Expand all Collapse all | Results 26 - 50 of 55 |
26. CMB 2006 (vol 49 pp. 55)
Non Abelian Twisted Reidemeister Torsion for Fibered Knots In this article, we give an explicit formula to compute the
non abelian twisted sign-deter\-mined Reidemeister torsion of the
exterior of a fibered knot in terms of its monodromy. As an
application, we give explicit formulae for the non abelian
Reidemeister torsion of torus knots and of the figure eight knot.
Keywords:Reidemeister torsion, Fibered knots, Knot groups, Representation space, $\SU$, $\SL$, Adjoint representation, Monodromy Categories:57Q10, 57M27, 57M25 |
27. CMB 2005 (vol 48 pp. 547)
Degeneracy of 2-Forms and 3-Forms We study some global aspects of differential complex 2-forms and 3-forms
on complex manifolds.
We compute the cohomology classes represented by the sets of points
on a manifold where such a form degenerates in various senses,
together with other similar cohomological obstructions.
Based on these results and a formula for projective
representations, we calculate the degree of the projectivization
of certain orbits of the representation $\Lambda^k\C^n$.
Keywords:Classes of degeneracy loci, 2-forms, 3-forms, Thom polynomials, global singularity theory Categories:14N10, 57R45 |
28. CMB 2005 (vol 48 pp. 32)
Non-Left-Orderable 3-Manifold Groups We show that several torsion free 3-manifold groups
are not left-orderable.
Our examples are groups of cyclic branched coverings of $S^3$
branched along links.
The figure eight knot provides simple
nontrivial examples. The groups arising in these examples are known
as Fibonacci groups which we show not to be left-orderable.
Many other examples of non-orderable groups are obtained by taking
3-fold branched covers of $S^3$ branched along various hyperbolic
2-bridge knots.
%with various hyperbolic 2-bridge knots as branched sets.
The manifold obtained in such a way from the $5_2$ knot
is of special interest as it is conjectured to be the hyperbolic
3-manifold with the smallest volume.
Categories:57M25, 57M12, 20F60 |
29. CMB 2004 (vol 47 pp. 332)
Recurrent Geodesics in Flat Lorentz $3$-Manifolds Let $M$ be a complete flat Lorentz $3$-manifold $M$ with purely
hyperbolic holonomy $\Gamma$. Recurrent geodesic rays are completely
classified when $\Gamma$ is cyclic. This implies that for any pair of
periodic geodesics $\gamma_1$, $\gamma_2$, a unique geodesic forward
spirals towards $\gamma_1$ and backward spirals towards $\gamma_2$.
Keywords:geometric structures on low-dimensional manifolds, notions of recurrence Categories:57M50, 37B20 |
30. CMB 2004 (vol 47 pp. 439)
On the Stable Basin Theorem The stable basin theorem was introduced by Basmajian and Miner as a
key step in their necessary condition for the discreteness of a
non-elementary group of complex hyperbolic isometries. In this
paper we improve several of Basmajian and Miner's key estimates and
so give a substantial improvement on the main inequality in the
stable basin theorem.
Categories:22E40, 20H10, 57S30 |
31. CMB 2004 (vol 47 pp. 60)
Rational Integer Invariants of Regular Cyclic Actions Let $g\colon M^{2n}\rightarrow M^{2n}$ be a smooth map of period $m>2$ which
preserves orientation. Suppose that the cyclic action defined by $g$ is regular
and that the normal bundle of the fixed point set $F$ has a $g$-equivariant
complex structure. Let $F\pitchfork F$ be the transverse self-intersection of
$F$ with itself. If the $g$-signature $\Sign (g,M)$ is a rational integer and
$n<\phi (m)$, then there exists a choice of orientations such that $\Sign(g,M)=
\Sign F=\Sign(F\pitchfork F)$.
Category:57S17 |
32. CMB 2003 (vol 46 pp. 617)
On Harmonic Theory in Flows Recently [8], a harmonic theory was developed for a compact
contact manifold from the viewpoint of the transversal geometry of
contact flow. A contact flow is a typical example of geodesible
flow. As a natural generalization of the contact flow, the present
paper develops a harmonic theory for various flows on compact
manifolds. We introduce the notions of $H$-harmonic and
$H^*$-harmonic spaces associated to a H\"ormander flow. We also
introduce the notions of basic harmonic spaces associated to a weak
basic flow. One of our main results is to show that in the special
case of isometric flow these harmonic spaces are isomorphic to the
cohomology spaces of certain complexes. Moreover, we find an
obstruction for a geodesible flow to be isometric.
