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1. CMB 2018 (vol 61 pp. 754)

Lidman, Tye; Tweedy, Eamonn
 A Note on Concordance Properties of Fibers in Seifert Homology Spheres In this note, we collect various properties of Seifert homology spheres from the viewpoint of Dehn surgery along a Seifert fiber. We expect that many of these are known to various experts, but include them in one place which we hope to be useful in the study of concordance and homology cobordism. Keywords:Seifert fibered, homology sphere, 3-manifold, concordance, cobordism, Heegaard FloerCategories:57M27, 57N70

2. CMB 2018 (vol 61 pp. 518)

Hong, Kyungpyo; Oh, Seungsang
 Bounds on multiple self-avoiding polygons A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problem of this study, we consider multiple self-avoiding polygons in a confined region, as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds of the number $p_{m \times n}$ of distinct multiple self-avoiding polygons in the $m \times n$ rectangular grid on the square lattice. For $m=2$, $p_{2 \times n} = 2^{n-1}-1$. And, for integers $m,n \geq 3$, $$2^{m+n-3} \left(\tfrac{17}{10} \right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \ 2^{m+n-3} \left(\tfrac{31}{16} \right)^{(m-2)(n-2)}.$$ Keywords:ring polymer, self-avoiding polygonCategories:57M25, 82B20, 82B41, 82D60

3. CMB 2017 (vol 61 pp. 211)

Tran, Anh T.; Yamaguchi, Yoshikazu
 The asymptotics of the higher dimensional Reidemeister torsion for exceptional surgeries along twist knots We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible $\operatorname{SL}_2(\mathbb{C})$-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coefficients in the higher dimensional Reidemeister torsion explicitly. Keywords:Reidemeister torsion, graph manifold, asymptotic behavior, exceptional surgeryCategories:57M27, 57M50

4. CMB 2017 (vol 61 pp. 85)

Ding, Fan; Geiges, Hansjörg; Zhang, Guangjian
 On subcritically Stein fillable 5-manifolds 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, 5-manifold, almost contact structure, thickeningCategories:53D35, 32Q28, 57M20, 57Q10, 57R17

5. CMB 2017 (vol 60 pp. 830)

Motegi, Kimihiko; Teragaito, Masakazu
 Generalized Torsion Elements and Bi-orderability of 3-manifold Groups It is known that a bi-orderable 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 non-hyperbolic, 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, bi-ordering, 3-manifold groupCategories:57M25, 57M05, 06F15, 20F05

6. CMB 2017 (vol 60 pp. 283)

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 detectionCategory:57M27

7. CMB 2017 (vol 60 pp. 235)

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 sequenceCategories:55R20, 55R25, 57R20

8. 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 groupCategories:19A22, 57S17

9. 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, causticsCategories:58K05, 57R45, 58K60

10. 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 groupsCategories:57M25, 57M27, 06F15

11. 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 groupsCategories:20F67, 05C10, 20J05, 57M60

12. 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 polynomialCategories:57M27, 57M25

13. 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 groupCategories:57M27, 32S55, 29F60

14. 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 domainCategories:20H25, 57M60

15. 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 groupsCategories:58B25, 17B65, 81R10, 57P99

16. 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 invariantCategories:57M27, 57M25

17. 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 setsCategories:57N10, 55M30, 57M27, 57N16

18. 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 surgeryCategories:57M25, 06F15

19. 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 surgeryCategories:57M25, 06F15

20. 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 surgeryCategories:57M25, 06F15

21. 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 surgeryCategories:57M25, 06F15

22. 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 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 fieldsCategories:57R20, 57R25

23. 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-upCategory:57R20

24. 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 $2 \le j Keywords:involution, projective space bundle, indecomposable manifold, splitting principle, Stiefel-Whitney class, characteristic numberCategory:57R85 25. CMB 2011 (vol 54 pp. 693) Lusala, Tsasa; Śniatycki, Jędrzej  Stratified Subcartesian Spaces We show that if the family$\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 ConditionsCategories:58A40, 57N80
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