Expand all Collapse all | Results 1 - 25 of 58 |
1. CMB Online first
Generalized Torsion in Knot Groups In a group, a nonidentity element is called
a generalized torsion element if some product of its conjugates
equals the identity. We show that for many classical knots one
can find generalized torsion in the fundamental group of its
complement, commonly called the knot group. It follows that
such a group is not bi-orderable. Examples include all torus
knots, the (hyperbolic) knot $5_2$ and algebraic knots in the
sense of Milnor.
Keywords:knot group, generalized torsion, ordered group Categories:57M27, 32S55, 29F60 |
2. CMB 2014 (vol 58 pp. 196)
Dihedral Groups of order $2p$ of Automorphisms of Compact Riemann Surfaces of Genus $p-1$ In this paper we prove that there is only one conjugacy class of
dihedral group of order $2p$ in the $2(p-1)\times 2(p-1)$ integral
symplectic group can be realized by an analytic automorphism
group
of compact connected Riemann surfaces of genus $p-1$. A pair of
representative generators of the realizable class is also given.
Keywords:dihedral group, automorphism group, Riemann surface, integral symplectic matrix, fundamental domain Categories:20H25, 57M60 |
3. CMB 2014 (vol 58 pp. 69)
Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies" The Theorem below is a correction to Theorem
3.5 in the article
entitled " Infinite Dimensional DeWitt Supergroups and Their
Bodies" published
in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283-288. Only part
(iii) of that Theorem
requires correction. The proof of Theorem 3.5 in the original
article failed to separate
the proof of (ii) from the proof of (iii). The proof of (ii)
is complete once it is established
that $ad_a$ is quasi-nilpotent for each $a$ since it immediately
follows that $K$
is quasi-nilpotent. The proof of (iii) is not complete
in the original article. The revision appears as the proof of
(iii) of the revised Theorem below.
Keywords:super groups, body of super groups, Banach Lie groups Categories:58B25, 17B65, 81R10, 57P99 |
4. CMB 2014 (vol 57 pp. 431)
The Rasmussen Invariant, Four-genus and Three-genus of an Almost Positive Knot Are Equal An oriented link is positive if it has a link diagram whose crossings are all positive.
An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing.
It is known that the Rasmussen invariant, $4$-genus and $3$-genus of a positive knot are equal.
In this paper, we prove that the Rasmussen invariant, $4$-genus and $3$-genus of an almost positive knot are equal.
Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram.
As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of $4$-genus one.
Keywords:almost positive knot, four-genus, Rasmussen invariant Categories:57M27, 57M25 |
5. CMB 2013 (vol 57 pp. 526)
On $3$-manifolds with Torus or Klein Bottle Category Two A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$
is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors
homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any
base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a
subgroup of a quotient group of $\pi_1 (K)$. Subsets of $M$ with this
latter property are called $\mathcal{G}_K$-contractible. We obtain a
list of the closed $3$-manifolds that can be covered by two open
$\mathcal{G}_K$-contractible subsets. This is applied to obtain a list
of the possible closed prime $3$-manifolds that can be covered by two
open $K$-contractible subsets.
Keywords:Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible sets Categories:57N10, 55M30, 57M27, 57N16 |
6. CMB Online first
Left-orderable fundamental group and Dehn surgery on the knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is
the two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
7. CMB Online first
Left-orderable fundamental group and Dehn surgery on the knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is
the two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
8. CMB 2013 (vol 57 pp. 310)
Left-orderable Fundamental Group and Dehn Surgery on the Knot $5_2$ We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is
the two-bridge knot corresponding to the rational number $3/7$, has left-orderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:left-ordering, Dehn surgery Categories:57M25, 06F15 |
9. CMB 2012 (vol 56 pp. 850)
Left-orderability and Exceptional Dehn Surgery on Twist Knots We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$-manifold whose fundamental group is left-orderable.
