Mostrando recursos 1 - 20 de 1.595

  1. Link invariants derived from multiplexing of crossings

    Miyazawa, Haruko Aida; Wada, Kodai; Yasuhara, Akira
    We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with a mixture of classical and virtual crossings. ¶ For integers  [math] [math] and an ordered [math] –component virtual link diagram [math] , a new virtual link diagram [math] is obtained from [math] by the multiplexing of all crossings. For welded isotopic virtual link diagrams [math] and [math] , the virtual link diagrams [math] and [math] are welded isotopic. From the point of view of classical link theory, it seems very interesting that new classical link invariants are obtained from welded link invariants via the...

  2. On the virtually cyclic dimension of mapping class groups of punctured spheres

    Aramayona, Javier; Juan-Pineda, Daniel; Trujillo-Negrete, Alejandra
    We calculate the virtually cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface. ¶ For spheres with an arbitrary number of punctures, we give a new upper bound for the virtually cyclic dimension of their mapping class group, improving the recent bound of Degrijse and Petrosyan (2015).

  3. The homology of configuration spaces of trees with loops

    Chettih, Safia; Lütgehetmann, Daniel
    We show that the homology of ordered configuration spaces of finite trees with loops is torsion-free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the [math] –page and differentials of Mayer–Vietoris spectral sequences for configuration spaces.

  4. On some adjunctions in equivariant stable homotopy theory

    Hu, Po; Kriz, Igor; Somberg, Petr
    We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This leads to a variety of interesting other adjunctions, including a chain of six (sometimes seven) adjoints involving the restriction functor to a subgroup of a finite group on equivariant spectra indexed over the trivial universe.

  5. On hyperbolic knots in $S^3$ with exceptional surgeries at maximal distance

    Audoux, Benjamin; Lecuona, Ana G; Roukema, Fionntan
    Baker showed that [math] of the [math] classes of Berge knots are obtained by surgery on the minimally twisted [math] –chain link. We enumerate all hyperbolic knots in [math] obtained by surgery on the minimally twisted [math] –chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal) and (lens, Seifert fibred) pairs. In light of Baker’s work, the classification in this paper conjecturally accounts for “most” hyperbolic knots in [math] realizing the maximal distance between these exceptional pairs. As a byproduct, we obtain that all examples that arise from the [math] –chain link actually...

  6. Quasiautomorphism groups of type $F_\infty$

    Audino, Samuel; Aydel, Delaney R; Farley, Daniel S
    The groups [math] , [math] , [math] , [math] and [math] are groups of quasiautomorphisms of the infinite binary tree. Their names indicate a similarity with Thompson’s well-known groups [math] , [math] and [math] . ¶ We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type [math] . Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on [math] cubical complexes.

  7. Stability phenomena in the homology of tree braid groups

    Ramos, Eric
    For a tree [math] , we study the changing behaviors in the homology groups [math] as [math] varies, where [math] . We prove that the ranks of these homologies can be described by a single polynomial for all [math] , and construct this polynomial explicitly in terms of invariants of the tree [math] . To accomplish this we prove that the group [math] can be endowed with the structure of a finitely generated graded module over an integral polynomial ring, and further prove that it naturally decomposes as a direct sum of graded shifts of squarefree monomial ideals. Following this,...

  8. Refinements of the holonomic approximation lemma

    Álvarez-Gavela, Daniel
    The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the [math] –principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov’s idea from convex integration of working “one pure partial derivative at a time” is central to the discussion. We give applications of our results to flexible symplectic and contact topology.

  9. The number of fiberings of a surface bundle over a surface

    Chen, Lei
    For a closed manifold [math] , let [math] be the number of ways that [math] can be realized as a surface bundle, up to [math] –fiberwise diffeomorphism. We consider the case when [math] . We give the first computation of [math] where [math] but [math] is not a product. In particular, we prove [math] for the Atiyah–Kodaira manifold and any finite cover of a trivial surface bundle. We also give an example where [math] .

  10. Equivariant dendroidal sets

    Pereira, Luís Alexandre
    We extend the Cisinski–Moerdijk–Weiss theory of [math] –operads to the equivariant setting to obtain a notion of [math] - [math] –operads that encode “equivariant operads with norm maps” up to homotopy. At the root of this work is the identification of a suitable category of [math] –trees together with a notion of [math] –inner horns capable of encoding the compositions of norm maps. ¶ Additionally, we follow Blumberg and Hill by constructing suitable variants associated to each of the indexing systems featured in their work.

