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We give a constructive proof that the Regge symmetry is a scissors congruence in hyperbolic space. The main tool is Leibon’s construction for computing the volume of a general hyperbolic tetrahedron. The proof consists of identifying the key elements in Leibon’s construction and permuting them.
The HKR (Hennings–Kauffman–Radford) framework is used to construct invariants of 4–thickenings of 2–dimensional CW complexes under 2–deformations (1– and 2– handle slides and creations and cancellations of 1–2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra and an element in a quotient of its center, which determines a trace function on . We study the subset of trace elements which define invariants of 4–thickenings under 2–deformations. In two subsets are identified : , which produces invariants of 4–thickenings normalizable to invariants of the boundary, and , which produces invariants of 4–thickenings depending only on the 2–dimensional spine and the second Whitney number of the 4–thickening. The case of the quantum is studied in details. We conjecture that leads to four HKR–type invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum is identified as a subalgebra of a quotient of its center.
It is well-known that self-linking is the only –valued Vassiliev invariant of framed knots in . However for most –manifolds, in particular for the total spaces of –bundles over an orientable surface , the space of –valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant of framed knots in orientable total spaces of –bundles over an orientable not necessarily compact surface . We show that if then is the universal order one invariant, i.e. it distinguishes every two framed knots that can be distinguished by order one invariants with values in an Abelian group.
In this paper we generalize the notion of strongly poly-free group to a larger class of groups, we call them strongly poly-surface groups and prove that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for any virtually strongly poly-surface group. A consequence is that the Whitehead group of a torsion free subgroup of any virtually strongly poly-surface group vanishes.
We prove the existence of a finite set of moves sufficient to relate any two representations of the same –manifold as a –fold simple branched covering of . We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degree and can be viewed as a second step in a program to establish similar results for arbitrary degree coverings of .
We demonstrate that the operation of taking disjoint unions of –holomorphic curves (and thus obtaining new –holomorphic curves) has a Seiberg–Witten counterpart. The main theorem asserts that, given two solutions , of the Seiberg–Witten equations for the –structures (with certain restrictions), there is a solution of the Seiberg–Witten equations for the –structure with , obtained by “grafting” the two solutions .
Given two measured laminations and in a hyperbolic surface which fill up the surface, Kerckhoff [Lines of Minima in Teichmueller space, Duke Math J. 65 (1992) 187–213] defines an associated line of minima along which convex combinations of the length functions of and are minimised. This is a line in Teichmüller space which can be thought as analogous to the geodesic in hyperbolic space determined by two points at infinity. We show that when is uniquely ergodic, this line converges to the projective lamination , but that when is rational, the line converges not to , but rather to the barycentre of the support of . Similar results on the behaviour of Teichmüller geodesics have been proved by Masur [Two boundaries of Teichmueller space, Duke Math. J. 49 (1982) 183–190].
The twisted face-pairing construction of our earlier papers gives an efficient way of generating, mechanically and with little effort, myriads of relatively simple face-pairing descriptions of interesting closed 3–manifolds. The corresponding description in terms of surgery, or Dehn-filling, reveals the twist construction as a carefully organized surgery on a link. In this paper, we work out the relationship between the twisted face-pairing description of closed 3–manifolds and the more common descriptions by surgery and Heegaard diagrams. We show that all Heegaard diagrams have a natural decomposition into subdiagrams called Heegaard cylinders, each of which has a natural shape given by the ratio of two positive integers. We characterize the Heegaard diagrams arising naturally from a twisted face-pairing description as those whose Heegaard cylinders all have integral shape. This characterization allows us to use the Kirby calculus and standard tools of Heegaard theory to attack the problem of finding which closed, orientable 3–manifolds have a twisted face-pairing description.
We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491–511]. As an application we extend the Dold–Kan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [Stable model categories are categories of modules, Topology, 42 (2003) 103–153] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra.