This is a generalization of a result of Clay, Lidman and Watson, and
also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
Keywords:left-ordering, twist knot, Dehn surgery Categories:57M25, 06F15 |
10. CMB 2011 (vol 55 pp. 586)
On Sha's Secondary Chern-Euler Class For a manifold with boundary, the restriction of Chern's transgression
form of the Euler curvature form over the boundary is closed. Its
cohomology class is called the secondary Chern-Euler class and was
used by Sha to formulate a relative PoincarÃ©-Hopf theorem under the
condition that the metric on the manifold is locally product near the
boundary. We show that the secondary Chern-Euler form is exact away
from the outward and inward unit normal vectors of the boundary by
explicitly constructing a transgression form. Using Stokes' theorem,
this evaluates the boundary term in Sha's relative PoincarÃ©-Hopf
theorem in terms of more classical indices of the tangential
projection of a vector field. This evaluation in particular shows
that Sha's relative PoincarÃ©-Hopf theorem is equivalent to the more
classical law of vector fields.
Keywords:transgression, secondary Chern-Euler class, locally product metric, law of vector fields Categories:57R20, 57R25 |
11. CMB 2011 (vol 55 pp. 368)
The Secondary Chern-Euler Class for a General Submanifold We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.
Keywords:secondary Chern-Euler class, normal sphere bundle, Euler characteristic, index, non-isolated singularities, blow-up Category:57R20 |
12. CMB 2011 (vol 55 pp. 164)
Involutions Fixing $F^n \cup \{\text{Indecomposable}\}$ Let $M^m$ be an $m$-dimensional, closed and smooth manifold, equipped with a smooth involution $T\colon M^m \to M^m$ whose fixed point set has the form $F^n \cup F^j$, where $F^n$ and $F^j$ are submanifolds with dimensions $n$ and $j$, $F^j$ is indecomposable and $ n >j$. Write $n-j=2^pq$, where $q \ge 1$ is odd and $p \geq 0$, and set $m(n-j) = 2n+p-q+1$ if $p \leq q + 1$
and $m(n-j)= 2n + 2^{p-q}$ if $p \geq q$. In this paper we show that $m \le m(n-j) + 2j+1$. Further, we show that this bound is \emph{almost} best possible, by exhibiting examples $(M^{m(n-j) +2j},T)$ where the fixed point set of
$T$ has the form $F^n \cup F^j$ described above, for every $2 \le j Keywords:involution, projective space bundle, indecomposable manifold, splitting principle, Stiefel-Whitney class, characteristic number Category:57R85 |
13. CMB 2011 (vol 54 pp. 693)
Stratified Subcartesian Spaces We show that if the family $\mathcal{O}$ of orbits of all vector fields on
a subcartesian space $P$ is locally finite and each orbit in $\mathcal{O}$
is locally closed, then $\mathcal{O}$ defines a smooth Whitney A
stratification of $P$. We also show that the stratification by orbit type of
the space of orbits $M/G$ of a proper action of a Lie group $G$ on a smooth
manifold $M$ is given by orbits of the family of all vector fields on $M/G$.
Keywords:Subcartesian spaces, orbits of vector fields, stratifications, Whitney Conditions Categories:58A40, 57N80 |
14. CMB 2011 (vol 54 pp. 283)
Surgery on $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$-Manifolds We show that closed $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$-manifolds
are topologically rigid if $n\geq 2$, and are rigid up to
$s$-cobordism, if $n=1$.
Keywords:topological rigidity, geometric structure, surgery groups Categories:57R67, 57N16 |
15. CMB 2010 (vol 54 pp. 556)
Cyclic Surgery Between Toroidal Surgeries
We show that there is an infinite family of hyperbolic knots such that
each knot admits a cyclic surgery $m$ whose adjacent surgeries $m-1$
and $m+1$ are toroidal. This gives an affirmative answer to a
question asked by Boyer and Zhang.
Keywords:cyclic surgery, toroidal surgery Category:57M25 |
16. CMB 2010 (vol 54 pp. 147)
Generalized Quandle Polynomials
We define a family of generalizations of the two-variable quandle polynomial.