  11. Symplectic embeddings of four-dimensional polydisks into balls

    Christianson, Katherine; Nelson, Jo
    We obtain new obstructions to symplectic embeddings of the four-dimensional polydisk [math] into the ball [math] for [math] , extending work done by Hind and Lisi and by Hutchings. Schlenk’s folding construction permits us to conclude our bound on [math] is optimal. Our proof makes use of the combinatorial criterion necessary for one “convex toric domain” to symplectically embed into another introduced by Hutchings (2016). We also observe that the computational complexity of this criterion can be reduced from [math] to [math] .

  12. The double $n$–space property for contractible $n$–manifolds

    Sparks, Peter
    Motivated by a recent paper of Gabai (J. Topol. 4 (2011) 529–534) on the Whitehead contractible 3–manifold, we investigate contractible manifolds [math] which decompose or split as [math] with [math] or [math] . Of particular interest to us is the case [math] . Our main results exhibit large collections of 4–manifolds that split in this manner.

  13. Symplectic homology and the Eilenberg–Steenrod axioms

    Cieliebak, Kai; Oancea, Alexandru
    We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg–Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer–Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology...

  14. The concordance invariant tau in link grid homology

    Cavallo, Alberto
    We introduce a generalization of the Ozsváth–Szabó [math] –invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus links and we also give an application to Legendrian link invariants in the standard contact [math] –sphere.

  15. Algebraic ending laminations and quasiconvexity

    Mj, Mahan; Rafi, Kasra
    We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence ¶ 1 H G Q 1 ¶ of hyperbolic groups. These laminations arise in different contexts: existence of Cannon–Thurston maps; closed geodesics exiting ends of manifolds; dual to actions on [math] –trees. ¶ We use the relationship between these laminations to prove quasiconvexity results for finitely generated infinite-index subgroups of [math] , the normal subgroup in the exact sequence above.

  16. The eta-inverted sphere over the rationals

    Wilson, Glen Matthew
    We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map [math] over fields of cohomological dimension at most [math] with characteristic different from [math] (this includes the [math] –adic fields [math] and the finite fields [math] of odd characteristic) and the field of rational numbers; the ring structure is also determined.

  17. The nonorientable $4$–genus for knots with $8$ or $9$ crossings

    Jabuka, Stanislav; Kelly, Tynan
    The nonorientable [math] –genus of a knot in the [math] –sphere is defined as the smallest first Betti number of any nonorientable surface smoothly and properly embedded in the [math] –ball with boundary the given knot. We compute the nonorientable [math] –genus for all knots with crossing number [math] or [math] . As applications we prove a conjecture of Murakami and Yasuhara and compute the clasp and slicing number of a  knot.

  18. Comparing $4$–manifolds in the pants complex via trisections

    Islambouli, Gabriel
    Given two smooth, oriented, closed [math] –manifolds, [math] and [math] , we construct two invariants, [math] and [math] , coming from distances in the pants complex and the dual curve complex, respectively. To do this, we adapt work of Johnson on Heegaard splittings of [math] –manifolds to the trisections of [math] –manifolds introduced by Gay and Kirby. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of “nearby” manifolds. This naturally leads to various graphs of [math] –manifolds coming from unbalanced trisections, and we briefly explore their properties.

  19. The relative lattice path operad

    Quesney, Alexandre
    We construct a set-theoretic coloured operad [math] that may be thought of as a combinatorial model for the Swiss cheese operad. This is the relative (or Swiss cheese) version of the lattice path operad constructed by Batanin and Berger. By adapting their condensation process we obtain a topological (resp.  chain) operad that we show to be weakly equivalent to the topological (resp.  chain) Swiss cheese operad.

  20. Compact Stein surfaces as branched covers with same branch sets

    Oba, Takahiro
    For each integer [math] , we construct a braided surface [math] in [math] and simple branched covers of [math] branched along [math] such that the covers have the same degrees and are mutually diffeomorphic, but Stein structures associated to the covers are mutually not homotopic. As a corollary, for each integer [math] , we also construct a transverse link [math] in the standard contact [math] –sphere and simple branched covers of [math] branched along [math] such that the covers have the same degrees and are mutually diffeomorphic, but contact manifolds associated to the covers are mutually not contactomorphic.

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