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Search: MSC category 57 ( Manifolds and cell complexes )

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1. CMB Online first

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

2. 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

3. 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

4. CMB Online first

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

5. 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

6. 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

7. 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

8. 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 $2 \le j Keywords:involution, projective space bundle, indecomposable manifold, splitting principle, Stiefel-Whitney class, characteristic numberCategory:57R85 9. CMB 2011 (vol 54 pp. 693) Lusala, Tsasa; Śniatycki, Jędrzej  Stratified Subcartesian Spaces We show that if the family$\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$. Keywords:Subcartesian spaces, orbits of vector fields, stratifications, Whitney ConditionsCategories:58A40, 57N80 10. CMB 2011 (vol 54 pp. 283) Hillman, J. A.; Roushon, S. K.  Surgery on$\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$. Keywords:topological rigidity, geometric structure, surgery groups Categories:57R67, 57N16 11. CMB 2010 (vol 54 pp. 556) Teragaito, Masakazu  Cyclic Surgery Between Toroidal Surgeries We show that there is an infinite family of hyperbolic knots such that each knot admits a cyclic surgery$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 surgeryCategory:57M25 12. 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 invariantsCategories:57M27, 76D99 13. 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 14. 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 15. CMB 2009 (vol 52 pp. 257) Ikeda, Toru  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 surfaceCategory:57M25 16. CMB 2008 (vol 51 pp. 508) Cavicchioli, Alberto; Spaggiari, Fulvia  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 mapCategories:57N65, 57R67, 57Q10 17. CMB 2008 (vol 51 pp. 535) Csorba, Péter  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, universalityCategories:57Q10, 05C10, 55P10 18. CMB 2007 (vol 50 pp. 481) Blanlœil, Vincent; Saeki, Osamu  Concordance des nÅuds 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©eCategories:57Q45, 57Q60, 57R40, 57R65, 57N13 19. CMB 2007 (vol 50 pp. 365) Godinho, Leonor  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 20. CMB 2007 (vol 50 pp. 390) Hebda, James J.; Hsieh, Chun-Chung; Tsau, Chichen M.  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 21. CMB 2007 (vol 50 pp. 206) Golasiński, Marek; Gonçalves, Daciberg Lima  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 formCategories:55M35, 55P15, 20E22, 20F28, 57S17 22. CMB 2006 (vol 49 pp. 624) Teragaito, Masakazu  On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces For a non-trivial knot in the$3$-sphere, only integral Dehn surgery can create a closed$3$-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two. Keywords:knot, Dehn surgery, non-orientable surfaceCategory:57M25 23. CMB 2006 (vol 49 pp. 337) Berlanga, R.  Homotopy Equivalence and Groups of Measure-Preserving Homeomorphisms It is shown that the group of compactly supported, measure-preserving homeomorphisms of a connected, second countable manifold is locally contractible in the direct limit topology. Furthermore, this group is weakly homotopically equivalent to the more general group of compactly supported homeomorphisms. Categories:57S05, 58F11 24. CMB 2006 (vol 49 pp. 36) Daskalopoulos, Georgios D.; Wentworth, Richard A.  Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles Using a modification of Webster's proof of the Newlander--Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class. Categories:57M50, 58E20, 53C24 25. CMB 2006 (vol 49 pp. 55) Dubois, Jérôme  Non Abelian Twisted Reidemeister Torsion for Fibered Knots In this article, we give an explicit formula to compute the non abelian twisted sign-deter\-mined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot. Keywords:Reidemeister torsion, Fibered knots, Knot groups, Representation space,$\SU$,$\SL\$, Adjoint representation, MonodromyCategories:57Q10, 57M27, 57M25
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