These polynomial invariants generalize in a natural way to eight-variable
polynomial invariants of finite biquandles. We use these polynomials to define
a family of link invariants that further generalize the quandle counting
invariant.
Keywords:finite quandles, finite biquandles, link invariants Categories:57M27, 76D99 |
17. CMB 2010 (vol 53 pp. 706)
Non-Right-Orderable 3-Manifold Groups
We exhibit infinitely many hyperbolic $3$-manifold
groups that are not right-orderable.
Categories:20F60, 57M05, 57M50 |
18. CMB 2010 (vol 53 pp. 542)
Smooth Mappings with Higher Dimensional Critical Sets In this paper we provide lower bounds for the dimension of various critical sets, and we point out some differential maps with high dimensional critical sets.
Categories:58K05, 57R70 |
19. CMB 2009 (vol 52 pp. 257)
Essential Surfaces in Graph Link Exteriors An irreducible graph manifold $M$ contains an essential torus if
it is not a special Seifert manifold.
Whether $M$ contains a closed essential surface of
negative Euler characteristic or not
depends on the difference of Seifert fibrations from the two sides
of a torus system which splits $M$ into Seifert manifolds.
However,
it is not easy to characterize geometrically the class of
irreducible graph manifolds which contain such surfaces.
This article studies this problem in the case of graph link exteriors.
Keywords:Graph link, Graph manifold, Seifert manifold, Essential surface Category:57M25 |
20. CMB 2008 (vol 51 pp. 508)
A Result in Surgery Theory We study the topological $4$-dimensional surgery problem
for a closed connected orientable
topological $4$-manifold $X$ with vanishing
second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has
one end and $F(r)$ is the free group of rank $r\ge 1$.
Our result is related to a theorem of Krushkal and Lee, and
depends on the validity of the Novikov conjecture for
such fundamental groups.
Keywords:four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly map Categories:57N65, 57R67, 57Q10 |
21. CMB 2008 (vol 51 pp. 535)
On the Simple $\Z_2$-homotopy Types of Graph Complexes and Their Simple $\Z_2$-universality We prove that the neighborhood complex $\N(G)$,
the box complex $\B(G)$, the homomorphism complex
$\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the
same simple $\Z_2$-homotopy type in the sense of
Whitehead. We show that these graph complexes
are simple $\Z_2$-universal.
Keywords:graph complexes, simple $\Z_2$-homotopy, universality Categories:57Q10, 05C10, 55P10 |
22. CMB 2007 (vol 50 pp. 481)
Concordance des nÅuds de dimension $4$ We prove that for a simply connected closed
$4$-dimensional manifold, its embeddings
into the sphere of dimension $6$ are all
concordant to each other.
Keywords:concordance, cobordisme, n{\oe}ud de dimension $4$, chirurgie plongÃ©e Categories:57Q45, 57Q60, 57R40, 57R65, 57N13 |
23. CMB 2007 (vol 50 pp. 365)
Equivariant Cohomology of $S^{1}$-Actions on $4$-Manifolds Let $M$ be a symplectic $4$-dimensional manifold equipped with a
Hamiltonian circle action with isolated fixed points. We describe a
method for computing its integral equivariant cohomology in terms of
fixed point data. We give some examples of these computations.
Categories:53D20, 55N91, 57S15 |
24. CMB 2007 (vol 50 pp. 390)
Linking Number of Singular Links and the Seifert Matrix We extend the notion of linking number of an
ordinary link of two components to that of a singular link
with transverse intersections in which case the linking
number is a half-integer. We then apply it to simplify
the construction of the Seifert matrix, and therefore
the Alexander polynomial, in a natural way.
Category:57M25 |
25. CMB 2007 (vol 50 pp. 206)
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ |
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional
$CW$-complex of the homotopy type of an $n$-sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with
free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as
well.
Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space form Categories:55M35, 55P15, 20E22, 20F28, 57S17 |