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Suppose is a finite group and a prime, such that none of the prime divisors of are congruent to modulo . We prove an equivariant analogue of Adams’ result that . We use this to show that the –connected cover of , when completed at , splits up to homotopy as a product, where one of the factors of the splitting contains the image of the classical equivariant –homomorphism on equivariant homotopy groups.
We prove that the Seidel morphism of is naturally related to the Seidel morphisms of and , when these manifolds are monotone. We deduce that any homotopy class of loops of Hamiltonian diffeomorphisms of one component, with nontrivial image via Seidel’s morphism, leads to an injection of the fundamental group of the group of Hamiltonian diffeomorphisms of the other component into the fundamental group of the group of Hamiltonian diffeomorphisms of the product. This result was inspired by and extends results obtained by Pedroza [Int. Math. Res. Not. (2008) Art. ID rnn049].
The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of this graph for codimension one subgroups leads to splittings. We also reprove some results of Peter Kropholler on splittings of groups over malnormal subgroups and variants of them.
The purpose of this paper is to interpret polynomial invariants of strongly invertible links in terms of Khovanov homology theory. To a divide, that is a proper generic immersion of a finite number of copies of the unit interval and circles in a –disc, one can associate a strongly invertible link in the –sphere. This can be generalized to signed divides: divides with or sign assignment to each crossing point. Conversely, to any link that is strongly invertible for an involution , one can associate a signed divide. Two strongly invertible links that are isotopic through an isotopy respecting the involution are called strongly equivalent. Such isotopies give rise to moves on divides. In a previous paper [Topology 47 (2008) 316-350], the author finds an exhaustive list of moves that preserves strong equivalence, together with a polynomial invariant for these moves, giving therefore an invariant for strong equivalence of the associated strongly invertible links. We prove in this paper that this polynomial can be seen as the graded Euler characteristic of a graded complex of –vector spaces. Homology of such complexes is invariant for the moves on divides and so is invariant through strong equivalence of strongly invertible links.
We prove the Kauffman–Harary Conjecture, posed in 1999: given a reduced, alternating diagram of a knot with prime determinant , every nontrivial Fox –coloring of will assign different colors to different arcs.
Let be a compact, orientable, –irreducible –manifold and be a connected closed essential surface in with which cuts into and . In the present paper, we show the following theorem: Suppose that there is no essential surface with boundary in satisfying , . Then . As a consequence, we further show that if has a Heegaard splitting with distance , , then .
The main results follow from a new technique which is a stronger version of Schultens’ Lemma.
We prove that each injective simplicial map from the complex of arcs of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the arc complex is isomorphic to the quotient of the mapping class group of the surface by its center.
We provide a pure algebraic version of the first-named author’s dynamical characterization of the Conrad property for group orderings. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given by the third author in the Appendix.
We construct examples of , free-by-cyclic, hyperbolic groups which fiber in infinitely many ways over . The construction involves adding a specialized square 2–cell to a non-positively curved, squared 2–complex defined by labeled oriented graphs. The fundamental groups of the resulting complexes are , hyperbolic, free-by-cyclic and can be mapped onto in infinitely many ways.
The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3–manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson’s Q–theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.
Let be a closed essential surface in a hyperbolic –manifold with a toroidal cusp . The depth of in is the maximal distance from points of in to the boundary of . It will be shown that if is an essential pleated surface which is not coannular to the boundary torus of then the depth of in is bounded above by a constant depending only on the genus of . The result is used to show that an immersed closed essential surface in which is not coannular to the torus boundary components of will remain essential in the Dehn filling manifold after excluding curves from each torus boundary component of , where is a constant depending only on the genus of the surface.
We produce a nonpositively curved square complex containing exactly four squares. Its universal cover is isomorphic to the product of two –valent trees. The group is a lattice in but is not virtually a nontrivial product of free groups. There is no such example with fewer than four squares. The main ingredient in our analysis is that contains an “anti-torus” which is a certain aperiodically tiled plane.
We show that every negative definite configuration of symplectic surfaces in a symplectic –manifold has a strongly symplectically convex neighborhood. We use this to show that if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic –manifolds.
Using the combinatorial approach to Heegaard Floer homology, we obtain a relatively easy formula for computation of the groups , where is the three-manifold obtained by –surgery on a knot inside a homology sphere .
We develop topological methods for analyzing difference topology experiments involving –string tangles. Difference topology is a novel technique used to unveil the structure of stable protein-DNA complexes. We analyze such experiments for the Mu protein-DNA complex. We characterize the solutions to the corresponding tangle equations by certain knotted graphs. By investigating planarity conditions on these graphs we show that there is a unique biologically relevant solution. That is, we show there is a unique rational tangle solution, which is also the unique solution with small crossing number.
In terms of category theory, the Gromov homotopy principle for a set valued functor asserts that the functor can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor holds if the functor can be induced from a (co)homology functor.
We examine the bordism principle in the case of functors given by (co)bordism groups of maps with prescribed singularities. Our main result implies that if a family of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space such that for each smooth manifold the cobordism group of maps into with only –singularities is isomorphic to the group of homotopy classes of maps . The spaces are relatively simple, which makes explicit computations possible even in the case where the dimension of the source manifold is bigger than the dimension of the target manifold.
We use the uniformly finite homology developed by Block and Weinberger to study the geometry of the Cayley graph of Thompson’s group . In particular, a certain class of subgraph of is shown to be nonamenable (in the Følner sense). This shows that if the Cayley graph of is amenable, these subsets, which include every finitely generated submonoid of the positive monoid of , must necessarily have measure zero.
We call a closed, connected, orientable manifold in one of the categories TOP, PL or DIFF chiral if it does not admit an orientation-reversing automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly chiral if it does not admit a self-map of degree . We prove that there are strongly chiral, smooth manifolds in every oriented bordism class in every dimension . We also produce simply-connected, strongly chiral manifolds in every dimension . For every , we exhibit lens spaces with an orientation-reversing self-diffeomorphism of order but no self-map of degree of smaller order.
We show that the –theory construction of our paper [Adv. Math 205 (2006) 163-228], which preserves multiplicative structure, extends to a symmetric monoidal closed bicomplete source category, with the multiplicative structure still preserved. The source category of [op cit], whose objects are permutative categories, maps fully and faithfully to the new source category, whose objects are (based) multicategories.
We extend a result of Minsky to show that, for a map of a surface to a hyperbolic –manifold, which is –incompressible rel a geodesic link with a definite tube radius, the set of noncontractible simple loops with bounded length representatives is quasi-convex in the complex of curves of the surface. We also show how wide product regions can be used to find a geodesic link with a definite tube radius with respect to which a map is –incompressible.
A real Bott manifold is the total space of an iterated –bundle over a point, where each –bundle is the projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with –coefficients are isomorphic.
A real Bott manifold is a real toric manifold and admits a flat Riemannian metric invariant under the natural action of an elementary abelian 2–group. We also prove that the converse is true, namely a real toric manifold which admits a flat Riemannian metric invariant under the action of an elementary abelian 2–group is a real Bott manifold.