexible classification of the original contact manifold. Moreover, Bourgeois manifolds associated to suitable monodromies provide new examples of weakly but not strongly fillable contact 5-manifolds. We also present the following application in any dimension: the standard contact structure in the unit cotangent bundle of the n-torus, which is a Bourgeois manifold, admits a unique aspherical filling up to diffeomorphism. This is joint work with Jonathan Bowden and Fabio Gironella.

**October 18: **Laurent Côté (Stanford)

*Title: *Rigidity of cotangent fibers of Riemann surfaces

*Abstract:* I will discuss joint work with Georgios Dimitroglou Rizell which studies symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our work is motivated by analogies to Arnold’s celebrated Nearby Lagrangian Conjecture and by work of Ekholm and Smith in higher dimensions. We crucially use several special properties of pseudo-holomorphic curves in dimension 4 as well as some fairly recent developments in the analysis of punctured pseudo-holomorphic curves. I will review these developments, highlighting in particular why certain pseudo-holomorphic curve techniques are particularly effective in dimension 4.

**October 25: **Alex Waldron (Michigan State)

*Title: *G_2-instantons on the 7-sphere

*Abstract:* I'll discuss a forthcoming paper studying families of G_2-instantons on S^7, focusing on those which are obtained by pulling back asd instantons on S^4 via the quaternionic Hopf fibration. In the charge-1 case this yields a smooth and complete 15-dimensional family. The situation for higher charge is more complicated, but we are able to compute all the infinitesimal deformations.

**November 1: **Ko Honda (UCLA)

*Title: *Convex hypersurface theory in higher-dimensional contact topology

*Abstract:* Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds. In particular they have been successfully applied to many classification problems. After briefly reviewing convex surface theory in dimension three, we explain how to generalize many of their properties to higher dimensions. This is joint work with Yang Huang.

**November 8: **Lisa Piccirillo (Brandeis+MIT)

*Title: *Knot traces and PL embeddings

*Abstract:* It remains at the forefront of 4-manifold topology to construct simple closed 4-manifolds with distinct smooth structures. Towards that end, it is of interest to construct simple 4-manifolds with boundary with very distinct smooth structures. For all genera g, we produce pairs of homeomorphic smooth 4-manifolds Z and Z' which are homotopy equivalent to a genus g surface and which have smooth structures distinguished by several formal properties: Z is diffeomorphic to a genus g knot trace and Z' is not, Z admits a smooth spine and Z' does not admit a piecewise linear spine. When g = 0, Z is geometrically simply connected and Z' is not. In particular our Z' are simple spineless 4-manifolds, which gives an alternative to Levine and Lidman’s recent solution to Problem 4.25 on Kirby’s list. This is joint work in progress with Kyle Hayden.

**November 15: **Dan Cristofaro-Gardiner (Santa Cruz)

*Title: *Refined asymptotics for the ECH spectrum

*Abstract:* In previous work, Hutchings, Ramos and I studied the embedded contact homology (ECH) spectrum for any closed three-manifold with a contact form, and proved a "volume identity" showing that the leading order asymptotics recover the contact volume. I will explain recent joint work with Savale that sharpens this asymptotic formula by better understanding the subleading term. I will then discuss some conjectures about what an optimal bound on the subleading asymptotics might be: for example, for certain generic domains in R^4, Hutchings has conjectured that the subleading asymptotics recover a contact "Ruelle invariant".

**November 22:** Harvard's "Current Developments in Mathematics" conference

**November 29:** Thanksgiving break

**December 6:** Song Sun (Berkeley)

*Title: *Singular Hermitian-Yang-Mills connections and reflexive sheaves

*Abstract:* The Donaldson-Uhlenbeck-Yau theorem relates the existence of Hermitian-Yang-Mills connections over a compact Kahler manifold with algebraic stability of a holomorphic vector bundle. This has been extended by Bando-Siu in 1994 to a class of singular Hermitian-Yang-Mills connections on reflexive sheaves. We study tangent cones of these singular connections in the geometric analytic sense, and show that they can be characterized in terms of certain algebro-geometric invariants of reflexive sheaves. In a sense, this can be viewed as a ``local” version of the Donaldson-Uhlenbeck-Yau correspondence. Based on joint work with Xuemiao Chen (University of Maryland).

--------------------------------------------------------------------------------------------------------------------------

__Spring 2019__

**February 1: **Zoltán Szabó (Princeton)

*Title: *Algebraic methods in knot Floer homology

*Abstract:* The aim of this talk is to present an algebraic description of knot Floer homology, discovered in a joint work with Peter Ozsváth.