In this paper we consider two connected closed Haken manifolds denoted by and , with the same Gromov simplicial volume. We give a simple homological criterion to decide when a given map between and can be changed by a homotopy to a homeomorphism. We then give a convenient process for constructing maps between and satisfying the homological hypothesis of the map .
The mod cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211–227] proved a conjecture due to N. Kuhn [On topologicaly realizing modules over the Steenrod algebra, Annals of Mathematics, 141 (1995) 321–347] stating that if the mod cohomology of a space is in a finite stage of the Krull filtration of the category of unstable modules over the Steenrod algebra then it is locally finite. Nevertheless his proof involves some finiteness hypotheses. We show how one can remove those finiteness hypotheses by using the homotopy theory of profinite spaces introduced by F. Morel [Ensembles profinis simpliciaux et interpretation geometrique du foncteur T, Bull. Soc. Math. France, 124 (1996) 347–373], thus obtaining a complete proof of the conjecture. For that purpose we build the Eilenberg–Moore spectral sequence and show its convergence in the profinite setting.
In this paper, we define the primitive/Seifert-fibered property for a knot in . If satisfied, the property ensures that the knot has a Dehn surgery that yields a small Seifert-fibered space (i.e. base and three or fewer critical fibers). Next we describe the twisted torus knots, which provide an abundance of examples of primitive/Seifert-fibered knots. By analyzing the twisted torus knots, we prove that nearly all possible triples of multiplicities of the critical fibers arise via Dehn surgery on primitive/Seifert-fibered knots.
For finite coverings we elucidate the interaction between transferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of various homotopy orbit spaces. In turn these results provide universal examples for computing the stable Euler classes (ie ) and transferred Chern classes for –fold covers. Applications to the classifying spaces of –groups are given.
In a recent paper, Dimca and Némethi pose the problem of finding a homogeneous polynomial such that the homology of the complement of the hypersurface defined by is torsion-free, but the homology of the Milnor fiber of has torsion. We prove that this is indeed possible, and show by construction that, for each prime , there is a polynomial with –torsion in the homology of the Milnor fiber. The techniques make use of properties of characteristic varieties of hyperplane arrangements.
We use skein theory to compute the coefficients of certain power series considered by Habiro in his theory of invariants of integral homology –spheres. Habiro originally derived these formulas using the quantum group . As an application, we give a formula for the colored Jones polynomial of twist knots, generalizing formulas of Habiro and Le for the trefoil and the figure eight knot.
Milnor’s triple linking numbers of a link in the 3–sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.
We study neighborhoods of configurations of symplectic surfaces in symplectic 4–manifolds. We show that suitably “positive” configurations have neighborhoods with concave boundaries and we explicitly describe open book decompositions of the boundaries supporting the associated negative contact structures. This is used to prove symplectic nonfillability for certain contact 3–manifolds and thus nonpositivity for certain mapping classes on surfaces with boundary. Similarly, we show that certain pairs of contact 3–manifolds cannot appear as the disconnected convex boundary of any connected symplectic 4–manifold. Our result also has the potential to produce obstructions to embedding specific symplectic configurations in closed symplectic 4–manifolds and to generate new symplectic surgeries. From a purely topological perspective, the techniques in this paper show how to construct a natural open book decomposition on the boundary of any plumbed 4–manifold.
Several authors have recently studied virtual knots and links because they admit invariants arising from –matrices. We prove that every virtual link is uniquely represented by a link in a thickened, compact, oriented surface such that the link complement has no essential vertical cylinder.
In this paper we investigate Uludağ’s method for constructing new curves whose fundamental groups are central extensions of the fundamental group of the original curve by finite cyclic groups.
In the first part, we give some generalizations to his method in order to get new families of curves with controlled fundamental groups. In the second part, we discuss some properties of groups which are preserved by these methods. Afterwards, we describe precisely the families of curves which can be obtained by applying the generalized methods to several types of plane curves. We also give an application of the general methods for constructing new Zariski pairs.