Representations Of Π1 Research Articles

Flat connections in three-manifolds and classical Chern–Simons invariant

A general method for the construction of smooth flat connections on 3-manifolds is introduced. The procedure is strictly connected with the deduction of the fundamental group of a manifold M by means of a Heegaard splitting presentation of M. For any given matrix representation of the fundamental group of M, a corresponding flat connection A on M is specified. It is shown that the associated classical Chern–Simons invariant assumes then a canonical form which is given by the sum of two contributions: the first term is determined by the intersections of the curves in the Heegaard diagram, and the second term is the volume of a region in the representation group which is determined by the representation of π1(M) and by the Heegaard gluing homeomorphism. Examples of flat connections in topologically nontrivial manifolds are presented and the computations of the associated classical Chern–Simons invariants are illustrated.

Unitary representations of the fundamental group of orbifolds

Let X be a smooth complex projective variety of dimension n and \(\mathcal {L}\) an ample line bundle on it. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles E on X with \(c_{1}(E) = 0 = c_{2} (E) \cdot c_{1} (\mathcal {L})^{n-2}\) and the equivalence classes of unitary representations of π1(X). We show that this bijective correspondence extends to smooth orbifolds.

Traceless SU(2) representations of 2-stranded tangles

AbstractGiven a 2-stranded tangle T contained in a ℤ-homology ball Y, we investigate the character variety R(Y, T) of conjugacy classes of traceless SU(2) representations of π1(Y \ T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y, T) for many tangles naturally associated to knots in S3. Moreover, we determine the image of the restriction map from R(T, Y) to the traceless SU(2) character variety of the 4-punctured 2-sphere (the pillowcase). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.

Canonical polygons for the hyperbolic structures on the torus with a single cone point

Let T0 be the once-punctured torus and θ a real number with 0<θ<2π. Let ρ˜ be a representation of π1(T0) to SL(2,R) which sends the peripheral loop to an elliptic element with trace −2cos⁡(θ/2). Let ρ be the PSL(2,R)-representation induced from ρ˜, and assume it satisfies Bowditch's Q-condition. In this paper, we construct a certain polyhedron, which is obtained as a variation of Jorgensen's theory to cone manifolds, and construct a complete cone hyperbolic structure on the 3-dimensional cone manifold obtained as the product of the torus with a single cone point and R which induces ρ as the holonomy representation.

Cohomology jump loci of quasi-projective varieties

We prove that the cohomology jump loci in the space of rank one local systems over a smooth quasi-projective variety are finite unions of torsion translates of subtori. The main ingredients are a recent result of Dimca-Papadima, some techniques introduced by Simpson, together with properties of the moduli space of logarithmic connections constructed by Nitsure and Simpson. MB(X) = Hom(π1(X),C ∗ ) to be the variety of C ∗ representations of π1(X). Then MB(X) is a direct product of (C ∗ ) b1(X) and a finite abelian group. For each point ρ ∈ MB(X), there exists a unique rank one local system Lρ, whose monodromy representation is ρ. The cohomology jump loci of X are the natural strata � i (X) = {ρ ∈ MB(X) | dimC H i (X,Lρ) ≥ k}. � i (X) is a Zariski closed subset of MB(X). A celebrated result of Simpson says that if X is a smooth projective variety defined over C, theni (X) is a union of torsion

The Fixed Point Locus of the Verschiebung on ℳX(2, 0) for Genus-2 Curves X in Charateristic 2

Abstract.We prove that for every ordinary genus-2 curve X over a finite field κ of characteristic 2 with Aut(X/κ) = ℤ/2ℤ × S3 there exist SL(2; κ[[s]])-representations of π1(X) such that the image of π1(X̄) is infinite. This result produces a family of examples similar to Y. Laszlo’s counterexample to A. J. de Jong’s question regarding the finiteness of the geometric monodromy of representations of the fundamental group.

On the geometry of the slice of trace-free $${{SL_2(\mathbb{C})}}$$ -characters of a knot group

Let K be a knot in an integral homology 3-sphere Σ with exterior EK, and let B2 denote the two-fold branched cover of Σ branched along K. We construct a map Φ from the slice of trace-free \({{{\rm SL}_2(\mathbb{C})}}\) -characters of π1(EK) to the \({{{\rm SL}_2(\mathbb{C})}}\)-character variety of π1(B2). When this map is surjective, it describes the slice as the two-fold branched cover over the \({{{\rm SL}_2(\mathbb{C})}}\)-character variety of B2 with branched locus given by the abelian characters, whose preimage is precisely the set of metabelian characters. We show that each metabelian character can be represented as the character of a binary dihedral representation of π1(EK). The map Φ is shown to be surjective for all 2-bridge knots and all pretzel knots of type (p, q, r). An extension of this framework to n-fold branched covers is also described.

Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

Given a knot complement X and its p-fold cyclic cover X p → X, we identify twisted polynomials associated to $${GL_1\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π 1(X p ) with twisted polynomials associated to related $${GL_p\left({\bf F}[t^{\pm 1}]\right)}$$ representations of π 1(X) which factor through metabelian representations. This provides a simpler and faster algorithm to compute these polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3, 7, 9, 11, 15), corresponding to permutations of (7, 9, 11, 15), represent distinct concordance classes.