Keywords:contact structure, geodesible flow, isometric flow, basic cohomology Categories:53C20, 57R30 |
33. CMB 2003 (vol 46 pp. 356)
Branched Covers of Tangles in Three-balls We give an algorithm for a surgery description of a $p$-fold cyclic branched
cover of $B^3$ branched along a tangle. We generalize constructions of
Montesinos and Akbulut-Kirby.
Keywords:tangle, branched cover, surgery, Heegaard decomposition Categories:57M25, 57M12 |
34. CMB 2003 (vol 46 pp. 265)
Reducing Spheres and Klein Bottles after Dehn Fillings Let $M$ be a compact, connected, orientable, irreducible 3-manifold with a
torus boundary. It is known that if two Dehn fillings on $M$ along the
boundary produce a reducible manifold and a manifold containing a Klein
bottle, then the distance between the filling slopes is at most three. This
paper gives a remarkably short proof of this result.
Keywords:Dehn filling, reducible, Klein bottle Category:57M50 |
35. CMB 2003 (vol 46 pp. 310)
Second Order Dehn Functions of Asynchronously Automatic Groups Upper bounds of second order Dehn functions of asynchronously
automatic groups are obtained.
Keywords:second order Dehn function, combing, asynchronously automatic group Categories:20E06, 20F05, 57M05 |
36. CMB 2003 (vol 46 pp. 122)
On Certain Finitely Generated Subgroups of Groups Which Split Define a group $G$ to be in the class $\mathcal{S}$ if for any
finitely generated subgroup $K$ of $G$ having the property that
there is a positive integer $n$ such that $g^n \in K$ for all
$g\in G$, $K$ has finite index in $G$. We show that a free
product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong
to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of
$C$ is finitely generated.
Keywords:free product with amalgamation, $\HNN$ group, graph of groups, fundamental group Categories:20E06, 20E08, 57M07 |
37. CMB 2002 (vol 45 pp. 231)
Erratum:~~The Lehmer Polynomial and Pretzel Links Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440--451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 |
38. CMB 2002 (vol 45 pp. 131)
The Topological Interpretation of the Core Group of a Surface in $S^4$ We give a topological interpretation of the core group invariant of a
surface embedded in $S^4$ \cite{F-R}, \cite{Ro}. We show that the
group is isomorphic to the free product of the fundamental group of
the double branch cover of $S^4$ with the surface as a branched set,
and the infinite cyclic group. We present a generalization for
unoriented surfaces, for other cyclic branched covers, and other
codimension two embeddings of manifolds in spheres.
Categories:57Q45, 57M12, 57M05 |
39. CMB 2001 (vol 44 pp. 440)
The Lehmer Polynomial and Pretzel Links In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the well-known Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 |
40. CMB 2000 (vol 43 pp. 268)
Cockcroft Properties of Thompson's Group In a study of the word problem for groups, R.~J.~Thompson
considered a certain group $F$ of self-homeomorphisms of the Cantor
set and showed, among other things, that $F$ is finitely presented.
Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed
that $F$ is the fundamental group of a finite two-complex $Z^2$
having Euler characteristic one and which is {\em Cockcroft}, in
the sense that each map of the two-sphere into $Z^2$ is
homologically trivial. We show that no proper covering complex of
$Z^2$ is Cockcroft. A general result on Cockcroft properties
implies that no proper regular covering complex of any finite
two-complex with fundamental group $F$ is Cockcroft.
Keywords:two-complex, covering space, Cockcroft two-complex, Thompson's group Categories:57M20, 20F38, 57M10, 20F34 |
41. CMB 2000 (vol 43 pp. 343)
Controlled Homeomorphisms Over Nonpositively Curved Manifolds We obtain a homotopy splitting of the forget control map for
controlled homeomorphisms over closed manifolds of nonpositive
curvature.
Keywords:controlled topology, controlled homeomorphism, nonpositive curvature, Novikov conjectures Categories:57N15, 53C20, 55R65, 57N37 |
42. CMB 2000 (vol 43 pp. 145)
On the 2-Parallel Versions of Links In this paper, we show that the absolute value of the signature of
the $2$-parallel version of a link is less than or equal to the
nullity of it and show that the signature, nullity, and Minkowski
units of the $2$-parallel version of a certain class of links are
always equal to $0$, $2$, and $1$ respectively.
