1. CMB Online first
 Motegi, Kimihiko; Teragaito, Masakazu

Generalized torsion elements and biorderability of 3manifold groups
It is known that a biorderable group has no generalized torsion
element,
but the converse does not hold in general.
We conjecture that the converse holds for the fundamental groups
of $3$manifolds,
and verify the conjecture for nonhyperbolic, geometric $3$manifolds.
We also confirm the conjecture for some infinite families of
closed hyperbolic $3$manifolds.
In the course of the proof,
we prove that each standard generator of the Fibonacci group
$F(2, m)$ ($m \gt 2$) is a generalized torsion element.
Keywords:generalized torsion element, biordering, 3manifold group Categories:57M25, 57M05, 06F15, 20F05 

2. CMB Online first
 Ding, Fan; Geiges, Hansjörg; Zhang, Guangjian

On subcritically Stein fillable 5manifolds
We make some elementary observations concerning subcritically
Stein
fillable contact structures on $5$manifolds.
Specifically, we determine the diffeomorphism type of such
contact manifolds in the case the fundamental group is finite
cyclic,
and we show that on the $5$sphere the standard contact structure
is the unique subcritically fillable one. More generally,
it is shown that subcritically fillable contact structures
on simply connected $5$manifolds are determined by their
underlying almost contact structure. Along the way, we discuss
the
homotopy classification of almost contact structures.
Keywords:subcritically Stein fillable, 5manifold, almost contact structure, thickening Categories:53D35, 32Q28, 57M20, 57Q10, 57R17 

3. CMB Online first
 Friedl, Stefan; Vidussi, Stefano

Twisted Alexander invariants detect trivial links
It follows from earlier work of SilverWilliams and the authors
that twisted Alexander polynomials detect the unknot and the
Hopf link.
We now show that twisted Alexander polynomials also detect the
trefoil and the figure8 knot,
that twisted Alexander polynomials detect whether a link is split
and that twisted Alexander modules detect trivial links. We use
this result to provide algorithms for detecting whether a link
is the unlink, whether it is split and whether it is totally
split.
Keywords:twisted Alexander polynomial, virtual fibering theorem, unlink detection Category:57M27 

4. CMB Online first
 Basu, Samik; Subhash, B

Topology of certain quotient spaces of Stiefel manifolds
We compute the cohomology of the right generalised projective
Stiefel manifolds. Following this, we discuss some easy applications
of the computations to the ranks of complementary bundles, and
bounds on the span and immersibility.
Keywords:projective Stiefel manifold, span, spectral sequence Categories:55R20, 55R25, 57R20 

5. CMB 2016 (vol 60 pp. 165)
 Morimoto, Masaharu

Cokernels of Homomorphisms from Burnside Rings to Inverse Limits
Let $G$ be a finite group and
let $A(G)$ denote the Burnside ring of $G$.
Then an inverse limit $L(G)$ of the groups $A(H)$ for
proper subgroups $H$ of $G$ and a homomorphism
${\operatorname{res}}$ from $A(G)$ to $L(G)$ are obtained in a natural
way.
Let $Q(G)$ denote the cokernel of ${\operatorname{res}}$.
For a prime $p$,
let $N(p)$ be the minimal
normal subgroup of $G$ such that the order of $G/N(p)$ is
a power of $p$, possibly $1$.
In this paper we prove that $Q(G)$ is isomorphic to
the cartesian product of the groups $Q(G/N(p))$, where $p$
ranges over the primes dividing the order of $G$.
Keywords:Burnside ring, inverse limit, finite group Categories:19A22, 57S17 

6. CMB 2016 (vol 59 pp. 806)
 Izumiya, Shyuichi

Geometric Interpretation of Lagrangian Equivalence
As an application of the theory of
graphlike Legendrian unfoldings, relations of the hidden structures
of caustics and wave front propagations are revealed.
Keywords:wave front propagations, big wave fronts, graphlike Legendrian unfoldings, caustics Categories:58K05, 57R45, 58K60 

7. CMB 2016 (vol 59 pp. 472)
 Clay, Adam; Desmarais, Colin; Naylor, Patrick

Testing Biorderability of Knot Groups
We investigate the biorderability of twobridge knot groups
and the groups of knots with 12 or fewer crossings by applying
recent theorems of Chiswell, Glass and Wilson.
Amongst all knots with 12 or fewer crossings (of which there
are 2977), previous theorems were only able to determine biorderability
of 499 of the corresponding knot groups. With our methods we
are able to deal with 191 more.
Keywords:knots, fundamental groups, orderable groups Categories:57M25, 57M27, 06F15 