**February 8:** no meeting

**February 15:** Cagatay Kutluhan (Buffalo)

*Title: *Can Floer-theoretic invariants detect overtwisted contact structures?

*Abstract:* In joint work with Matic, Van Horn-Morris, and Wand, we seek an answer to this question. We define a refinement of the contact invariant in Heegaard Floer homology that takes values in Z_{\ge 0} \cup {\infty}, called (spectral) order. Among other things, we prove that overtwisted contact structures have zero order, whereas Stein fillable contact structures have infinite order. Furthermore, we show that a strictly increasing sequence of positive integers is realized as the order of a family of contact structures with vanishing Heegaard Floor contact invariant. After defining our contact invariant and discussing some of its key properties, I will talk about its computability and some problems that are content of work in progress.

**February 22: **John Baldwin (Boston College)

*Title:* Instanton L-space knots

*Abstract:* A 3-manifold is said to be SU(2)-abelian if all homomorphisms from its fundamental group to SU(2) factor through the first homology of the manifold. Understanding which manifolds are SU(2)-abelian is a difficult and wide open problem, even in the case of 3-manifolds arising from surgery on a knot in the 3-sphere. Kronheimer and Mrowka showed, for instance, that n-surgery on a nontrivial knot is not SU(2)-abelian for n = 1 or 2, but it isn't even known whether the same is true for n = 3 or 4 (the same isn't true for n = 5 as 5-surgery on the right-handed trefoil is a lens space, which has abelian fundamental group). We approach this question by trying to understand which knots have instanton L-space surgeries. A rational homology sphere is said to be an instanton L-space if its framed instanton homology has the smallest rank possible, in analogy with Heegaard Floer homology; familiar examples include lens spaces and branched double covers of alternating knots. Moreover, it is generally the case that SU(2)-abelian manifolds are instanton L-spaces. We conjecture that if surgery on a knot K in the 3-sphere results in an instanton L-space then K is fibered and its Seifert genus equals its smooth slice genus, and we discuss a strategy for proving this.

**March 1: **Peter Ozsváth (Princeton)

*Title: *Computing knot Floer homology

*Abstract:* I will explain how to use bordered Floer homology to obtain a practical, algebraic computation of knot Floer homology. This is joint work with Zoltán Szabó.

**March 8: **Thomas Walpuski (Michigan State)

*Title: *Super-rigidity and Castelnuovo’s bound

*Abstract:* Castelnuovo’s bound is a very classical result in algebraic geometry. It asserts a sharp bound on the genus of a curve of degree d in n-dimensional projective space. It is an interesting question to ask whether analogues of Castelnuovo’s bound hold in almost complex geometry. There is a direct analogue in dimension four. In dimension at least eight genus bounds can be established for generic almost complex structures. These results leave open the case of dimension six.

Bryan and Panharipande introduced the notion of super-rigidity of an almost complex structure. They also speculated that this condition might hold for a generic almost complex structure (compatible with a fixed symplectic structure). It had been believed for a long time that super-rigidity will play an important role in the proof of the Gopakumar–Vafa conjecture. However, it turned that Ionel and Parker’s recent proof of this conjecture did not make use of it. Nevertheless, super-rigidity has important consequences. I will present one of these consequences, namely, a genus bound for index zero pseudo-holomorphic curves. This is joint work with Aleksander Doan and, heavily, relies on work by De Lellis, Spadaro, and Spolaor and ideas of Taubes.

There has been a lot of progress towards establishing Bryan and Pandharipande’s super-rigidity conjecture in the work of Wendl. In fact, based on his ideas, Aleksander Doan and I have developed an abstract framework for equivariant transversality/Brill–Noether type questions. Wendl’s work shows that the super-rigidity conjecture holds provided generic real Cauchy-Riemann operators satisfy an easy to state analytic condition. I will explain what this condition means and discuss a few cases in which this condition (or versions of it hold).

**March 15: **Emmy Murphy (Northwestern)

*Title: *Inductively collapsing Fukaya categories and flexibility

*Abstract: *A Weinstein manifold is a symplectic manifold which admits a Lagrangian skeleton, and associated to any Weinstein manifold is its wrapped Fukaya category, a powerful algebraic invariant. One important case is that the wrapped category of C^n is trivial. The talk will discuss a partial converse: if X is any Weinstein 6-manifold which is contractible, admitting an arboreal skeleton so that the wrapped category is inductively collapsing, then X is symplectomorphic to C^3. A large portion of the talk will be defining the notion of inductively collapsing: this is a purely algebraic condition, but it depends on a presentation of the wrapped category, which itself comes from a chosen Lagrangian skeleton.

