Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact email@example.com with any questions.
Let be a compact Lie group. We build a tower of –spectra over the suspension spectrum of the space of linear isometries from one –representation to another. The stable cofibres of the maps running down the tower are certain interesting Thom spaces. We conjecture that this tower provides an equivariant extension of Miller’s stable splitting of Stiefel manifolds. We provide a cohomological obstruction to the tower producing a splitting in most cases; however, this obstruction does not rule out a split tower in the case where the Miller splitting is possible. We claim that in this case we have a split tower which would then produce an equivariant version of the Miller splitting and prove this claim in certain special cases, though the general case remains a conjecture. To achieve these results we construct a variation of the functional calculus with useful homotopy-theoretic properties and explore the geometric links between certain equivariant Gysin maps and residue theory.
Let be the topological knot concordance group of knots under connected sum modulo slice knots. Cochran, Orr and Teichner defined a filtration:
The quotient is isomorphic to Levine’s algebraic concordance group; is the algebraically slice knots. The quotient contains all metabelian concordance obstructions.
Using chain complexes with a Poincaré duality structure, we define an abelian group , our second order algebraic knot concordance group. We define a group homomorphism which factors through , and we can extract the two stage Cochran–Orr–Teichner obstruction theory from our single stage obstruction group . Moreover there is a surjective homomorphism , and we show that the kernel of this homomorphism is nontrivial.
We discuss an infinite class of metabelian Von Neumann –invariants. Each one is a homomorphism from the monoid of knots to . In general they are not well defined on the concordance group. Nonetheless, we show that they pass to well defined homomorphisms from the subgroup of the concordance group generated by anisotropic knots. Thus, the computation of even one of these invariants can be used to conclude that a knot is of infinite order. We introduce a method to give a computable bound on these –invariants. Finally we compute this bound to get a new and explicit infinite set of twist knots which is linearly independent in the concordance group and whose every member is of algebraic order 2.
We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.
We give a direct interpretation of Neumann’s combinatorial formula for the Chern–Simons invariant of a 3–manifold with a representation in whose restriction to the boundary takes values in upper triangular matrices. Our construction does not involve group homology or Bloch group but is based on the construction of an explicit flat connection for each tetrahedron of a simplicial decomposition of the manifold.
Let be the field with elements, and let be a finite group. By exhibiting an –operad action on for a complete projective resolution of the trivial –module , we obtain power operations of Dyer–Lashof type on Tate cohomology . Our operations agree with the usual Steenrod operations on ordinary cohomology . We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups . We also show that the operations in negative degree are nontrivial.
As an application, we prove that at the prime these operations can be used to determine whether a Tate cohomology class is productive (in the sense of Carlson) or not.
We prove that every –local compact group is approximated by transporter systems over finite –groups. To do so, we use unstable Adams operations acting on a given –local compact group and study the structure of resulting fixed points.
The cohomology of the pure string motion group admits a natural action by the hyperoctahedral group . In recent work, Church and Farb conjectured that for each , the cohomology groups are uniformly representation stable; that is, the description of the decomposition of into irreducible –representations stabilizes for . We use a characterization of given by Jensen, McCammond and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group vanish for . We also prove that the subgroup of of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.
In [Hiroshima Math. J. 12 (1982) 611–626], Oka and the second author considered the cohomology of the second Morava stabilizer algebra to study nontriviality of the products of beta elements of the stable homotopy groups of spheres. In this paper, we use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres: , , and for with for a prime number .
We show that there are topologically slice links whose individual components are smoothly concordant to the unknot, but which are not smoothly concordant to any link with unknotted components. We also give generalizations in the topological category regarding components of prescribed Alexander polynomials. The main tools are covering link calculus, algebraic invariants of rational knot concordance theory, and the correction term of Heegaard Floer homology.
In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdös and Rényi model of a random graph and find precise threshold functions for hyperbolicity (or relative hyperbolicity). We also study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as , random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter is constant and satisfies .
The universal invariant of bottom tangles has a universality property for the colored Jones polynomial of links. A bottom tangle is called boundary if its components admit mutually disjoint Seifert surfaces. Habiro conjectured that the universal invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra of the Lie algebra . In the present paper, we prove an improved version of Habiro’s conjecture. As an application, we prove a divisibility property of the colored Jones polynomial of boundary links.
We distinguish the handlebody-knots and in the table, due to Ishii et al, of irreducible handlebody-knots up to six crossings. Furthermore, we construct two infinite families of handlebody-knots, each containing one of the pairs and , and show that any two handlebody-knots in each family have homeomorphic complements but they are not equivalent.
We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen –invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension . The basic properties of the –invariant all extend to the case of links; in particular, any orientable cobordism between links induces a map between their corresponding vector spaces which is filtered of degree . A corollary of this construction is that any component-preserving orientable cobordism from a –thin link to a link split into components must have genus at least . In particular, no quasi-alternating link is concordant to a split link.
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedded graph represents a link diagram. Furthermore, although a Tait graph describes a unique link diagram, the same embedded graph can represent many different link diagrams. One is then led to ask which embedded graphs represent link diagrams, and how link diagrams presented by the same embedded graphs are related to one another. Here we answer these questions by characterizing the class of embedded graphs that represent link diagrams, and then using this characterization to find a move that relates all of the link diagrams that are presented by the same set of embedded graphs.
We improve a bound of Borcherds on the virtual cohomological dimension of the nonreflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink’s result that the nonreflection part of a reflection centralizer is free. Namely, the nonreflection part of the normalizer of parabolic subgroup of type or is either free or has a free subgroup of index .
Under certain conditions of technical order, we show that closed connected Hamiltonian fibrations over symplectically uniruled manifolds are also symplectically uniruled. As a consequence, we partially extend to nontrivial Hamiltonian fibrations a result of Lu [Math. Res. Lett. 7 (2000) 383–387], stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurfaces of contact type. The proof of our result is based on the product formula for Gromov–Witten invariants of Hamiltonian fibrations derived by the author in [arXiv 0904.1492].
The protein recombinase can change the knot type of circular DNA. The action of a recombinase converting one knot into another knot is normally mathematically modeled by band surgery. Band surgeries on a –bridge knot yielding a –torus link are characterized. We apply this and other rational tangle surgery results to analyze Xer recombination on DNA catenanes using the tangle model for protein-bound DNA.
A group property made homotopical is a property of the corresponding classifying space. This train of thought can lead to a homotopical definition of normal maps between topological groups (or loop spaces).
In this paper we deal with such maps, called homotopy normal maps, which are topological group maps being “normal” in that they induce a compatible topological group structure on the homotopy quotient . We develop the notion of homotopy normality and its basic properties and show it is invariant under homotopy monoidal endofunctors of topological spaces, eg localizations and completions. In the course of characterizing normality, we define a notion of a homotopy action of a loop space on a space phrased in terms of Segal’s –fold delooping machine. Homotopy actions are “flexible” in the sense they are invariant under homotopy monoidal functors, but can also rigidify to (strict) group actions.
We study the augmentation quotients of the IA-automorphism group of a free group and a free metabelian group. First, for any group , we construct a lift of the –th Johnson homomorphism of the automorphism group of to the –th augmentation quotient of the IA-automorphism group of . Then we study the images of these homomorphisms for the case where is a free group and a free metabelian group. As a corollary, we detect a –free part in each of the augmentation quotients, which can not be detected by the abelianization of the IA-automorphism group.