Keywords:braid, Goeritz matrix, Minkowski unit, nullity, signature, 2-parallel version Category:57M25 |
43. CMB 2000 (vol 43 pp. 79)
Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of
cyclotomic Schur algebras and of cyclotomic Hecke algebras, over
discrete valuation rings.
Categories:20G05, 20C20, 16G30, 17B37, 57M25 |
44. CMB 1999 (vol 42 pp. 257)
Homotopy of Knots and the Alexander Polynomial Any knot in a 3-dimensional homology sphere is homotopic to a knot
with trivial Alexander polynomial.
Categories:57N10, 57M05, 57M25, 57N65 |
45. CMB 1999 (vol 42 pp. 190)
Topological Quantum Field Theory and Strong Shift Equivalence Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of
a closed $(d+1)$-dimensional manifold $M$, we define an invariant
taking values in a strong shift equivalence class of matrices. The
notion of strong shift equivalence originated in R.~Williams' work
in symbolic dynamics. The Turaev-Viro module associated to a TQFT
and an infinite cyclic covering is then given by the Jordan form of
this matrix away from zero. This invariant is also defined if the
boundary of $M$ has an $S^1$ factor and the infinite cyclic cover
of the boundary is standard. We define a variant of a TQFT
associated to a finite group $G$ which has been studied by Quinn.
In this way, we recover a link invariant due to D.~Silver and
S.~Williams. We also obtain a variation on the Silver-Williams
invariant, by using the TQFT associated to $G$ in its unmodified form.
Keywords:knot, link, TQFT, symbolic dynamics, shift equivalence Categories:57R99, 57M99, 54H20 |
46. CMB 1999 (vol 42 pp. 248)
The Classification of $\Pin_4$-Bundles over a $4$-Complex In this paper we show that the Lie-group $\Pin_4$ is isomorphic to
the semidirect product $(\SU_2\times \SU_2)\timesv \Z/2$ where
$\Z/2$ operates by flipping the factors. Using this structure
theorem we prove a classification theorem for $\Pin_4$-bundles over
a finite $4$-complex $X$.
Categories:55N25, 55R10, 57S15 |
47. CMB 1999 (vol 42 pp. 149)
A Note on Finite Dehn Fillings Let $M$ be a compact, connected, orientable 3-manifold whose
boundary is a torus and whose interior admits a complete hyperbolic
metric of finite volume. In this paper we show that if the minimal
Culler-Shalen norm of a non-zero class in $H_1(\partial M)$ is
larger than $8$, then the finite surgery conjecture holds for $M$.
This means that there are at most $5$ Dehn fillings of $M$ which
can yield manifolds having cyclic or finite fundamental groups and
the distance between any slopes yielding such manifolds is at most
$3$.
Categories:57M25, 57R65 |
48. CMB 1999 (vol 42 pp. 46)
Generic Partial Two-Point Sets Are Extendable It is shown that under $\ZFC$ almost all planar compacta that meet
every line in at most two points are subsets of sets that meet every
line in exactly two points. This result was previously obtained by the
author jointly with K.~Kunen and J.~van~Mill under the assumption that
Martin's Axiom is valid.
Category:57N05 |
49. CMB 1999 (vol 42 pp. 52)
Embedding Coverings in Bundles If $V\to X$ is a vector bundle of fiber dimension $k$ and $Y\to X$
is a finite sheeted covering map of degree $d$, the implications
for the Euler class $e(V)$ in $H^k(X)$ of $V$ implied by the
existence of an embedding $Y\to V$ lifting the covering map are
explored. In particular it is proved that $dd'e(V)=0$ where $d'$
is a certain divisor of $d-1$, and often $d'=1$.
Categories:57M10, 55R25, 55S40, 57N35 |
50. CMB 1998 (vol 41 pp. 374)
Normal invariants of lens spaces We show that normal and stable normal invariants of polarized
homotopy equivalences of lens spaces $M = L(2^m;\r)$ and
$N = L(2^m;\s)$ are determined by certain $\ell$-polynomials
evaluated on the elementary symmetric functions
$\sigma_i(\rsquare)$ and $\sigma_i(\ssquare)$. Each polynomial
$\ell_k$ appears as the homogeneous part of degree $k$ in the
Hirzebruch multiplicative $L$-sequence. When $n = 8$, the
elementary symmetric functions alone determine the relevant normal
invariants.
Categories:57R65, 57S25 |