**March 22: **Harvard's** **spring break

**March 29: **Chris Scaduto (Simons Center)

*Title: *Instantons and lattices of smooth 4-manifolds with boundary

*Abstract:* Given a 3-manifold Y, what are the possible definite intersection forms of smooth 4-manifolds with boundary Y? Donaldson's theorem says that if Y is the 3-sphere, then all such intersection forms are standard integer Euclidean lattices. I will survey some new progress on this problem, for other 3-manifolds, that comes from instanton Floer theory.

**April 5: **Melissa Liu (Columbia)

*Title: *The Yang-Mills Equations over Klein Surfaces

*Abstract:* In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory, and derived results on topology of the moduli space of algebraic bundles over a complex algebraic curve. In this talk, I will discuss Yang-Mills functional over a Klein surface (a 2-manifold equipped with a dianalytic structure) from the point of view of Morse theory, and derive results on topology of the moduli space of real or quaternionic vector bundles over a real algebraic curve. This is based on joint work with Florent Schaffhauser.

**April ****12: **Joel Fish (UMass Boston)

*Title: *Feral pseudoholomorphic curves and minimal sets

*Abstract:* I will discuss some recent joint work with Helmut Hofer, in which we define and establish properties of a new class of pseudoholomorhic curves (feral J-curves) to study certain divergence free flows in dimension three. In particular, we show that if H is a smooth, proper, Hamiltonian in R^4, then no regular non-empty energy level of H is minimal. That is, the flow of the associated Hamiltonian vector field has a trajectory which is not dense.

**April ****19: **Aliakbar Daemi (Simons Center)

*Title: *Exotic Structures, Homology Cobordisms and Chern-Simons Functional

*Abstract:* An exotic structure on a smooth manifold X is another smooth manifold which is homeomorphic but not diffeomorphic to X. There are many closed 4-manifolds which admit exotic smooth structures. However, it is still an open question whether there are exotic structures on simple closed 4-manifolds such as the 4-dimensional sphere (smooth Poincare conjecture) and S^1xS^3. Motivated by the latter case, Akbulut asked whether there are an integral homology sphere Y with non-trivial Rokhlin invariant and a simply connected homology cobordism from Y to itself. In this talk, I will introduce various invariants of homology cobordism classes of 3-manifolds and discuss some of their topological applications. In particular, we answer Akbulut’s question for various integral homology spheres and propose a plan to completely address his conjecture.

**April ****26:** Jennifer Hom (Georgia Tech)

*Title: *Heegaard Floer and homology cobordism

*Abstract:* We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. The proof relies on the involutive Heegaard Floer homology package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

**May 3: **S-T. Yau's birthday conference

__Fall 2018__

**September 14:** Denis Auroux (Harvard)

*Title: *An invitation to homological mirror symmetry

*Abstract:* We will give a gentle introduction to some recent developments in the area of homological mirror symmetry. We will use simple examples to illustrate Kontsevich's conjecture and its extension beyond the Calabi-Yau setting in which it was first formulated. We will mostly focus on a one-dimensional example, the pair of pants, to give a flavor of the geometric concepts involved in a general formulation of homological mirror symmetry. If time permits, we will then describe a program to prove homological mirror symmetry for essentially arbitrary complete intersections in toric varieties (joint work in progress with Mohammed Abouzaid).

**September 21: **Weiyi Zhang (Warwick)

*Title: *From smooth to almost complex

*Abstract:* An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. We will discuss differential topology of almost complex manifolds, explain how to use transversality statements for smooth manifolds to formulate and prove corresponding results for an arbitrary almost complex manifold. The examples include intersection of almost complex manifolds, structure of pseudoholomorphic maps and zero locus of certain harmonic forms.

One of the main technical tools is Taubes' notion of "positive cohomology assignment", which plays the role of local intersection number. I will begin with explaining its motivation through multiplicities of zeros of a smooth function.

Our results would lead to a notion of birational morphism between almost complex manifolds. Various birational invariants, including Kodaira dimension, for almost complex manifolds will be introduced and discussed (this part is joint with Haojie Chen).

