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1. CJM Online first

Lescop, Christine
On homotopy invariants of combings of three-manifolds
Combings of compact, oriented $3$-dimensional manifolds $M$ are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying Spin$^c$-structure. A combing is called torsion if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf introduced a $\mathbb Q$-valued invariant $\theta_G$ of torsion combings on closed $3$-manifolds, and he showed that $\theta_G$ distinguishes all torsion combings with the same Spin$^c$-structure. We give an alternative definition for $\theta_G$ and we express its variation as a linking number. We define a similar invariant $p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$ to the $\Theta$-invariant, which is the simplest configuration space integral invariant of rational homology $3$-balls, by the formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for $3$-manifolds.

Keywords:Spin$^c$-structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, Casson-Walker invariant, perturbative expansion of Chern-Simons theory, configuration space integrals
Categories:57M27, 57R20, 57N10

2. CJM 2013 (vol 66 pp. 141)

Caillat-Gibert, Shanti; Matignon, Daniel
Existence of Taut Foliations on Seifert Fibered Homology $3$-spheres
This paper concerns the problem of existence of taut foliations among $3$-manifolds. Since the contribution of David Gabai, we know that closed $3$-manifolds with non-trivial second homology group admit a taut foliation. The essential part of this paper focuses on Seifert fibered homology $3$-spheres. The result is quite different if they are integral or rational but non-integral homology $3$-spheres. Concerning integral homology $3$-spheres, we can see that all but the $3$-sphere and the Poincaré $3$-sphere admit a taut foliation. Concerning non-integral homology $3$-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology $3$-spheres.

Keywords:homology 3-spheres, taut foliation, Seifert-fibered 3-manifolds
Categories:57M25, 57M50, 57N10, 57M15

3. CJM 2013 (vol 66 pp. 453)

Vaz, Pedro; Wagner, Emmanuel
A Remark on BMW algebra, $q$-Schur Algebras and Categorification
We prove that the 2-variable BMW algebra embeds into an algebra constructed from the HOMFLY-PT polynomial. We also prove that the $\mathfrak{so}_{2N}$-BMW algebra embeds in the $q$-Schur algebra of type $A$. We use these results to suggest a schema providing categorifications of the $\mathfrak{so}_{2N}$-BMW algebra.

Keywords:tangle algebras, BMW algebra, HOMFLY-PT Skein algebra, q-Schur algebra, categorification
Categories:57M27, 81R50, 17B37, 16W99

4. CJM 2012 (vol 65 pp. 575)

Kallel, Sadok; Taamallah, Walid
The Geometry and Fundamental Group of Permutation Products and Fat Diagonals
Permutation products and their various ``fat diagonal'' subspaces are studied from the topological and geometric point of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We write down an expression for the fundamental group of any permutation product of a connected space $X$ having the homotopy type of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then prove that the fundamental group of the configuration space of $n$-points on $X$, of which multiplicities do not exceed $n/2$, coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included.

Keywords:symmetric products, fundamental group, orbit stratification
Categories:14F35, 57F80

5. CJM 2011 (vol 64 pp. 102)

Ishii, Atsushi; Iwakiri, Masahide
Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies
We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.

Keywords:quandle cocycle invariant, knotted handlebody, spatial graph
Categories:57M27, 57M15, 57M25

6. CJM 2010 (vol 63 pp. 436)

Mine, Kotaro; Sakai, Katsuro
Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces
Let $F$ be a non-separable LF-space homeomorphic to the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$, where $\aleph_0 < \tau_1 < \tau_2 < \cdots$. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $|K| \times F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$ with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$ (and $\operatorname{card} K^{(0)} \le \tau$), and, conversely, if $K$ is such a simplicial complex, then the product $|K| \times F$ can be embedded in $F$ as an open set, where $|K|$ is the polyhedron of $K$ with the metric topology.

Keywords:LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$
Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40

7. CJM 2010 (vol 62 pp. 1387)

Pamuk, Mehmetcik
Homotopy Self-Equivalences of 4-manifolds with Free Fundamental Group
We calculate the group of homotopy classes of homotopy self-equivalences of $4$-manifolds with free fundamental group and obtain a classification of such $4$-manifolds up to $s$-cobordism.

Categories:57N13, 55P10, 57R80

8. CJM 2010 (vol 62 pp. 994)

Breslin, William
Curvature Bounds for Surfaces in Hyperbolic 3-Manifolds
A triangulation of a hyperbolic $3$-manifold is \emph{L-thick} if each tetrahedron having all vertices in the thick part of $M$ is $L$-bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant $L$ such that every complete hyperbolic $3$-manifold has an $L$-thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of $\pi_1$-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic $3$-manifolds.

Category:57M50

9. CJM 2009 (vol 62 pp. 284)

Grbić, Jelena; Theriault, Stephen
Self-Maps of Low Rank Lie Groups at Odd Primes
Let G be a simple, compact, simply-connected Lie group localized at an odd prime~p. We study the group of homotopy classes of self-maps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.