Fox coloured knots and triangulations of $S^{3}$

A Fox coloured link is a pair (L,ω), where L is a link in S3 and ω a simple and transitive representation of π1(S3∖L) onto the symmetric group Σ3 on three elements. Here, a representation is called simple if it sends the meridians to transpositions. By works of the first two authors, any Fox coloured link (L,ω) gives rise to a closed orientable 3-manifold M(L,ω) equipped with a 3-fold simple covering p:M(L,ω)→S3 branched over L, and any closed orientable 3-manifold is homeomorphic to an M(K,ω) for some Fox coloured knot (K,ω) [see H. M. Hilden, Bull. Amer. Math. Soc. 80 (1974), 1243–1244; J. M. Montesinos, Bull. Amer. Math. Soc. 80 (1974), 845–846;]. In [Adv. Geom. 3 (2003), no. 2, 191–225;], I. V. Izmestʹev and M. Joswig proved that a triangulation of S3 gives rise in a natural way to some graph G on S3 and a representation of π1(S3∖G) into the symmetric group Σm for some m≤4. They also proved that any pair (L,ω), where L is a link in S3 and ω a simple (not necessarily transitive) representation of π1(S3∖L) into the symmetric group Σ4, can be obtained from a triangulation of S3. The proof that Izmestʹev and Joswig gave of this result is non-constructive. In the paper under review, the authors give a constructive proof of the same result. In particular, given a pair (L,ω) consisting of a link L in S3 and a simple (not necessarily transitive) representation of π1(S3∖L) onto the symmetric group Σ4, they construct a triangulation of S3 that gives rise to (L,ω) in a natural way.

Mod ℓ representations of arithmetic fundamental groups, I: An analog of Serre's conjecture for function fields

There is a well-known conjecture of Serre that any continuous, irreducible representation ρ̲:GQ→GL2(F̲ℓ) which is odd arises from a newform. Here GQ is the absolute Galois group of Q, and F̲ℓ is an algebraic closure of the finite field Fℓ of ℓ of ℓ elements. We formulate such a conjecture for n-dimensional mod ℓ representations of π1(X) for any positive integer n and where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p (p≠ℓ), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for ℓ>2 follows from a result announced by Gaitsgory in [G]. The methods are different.

Unipotent flat bundles and Higgs bundles over compact Kähler manifolds

We characterize those unipotent representations of the fundamental group π 1 ( X , x ) \pi _1(X,x) of a compact Kähler manifold X X , which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of π 1 ( X , x ) \pi _1(X,x) that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain’s result and ours follow from a careful study of Simpson’s correspondence between Higgs bundles and local systems.

Sur la convexit� holomorphe des rev�tements lin�aires r�ductifs d?une vari�t� projective alg�brique complexe

Let X be a smooth connected compact projective variety over ℂ. We study the Shafarevich conjecture concerning holomorphic convexity along the lines of Kollar’s approach, when the fundamental group of X admits large finite dimensionnal representations. We prove that, given n∈ℕ, the topological covering space $\widetilde{X}_{n}$ of a projective algebraic compact complex manifold X corresponding to the intersection of the kernels of all linear reductive representations of π1(X) to GL n (ℂ) is holomorphically convex. In the surface case, this is a corollary of a theorem due to Katzarkov and Ramachandran.

ARITHMETIC KNOTS IN CLOSED 3-MANIFOLDS

Let Od denote the ring of integers in [Formula: see text]. An orientable finite volume cusped hyperbolic 3-manifold M is called arithmetic if the faithful discrete representation of π1(M) into PSL(2,C) is conjugate to a group commensurable with some Bianchi group PSL (2,Od). If M is a closed orientable 3-manifold, we say a link L ⊂ M is arithmetic if M\L is arithmetic. In the paper, we show that there exist closed orientable 3-manifolds which do not contain anarithmetic knot. Our methods give much more precise informations for non-hyperbolic 3-manifolds. For certain Lens Spaces, we can give fairly complete statements.

Rigidity and Deformation Spaces of Strictly Convex Real Projective Structures on Compact Manifolds

In this paper we show that if two strictly convex, compact real projective manifolds have the same marked length spectrum with respect to the Hilbert metric, then they are projectively equivalent. This is a rigidity for Finsler metric with a special geometric structure. Furthermore we prove an analogue of a Hitchin's conjecture for hyperbolic 3-manifolds, namely the deformation space of convex real projective structures on a compact hyperbolic 3-manifold M is a component in the moduli space of PGL(4,ℍ)-representations of π1(M).

Homotopy invariance of relative eta-invariants and 𝐶*-algebra 𝐾-theory

We prove a close cousin of a theorem of Weinberger about the homotopy invariance of certain relative eta-invariants by placing the problem in operator K K -theory. The main idea is to use a homotopy equivalence h : M → M ′ h:M \to M’ to construct a loop of invertible operators whose “winding number" is related to eta-invariants. The Baum-Connes conjecture and a technique motivated by the Atiyah-Singer index theorem provides us with the invariance of this winding number under twistings by finite-dimensional unitary representations of π 1 ( M ) \pi _{1}(M) .

Geometric intersection numbers on a four-punctured sphere

Let G 4 \mathcal {G}_{4} be the space of all simple closed geodesics on the punctured sphere Σ 4 \Sigma _{4} . We construct an explicit homeomorphism of the completion of G 4 \mathcal {G}_{4} onto a circle by using geometric intersection numbers. Also, we relate these geometric intersection numbers to trace polynomials of transformations corresponding to geodesics in G 4 \mathcal {G}_{4} in a representation of π 1 ( Σ 4 ) \pi _{1}(\Sigma _{4}) into P S L ( 2 , C ) PSL(2,\mathbf {C}) .

Casson-Lin's Invariant and Floer Homology

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

Covering Linkage Invariants

Let K be a knot in a manifold M. Corresponding to a representation of Π1(M — K) into a transitive group of permutations there is a branched covering space of M. K is covered by which may be a link of several components. The set of linking numbers between the various components of has long been recognised as a useful knot invariant. Bankwitz and Schumann used this invariant in considering dihedral coverings of Viergeflechte.