**September 28: **no meeting

**October 5: **Anar Akhmedov (University of Minnesota, visiting scholar @Harvard)

*Title: *Construction of symplectic 4-manifolds and Lefschetz fibrations via Luttinger surgery

*Abstract:* Luttinger surgery is a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold. The surgery was introduced by Karl Luttinger in 1995, who used it to study Lagrangian tori in R^4. Luttinger's surgery has been very effective tool recently for constructing exotic smooth and symplectic structures on 4-manifolds. In this talk, using Luttinger surgery, I will present the constructions of smallest known closed exotic simply connected symplectic 4-manifolds and exotic Lefschetz fibrations over 2-sphere whose total spaces have arbitrary finitely presented group G as the fundamental group. If time permits, we will also discuss some applications to the geography of symplectic 4-manifolds and Lefschetz fibrations. Part of these are (separate) joint works with B. Doug Park, Burak Ozbagci, and Naoyuki Monden.

**October 12: **Paul Feehan (Rutgers)

*Title: *Lojasiewicz inequalities and Morse-Bott functions

*Abstract:* The Lojasiewicz gradient and distance inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman. We shall first describe a more direct proof of the Lojasiewicz gradient inequality that uses resolution of singularities for real analytic varieties to reduce to the case of functions with simple normal crossings, where the Lojasiewicz exponent may be computed explicitly — thus giving insight into its geometric meaning. It is well-known and easy to prove that if a function on a Banach space is Morse-Bott, then its Lojasiewicz exponent is 1/2. We show that the less obvious converse is also true: if the Lojasiewicz exponent of an analytic function on a Banach space is 1/2 at a critical point, then the function is Morse-Bott on a neighborhood of that point. We illustrate these phenomena with applications of Lojasiewicz inequalities to the Yang-Mills energy function near the critical set of flat connections on a principal G-bundle over a closed Riemannian manifold.

**October 19: **Aleksander Doan (Stony Brook)

*Title: *Harmonic Z/2 spinors and wall-crossing in Seiberg-Witten theory

*Abstract:* The notion of a harmonic Z/2 spinor was introduced by Taubes as an abstraction of various limiting objects appearing in compactifications of gauge-theoretic moduli spaces. I will explain this notion and discuss an existence result for harmonic Z/2 spinors on three-manifolds. The proof uses a wall-crossing formula for solutions of generalized Seiberg-Witten equations in dimension three, a result itself motivated by Yang-Mills theory on Riemannian manifolds with special holonomy G_2. The talk is based on joint work with Thomas Walpuski.

**October 26: **Michael Hutchings (Berkeley)

*Title: *Equivariant symplectic capacities

*Abstract:* We define a sequence of symplectic capacities using positive S^1-equivariant symplectic homology. These capacities are conjecturally equal to the Ekeland-Hofer capacities. However they satisfy axioms which allow them to be computed for many examples, such as convex toric domains. They also give sharp obstructions to symplectic embeddings of cubes into convex toric domains. The asymptotics of these capacities are conjecturally related to Lagrangian embeddings. This is a joint work with Jean Gutt.

**November 2: **Laure Flapan (Northeastern)

*Title: *Monodromy of Kodaira fibrations

*Abstract: *A long-standing question in studying the topology of complex algebraic varieties is the question of what groups can occur as the fundamental group of a smooth projective variety. One approximation of this question in the case of fibered varieties is to ask what groups can occur as the monodromy group of such a fibration. We use Hodge theory to investigate this question in the case of fibered algebraic surfaces, called Kodaira fibrations, whose fibers are all smooth and draw connections with questions about Shimura varieties and the moduli space of smooth algebraic curves.

**November 9:** Mohammed Abouzaid (Columbia)

*Title: *A formalism for Floer theory with coefficients

*Abstract:* I will discuss a formalism for defining (Lagrangian) Floer homology with coefficients which is a little more abstract than has been used so far in the literature, and is designed to generalise a recent construction of Rezchikov, in the exact setting, to the case of Lagrangians which may bound holomorphic discs. Various parts of this talk will report on joint work with Pardon, and with Blumberg and Kragh.

**November 16:** Harvard's "Current Developments in Mathematics" conference

**November 23:** Thanksgiving break

**November 30:** John Pardon (Princeton)

*Title: *Structural results in wrapped Floer theory

*Abstract:* I will discuss results relating different partially wrapped Fukaya categories. These include a Kunneth formula, a 'stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity. This is joint work with Sheel Ganatra and Vivek Shende.

**December 7:** Raphael Zentner (Regensburg)

*Title: *Irreducible SL(2,C)-representations of homology-3-spheres

*Abstract:* We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein, and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).