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Search: MSC category 57 ( Manifolds and cell complexes )

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26. CMB 2006 (vol 49 pp. 55)

Dubois, Jérôme
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)

Fehér, L. M.; Némethi, A.; Rimányi, R.
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)

Dąbkowski, Mieczysław K.; Przytycki, Józef H.; Togha, Amir A.
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)

Charette, Virginie; Goldman, William M.; Jones, Catherine A.
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)

Parker, John R.
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)

Little, Robert D.
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)

Pak, Hong Kyung
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)

Ishiwata, Makiko; Przytycki, Józef H.; Yasuhara, Akira
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)

Oh, Seungsang
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)

Wang, Xiaofeng
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)

Moon, Myoungho
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)

Hironaka, Eriko
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)

Przytycki, Józef H.; Rosicki, Witold
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)

Hironaka, Eriko
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)

Bogley, W. A.; Gilbert, N. D.; Howie, James
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)

Hughes, Bruce; Taylor, Larry; Williams, Bruce
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)

Chang, Jae-Ho; Lee, Sang Youl; Park, Chan-Young
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)

König, Steffen
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)

Austin, David; Rolfsen, Dale
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)

Gilmer, Patrick M.
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)

Weber, Christian
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)

Boyer, S.; Zhang, X.
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)

Dijkstra, Jan J.
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)

Edmonds, Allan L.
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)

Young, Carmen M.
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
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