On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A (G,n)-complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n−1. In case G is an n-dimensional group there is a unique (up to homotopy) (G,n)-complex on the minimal Euler-characteristic level χmin(G,n). For every n we give examples of n-dimensional groups G for which there exist homotopically distinct (G,n)-complexes on the level χmin(G,n)+1. In the case where n=2 these examples are algebraic.     

An effective algorithm to get linking numbers of 3-manifolds

Summary A colored triangulation of a 3-manifoldM ³ is a decomposition into tetrahedra so that each vertex of them receive one of the colors 0, 1, 2, 3 in such a way that each tetrahedron has four differently colored vertices. From the combinatorics of the dual of a colored triangulation forM ³ we provide an easy algorithm to get a special kind of intersection matrix; from this matrix and from the torsion coefficients of the firstZ-homology group ofM ³ we provide a formula which yields its linking numbers.     

Pre-Bloch invariants of 3-manifolds with boundary

Let M be an oriented hyperbolic 3-manifold with finite volume. In [W.D. Neumann, J. Yang, Bloch invariants of hyperbolic 3-manifolds, Duke Math. J. 96 (1999) 29-59. [9]], Neumann and Yang defined an element β(M) of Bloch group B(C) for M. For this β(M), volume and Chern-Simons invariant of M is represented by a transcendental function. In this paper, we define β(M,ρ,C,o)∈P(C) for an oriented 3-manifold M with boundary, a representation of its fundamental group , a pants decomposition C of ∂M and an orientation o on simple closed curves of C. Unlike in the case of finite volume, we construct an element of pre-Bloch group P(C), and we need essentially the pants decomposition on the boundary. The volume makes sense for β(M,ρ,C,o) and we can describe the variation of volume on the deformation space.     

On the height of 2-component links

Let p be a prime and let L be a 2-component link in S³. We define a numerical invariant, called p-height of L, using a tower of successive p-fold branched cyclic coverings of L, and show, in particular, 2-height is algorithmically determined for any 2-component link. Some relationships between p-height and known link type invariants are also established.     

On the vertex group of n-manifolds

In this paper we obtain the decomposition of the vertex group of n-manifolds, extending the one given by Kauffman and Lins for dimension 3 and solving the related conjecture. The result is obtained in the more general category of gems: the vertex group of a gem , representing an n-manifold M, is the free product of n copies of the fundamental group of M and a free group F of rank N–n, where N is the number of n-residues of . In particular, for crystallizations FZ and consequently the vertex group is an invariant of M.     

Surgery on 3-manifolds with \mathbb{S} 1-actions

We study the topological structure of all 3-manifolds obtained by surgery along principal fibers of a closed orientable -manifold. As a consequence, we give alternative proofs of some classical results due to W. Heil and L. Moser. Moreover, we completely specify the Seifert invariants for the considered manifolds. Finally we classify the manifolds obtained by surgery along certain Seifert links and determine geometric presentations of their fundamental groups.Work performed under the auspices of C.N.R. (National Research Council) of Italy and partially supported by Ministero della Ricerca Scientifica e Tecnologica within the projects Geometria Reale e Complessa and Topologia.     

Tits alternative for 3-manifold groups

Let M be an irreducible, orientable, closed 3-manifold with fundamental group G. We show that if the pro-p completion of G is infinite then G is either soluble-by-finite or contains a free subgroup of rank 2. Both authors are partially supported by “Bolsa de produtividade de pesquisa” from CNPq, Brazil. Received: 16 February 2006     

Twisted Alexander polynomials of twist knots for nonabelian representations

In this paper, we describe the twisted Alexander polynomial of twist knots for nonabelian SL(2,C)-representations and investigate in detail the coefficient of the highest degree term as a function on the representation space of the knot group. In particular, we introduce the notion of monic representation and discuss its relation to the fiberedness of knots.     

On pallets for Fox colorings of spatial graphs

We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements.     

The structure of 3-manifolds with two-generated fundamental group

We provide a structure theorem for 3-manifolds with 2-generated fundamental group and non-trivial JSJ-decomposition. We further give a number of applications.     

Homological characterization of the unknot

Given a knot K in the 3-sphere, let Q_K be its fundamental quandle as introduced by Joyce. Its first homology group is easily seen to be . We prove that H₂(Q_K)=0 if and only if K is trivial, and whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components.More detailed information can be derived from the conjugation quandle: let Q_K^π be the conjugacy class of a meridian in the knot group . We show that , where p is the number of prime summands in a connected sum decomposition of K.     

On cyclic bra nched coverings of torus knots

We prove that then-fold cyclic coverings of the 3-sphere branched over the torus knotsK(p,q), p>q2 (i.e. the Brieskorn manifolds in the sense of [12]) admit spines corresponding to cyclic presentations of groups ifp1 (modq). These presentations include as a very particular case the Sieradski groups, first introduced in [14] and successively obtained from geometric constructions in [4], [9], and [15]. So our main theorem answers in affirmative to an open question suggested by the referee in [14]. Then we discuss a question concerning cyclic presentations of groups and Alexander polynomials of knots.Work Performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and partially supported by the Ministero per la Ricerca Scientifica e Tecnologica of Italy Within the projectsGeometria Reale e Complessa andTopologia and by the Korean Science and Engineering Foundation.     

Universal coverings of PL-manifolds via coloured graphs

Summary By means of techniques and results concerning maps on surfaces [JS] and edge-coloured graphs representing PL-manifolds [FGG], we prove the existence of an infinite ball complexP(n), n > 1, such thatevery orientable PL-manifold of dimension n is a quotient of |P(n)| by the action of a finite index subgroup of a Fuchsian group with signature ,with h(2) = h(3) = 4 and h(n) = 2, for n > 3. The core of the proof is that all orientable PL-manifolds of dimensionn can be represented by edge-coloured graphs which are quotients of a universal graph, only depending onn.     

Hom complexes and hypergraph colorings

Babson and Kozlov (2006) [2] studied Hom-complexes of graphs with a focus on graph colorings. In this paper, we generalize Hom-complexes to r-uniform hypergraphs (with multiplicities) and study them mainly in connection with hypergraph colorings. We reinterpret a result of Alon, Frankl and Lovász (1986) [1] by Hom-complexes and show a hierarchy of known lower bounds for the chromatic numbers of r-uniform hypergraphs (with multiplicities) using Hom-complexes.