Keywords:Lie group, self-map, H-map
Categories:55P45, 55Q05, 57T20

10. CJM 2009 (vol 62 pp. 614)

Pronk, Dorette; Scull, Laura
Translation Groupoids and Orbifold Cohomology
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams.

Keywords:orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractions
Categories:57S15, 55N91, 19L47, 18D05, 18D35

11. CJM 2008 (vol 60 pp. 1240)

Beliakova, Anna; Wehrli, Stephan
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine--Tristram signature.

Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism
Categories:57M25, 57M27, 18G60

12. CJM 2008 (vol 60 pp. 164)

Lee, Sangyop; Teragaito, Masakazu
Boundary Structure of Hyperbolic $3$-Manifolds Admitting Annular and Toroidal Fillings at Large Distance
For a hyperbolic $3$-manifold $M$ with a torus boundary component, all but finitely many Dehn fillings yield hyperbolic $3$-manifolds. In this paper, we will focus on the situation where $M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling. For such a situation, Gordon gave an upper bound of $5$ for the distance between such slopes. Furthermore, the distance $4$ is realized only by two specific manifolds, and $5$ is realized by a single manifold. These manifolds all have a union of two tori as their boundaries. Also, there is a manifold with three tori as its boundary which realizes the distance $3$. We show that if the distance is $3$ then the boundary of the manifold consists of at most three tori.

Keywords:Dehn filling, annular filling, toroidal filling, knot
Categories:57M50, 57N10

13. CJM 2007 (vol 59 pp. 418)

Stoimenow, A.
On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials
It is known that the Brandt--Lickorish--Millett--Ho polynomial $Q$ contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from $Q$ is an open problem. We show that this is not so up to degree $9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degree $d \le 10$ are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

Categories:57M25, 57M27, 20F36, 57M50

14. CJM 2006 (vol 58 pp. 673)

Bart, Anneke; Scannell, Kevin P.
The Generalized Cuspidal Cohomology Problem
Let $\Gamma \subset \SO(3,1)$ be a lattice. The well known \emph{bending deformations}, introduced by \linebreak Thurston and Apanasov, can be used to construct non-trivial curves of representations of $\Gamma$ into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in $\H^1(\Gamma, \mink{4})$. Our main result generalizes this construction of cohomology to the context of ``branched'' totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in $S^3$ which is not infinitesimally rigid in $\SO(4,1)$. The first order deformations of this link complement are supported on a piecewise totally geodesic $2$-complex.

Categories:57M50, 22E40

15. CJM 2006 (vol 58 pp. 529)

Dijkstra, Jan J.; Mill, Jan van
On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself
In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\R$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\Q$ onto itself is homeomorphic to the infinite power of $\Q$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\Q$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to $\R^n$ for $n\ge 2$, by linking the groups in question with Erd\H os space.

Keywords:homeomorphism group, real line, countable dense set, pseudoboundary, Erd\H{o}s space, hyperspace
Category:57S05

16. CJM 2004 (vol 56 pp. 1022)

Matignon, D.; Sayari, N.
Non-Orientable Surfaces and Dehn Surgeries
Let $K$ be a knot in $S^3$. This paper is devoted to Dehn surgeries which create $3$-manifolds containing a closed non-orientable surface $\ch S$. We look at the slope ${p}/{q}$ of the surgery, the Euler characteristic $\chi(\ch S)$ of the surface and the intersection number $s$ between $\ch S$ and the core of the Dehn surgery. We prove that if $\chi(\hat S) \geq 15 - 3q$, then $s=1$. Furthermore, if $s=1$ then $q\leq 4-3\chi(\ch S)$ or $K$ is cabled and $q\leq 8-5\chi(\ch S)$. As consequence, if $K$ is hyperbolic and $\chi(\ch S)=-1$, then $q\leq 7$.

Keywords:Non-orientable surface, Dehn surgery, Intersection graphs
Categories:57M25, 57N10, 57M15

17. CJM 2003 (vol 55 pp. 1080)

Kellerhals, Ruth
Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds
We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$.

Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40

18. CJM 2003 (vol 55 pp. 766)

Kerler, Thomas
Homology TQFT's and the Alexander--Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory
We develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology of $U(1)$-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^* \mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL (2,\mathbb{R})$-equivariant functors and, as such, are isomorphic. The $\SL (2,\mathbb{R})$-action in the Hennings construction comes from the natural action on $\mathcal{N}$ and in the case of the Frohman--Nicas theory from the Hard--Lefschetz decomposition of the $U(1)$-moduli spaces given that they are naturally K\"ahler. The irreducible components of this TQFT, corresponding to simple representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg--Witten theories, Casson type theories for homology circles {\it \`a la} Donaldson, higher rank gauge theories following Frohman and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of Reshetikhin--Turaev theories over the cyclotomic integers $\mathbb{Z} [\zeta_p]$. We also conjecture that the Hennings TQFT for quantum-$\mathfrak{sl}_2$ is the product of the Reshetikhin--Turaev TQFT and such a homological TQFT.

Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27

19. CJM 2003 (vol 55 pp. 636)

Schwartzman, Sol
Higher Dimensional Asymptotic Cycles
Given a $p$-dimensional oriented foliation of an $n$-dimensional compact manifold $M^n$ and a transversal invariant measure $\tau$, Sullivan has defined an element of $H_p (M^n,R)$. This generalized the notion of a $\mu$-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure $\mu$. In this one-dimensional case there was a natural 1--1 correspondence between transversal invariant measures $\tau$ and invariant measures $\mu$ when one had a smooth flow without stationary points. For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

Categories:57R30, 57S20

20. CJM 2002 (vol 54 pp. 396)

Lebel, André
Framed Stratified Sets in Morse Theory
In this paper, we present a smooth framework for some aspects of the ``geometry of CW complexes'', in the sense of Buoncristiano, Rourke and Sanderson \cite{[BRS]}. We then apply these ideas to Morse theory, in order to generalize results of Franks \cite{[F]} and Iriye-Kono \cite{[IK]}. More precisely, consider a Morse function $f$ on a closed manifold $M$. We investigate the relations between the attaching maps in a CW complex determined by $f$, and the moduli spaces of gradient flow lines of $f$, with respect to some Riemannian metric on~$M$.

Categories:57R70, 57N80, 55N45

21. CJM 2001 (vol 53 pp. 1309)

Steer, Brian; Wren, Andrew
The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces
A theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact K\"ahler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.

Categories:14J17, 57R57

22. CJM 2001 (vol 53 pp. 780)

Nicolaescu, Liviu I.
Seiberg-Witten Invariants of Lens Spaces
We show that the Seiberg-Witten invariants of a lens space determine and are determined by its Casson-Walker invariant and its Reidemeister-Turaev torsion.

Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sums
Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25

23. CJM 2001 (vol 53 pp. 212)

Puppe, V.
Group Actions and Codes
A $\mathbb{Z}_2$-action with ``maximal number of isolated fixed points'' ({\it i.e.}, with only isolated fixed points such that $\dim_k (\oplus_i H^i(M;k)) =|M^{\mathbb{Z}_2}|, k = \mathbb{F}_2)$ on a $3$-dimensional, closed manifold determines a binary self-dual code of length $=|M^{\mathbb{Z}_2}|$. In turn this code determines the cohomology algebra $H^*(M;k)$ and the equivariant cohomology $H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary self-dual codes one gets information about the cohomology type of $3$-manifolds which admit involutions with maximal number of isolated fixed points. In particular, ``most'' cohomology types of closed $3$-manifolds do not admit such involutions. Generalizations of the above result are possible in several directions, {\it e.g.}, one gets that ``most'' cohomology types (over $\mathbb{F}_2)$ of closed $3$-manifolds do not admit a non-trivial involution.

Keywords:Involutions, $3$-manifolds, codes
Categories:55M35, 57M60, 94B05, 05E20

24. CJM 2000 (vol 52 pp. 293)

Collin, Olivier
Floer Homology for Knots and $\SU(2)$-Representations for Knot Complements and Cyclic Branched Covers
In this article, using 3-orbifolds singular along a knot with underlying space a homology sphere $Y^3$, the question of existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers of $Y^3$ along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian $\SU(2)$-representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in $S^3$.

Categories:57R57, 57M12, 57M25, 57M05

25. CJM 1999 (vol 51 pp. 1035)

Litherland, R. A.
The Homology of Abelian Covers of Knotted Graphs
Let $\tilde M$ be a regular branched cover of a homology 3-sphere $M$ with deck group $G\cong \zt^d$ and branch set a trivalent graph $\Gamma$; such a cover is determined by a coloring of the edges of $\Gamma$ with elements of $G$. For each index-2 subgroup $H$ of $G$, $M_H = \tilde M/H$ is a double branched cover of $M$. Sakuma has proved that $H_1(\tilde M)$ is isomorphic, modulo 2-torsion, to $\bigoplus_H H_1(M_H)$, and has shown that $H_1(\tilde M)$ is determined up to isomorphism by $\bigoplus_H H_1(M_H)$ in certain cases; specifically, when $d=2$ and the coloring is such that the branch set of each cover $M_H\to M$ is connected, and when $d=3$ and $\Gamma$ is the complete graph $K_4$. We prove this for a larger class of coverings: when $d=2$, for any coloring of a connected graph; when $d=3$ or $4$, for an infinite class of colored graphs; and when $d=5$, for a single coloring of the Petersen graph.

Categories:57M12, 57M25, 57M15
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