8. CMB 2015 (vol 59 pp. 170)
 MartínezPedroza, Eduardo

A Note on Fine Graphs and Homological Isoperimetric Inequalities
In the framework of homological characterizations of relative
hyperbolicity, Groves and Manning posed the question of whether
a simply connected $2$complex $X$ with a linear homological
isoperimetric inequality, a bound on the length of attaching
maps of $2$cells and finitely many $2$cells adjacent to any
edge must have a fine $1$skeleton. We provide a positive answer
to this question. We revisit a homological characterization
of relative hyperbolicity, and show that a group $G$ is hyperbolic
relative to a collection of subgroups $\mathcal P$ if and only if
$G$ acts cocompactly with finite edge stabilizers on an connected
$2$dimensional cell complex with a linear homological isoperimetric
inequality and $\mathcal P$ is a collection of representatives of
conjugacy classes of vertex stabilizers.
Keywords:isoperimetric functions, Dehn functions, hyperbolic groups Categories:20F67, 05C10, 20J05, 57M60 

9. CMB 2015 (vol 59 pp. 159)
 MacColl, Joseph

Rotors in Khovanov Homology
Anstee, Przytycki, and Rolfsen introduced the idea of rotants,
pairs of links related by a generalised form of link mutation.
We exhibit infinitely many pairs of rotants which can be distinguished
by Khovanov homology, but not by the Jones polynomial.
Keywords:geometric topology, knot theory, rotants, khovanov homology, jones polynomial Categories:57M27, 57M25 

10. CMB 2015 (vol 59 pp. 182)
 Naylor, Geoff; Rolfsen, Dale

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 biorderable. 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 

11. CMB 2014 (vol 58 pp. 196)
12. CMB 2014 (vol 58 pp. 69)
 Fulp, Ronald Owen

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. 283288. 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 quasinilpotent for each $a$ since it immediately
follows that $K$
is quasinilpotent. 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 

13. CMB 2014 (vol 57 pp. 431)
 Tagami, Keiji

The Rasmussen Invariant, Fourgenus and Threegenus 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, fourgenus, Rasmussen invariant Categories:57M27, 57M25 

14. CMB 2013 (vol 57 pp. 526)
 Heil, Wolfgang; Wang, Dongxu

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:LusternikSchnirelmann category, coverings of $3$manifolds by open $K$contractible sets Categories:57N10, 55M30, 57M27, 57N16 

15. CMB Online first


Leftorderable 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 twobridge knot corresponding to the rational number $3/7$, has leftorderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:leftordering, Dehn surgery Categories:57M25, 06F15 

16. CMB Online first


Leftorderable 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 twobridge knot corresponding to the rational number $3/7$, has leftorderable
fundamental group if the slope $r$ satisfies $0\le r \le 4$.
Keywords:leftordering, Dehn surgery Categories:57M25, 06F15 

17. CMB 2013 (vol 57 pp. 310)
18. CMB 2012 (vol 56 pp. 850)
 Teragaito, Masakazu

Leftorderability and Exceptional Dehn Surgery on Twist Knots
We show that any exceptional nontrivial Dehn surgery on a twist knot, except the trefoil,
yields a $3$manifold whose fundamental group is leftorderable.
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:leftordering, twist knot, Dehn surgery Categories:57M25, 06F15 

19. CMB 2011 (vol 55 pp. 586)
 Nie, Zhaohu

On Sha's Secondary ChernEuler 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 ChernEuler 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 ChernEuler 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 ChernEuler class, locally product metric, law of vector fields Categories:57R20, 57R25 

20. CMB 2011 (vol 55 pp. 368)
 Nie, Zhaohu

The Secondary ChernEuler Class for a General Submanifold
We define and study the secondary ChernEuler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with nonisolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.
Keywords:secondary ChernEuler class, normal sphere bundle, Euler characteristic, index, nonisolated singularities, blowup Category:57R20 

21. CMB 2011 (vol 55 pp. 164)
 Pergher, Pedro L. Q.

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 $nj=2^pq$, where $q \ge 1$ is odd and $p \geq 0$, and set $m(nj) = 2n+pq+1$ if $p \leq q + 1$
and $m(nj)= 2n + 2^{pq}$ if $p \geq q$. In this paper we show that $m \le m(nj) + 2j+1$. Further, we show that this bound is \emph{almost} best possible, by exhibiting examples $(M^{m(nj) +2j},T)$ 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, StiefelWhitney class, characteristic number Category:57R85 

22. CMB 2011 (vol 54 pp. 693)
 Lusala, Tsasa; Śniatycki, Jędrzej

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 

23. CMB 2011 (vol 54 pp. 283)
24. CMB 2010 (vol 54 pp. 556)
 Teragaito, Masakazu

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 $m1$
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 

25. CMB 2010 (vol 54 pp. 147)
 Nelson, Sam

Generalized Quandle Polynomials
We define a family of generalizations of the twovariable quandle polynomial.
These polynomial invariants generalize in a natural way to eightvariable
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 
