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1. 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

2. CMB Online first

Friedl, Stefan; Vidussi, Stefano
Twisted Alexander invariants detect trivial links
It follows from earlier work of Silver--Williams 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 figure-8 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

3. 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

4. CMB 2016 (vol 59 pp. 806)

Izumiya, Shyuichi
Geometric Interpretation of Lagrangian Equivalence
As an application of the theory of graph-like Legendrian unfoldings, relations of the hidden structures of caustics and wave front propagations are revealed.

Keywords:wave front propagations, big wave fronts, graph-like Legendrian unfoldings, caustics
Categories:58K05, 57R45, 58K60

5. CMB 2016 (vol 59 pp. 472)

Clay, Adam; Desmarais, Colin; Naylor, Patrick
Testing Bi-orderability of Knot Groups
We investigate the bi-orderability of two-bridge 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 bi-orderability 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

6. CMB 2015 (vol 59 pp. 170)

Martínez-Pedroza, 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

7. 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

8. 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 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

9. CMB 2014 (vol 58 pp. 196)

Yang, Qingjie; Zhong, Weiting
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

10. 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. 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

11. CMB 2014 (vol 57 pp. 431)

Tagami, Keiji
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

12. 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:Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible sets
Categories:57N10, 55M30, 57M27, 57N16

13. 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

14. 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

15. CMB 2013 (vol 57 pp. 310)

Hakamata, Ryoto; Teragaito, Masakazu
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

16. CMB 2012 (vol 56 pp. 850)

Teragaito, Masakazu
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

17. CMB 2011 (vol 55 pp. 586)

Nie, Zhaohu
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 Poincaré-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 Poincaré-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 Poincaré-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

18. CMB 2011 (vol 55 pp. 368)

Nie, Zhaohu
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

19. 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 $n-j=2^pq$, 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

20. 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

21. CMB 2011 (vol 54 pp. 283)

Hillman, J. A.; Roushon, S. K.
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

22. 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 $m-1$ and $m+1$ are toroidal. This gives an affirmative answer to a question asked by Boyer and Zhang.

Keywords:cyclic surgery, toroidal surgery

23. CMB 2010 (vol 54 pp. 147)

Nelson, Sam
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

24. CMB 2010 (vol 53 pp. 706)

Roberts, R.; Shareshian, J.
Non-Right-Orderable 3-Manifold Groups
We exhibit infinitely many hyperbolic $3$-manifold groups that are not right-orderable.

Categories:20F60, 57M05, 57M50

25. CMB 2010 (vol 53 pp. 542)

Pintea, Cornel
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
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