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We present a new method for computing fundamental groups of curve complements using a variation of the Zariski–van Kampen method on general ruled surfaces. As an application we give an alternative (computation-free) proof for the fundamental group of generic –torus curves.
We investigate Legendrian graphs in . We extend the Thurston–Bennequin number and the rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with and if and only if it does not contain as a minor. We show that the pair does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. When we restrict to planar spatial graphs, a pair determines two Legendrian isotopy classes of the lollipop graph and a pair determines four Legendrian isotopy classes of the handcuff graph.
We provide a description of the structure of the set of homomorphisms from a finitely generated group to any torsion-free (–dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jørgensen–Thurston Theorem in hyperbolic geometry.
Let be a finite group and be a family of subgroups of which is closed under conjugation and taking subgroups. Let be a –CW–complex whose isotropy subgroups are in and let be a compatible family of –spaces. A –fibration over with the fiber type is a –equivariant fibration where is –homotopy equivalent to for each . In this paper, we develop an obstruction theory for constructing –fibrations with the fiber type over a given –CW–complex . Constructing –fibrations with a prescribed fiber type is an important step in the construction of free –actions on finite CW–complexes which are homotopy equivalent to a product of spheres.
We show exponential growth of torsion numbers for links whose first nonzero Alexander polynomial has positive logarithmic Mahler measure. This extends a theorem of Silver and Williams to the case of a null first Alexander polynomial and provides a partial solution for a conjecture of theirs.
We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava –theory at height .
We construct a free and transitive action of the group of bilinear forms on the set of –products on , a regular quotient of an even –ring spectrum with . We show that this action induces a free and transitive action of the group of quadratic forms on the set of equivalence classes of –products on . The characteristic bilinear form of introduced by the authors in a previous paper is the natural obstruction to commutativity of . We discuss the examples of the Morava –theories and the –periodic Morava –theories .
A –dimensional –connected closed oriented manifold smoothly embedded in the sphere is called a –link. We introduce the notion of exact links, which admit Seifert surfaces with good homological conditions. We prove that for , two exact –links are cobordant if they have such Seifert surfaces with algebraically cobordant Seifert forms. In particular, two fibered –links are cobordant if and only if their Seifert forms with respect to their fibers are algebraically cobordant. With this broad class of exact links, we thus clarify the results of Blanlœil [Ann. Fac. Sci. Toulouse Math. 7 (1998) 185–205] concerning cobordisms of odd dimensional nonspherical links.
It is well-known that a point in the (unprojectivized) Culler–Vogtmann Outer space is uniquely determined by its translation length function . A subset of a free group is called spectrallyrigid if, whenever are such that for every then in . By contrast to the similar questions for the Teichmüller space, it is known that for there does not exist a finite spectrally rigid subset of .
In this paper we prove that for if is a subgroup that projects to a nontrivial normal subgroup in then the –orbit of an arbitrary nontrivial element is spectrally rigid. We also establish a similar statement for , provided that is not conjugate to a power of .
First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)operad of CW–complexes whose constituent spaces form a homotopy associative version of the cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyclic version of Deligne’s conjecture. This chain model contains the minimal operad of Kontsevich and Soibelman as a suboperad and restriction of the action to this suboperad recovers the results of Kontsevich and Soibelman [Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht (2000) 255–307] and Kaufmann and Schwell [Adv. Math. 223 (2010) 2166–2199] in the unframed case. Additionally this proof recovers the work of Kaufmann in the case of a strict Frobenius algebra. We then extend our results to the context of cyclic categories, with an eye toward the homotopy BV structure present on the Hochschild cochains of the Fukaya category of a suitable symplectic manifold.
We consider the outer automorphism group of the right-angled Artin group of a random graph on vertices in the Erdős–Rényi model. We show that the functions and bound the range of edge probability functions for which is finite: if the probability of an edge in is strictly between these functions as grows, then asymptotically is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically is almost surely infinite. This sharpens a result of Ruth Charney and Michael Farber.
We study algebraic structures of certain submonoids of the monoid of homology cylinders over a surface and the homology cobordism groups, using Reidemeister torsion with non-commutative coefficients. The submonoids consist of ones whose natural inclusion maps from the boundary surfaces induce isomorphisms on higher solvable quotients of the fundamental groups. We show that for a surface whose first Betti number is positive, the homology cobordism groups are other enlargements of the mapping class group of the surface than that of ordinary homology cylinders. Furthermore we show that for a surface with boundary whose first Betti number is positive, the submonoids consisting of irreducible ones as –manifolds trivially acting on the solvable quotients of the surface group are not finitely generated.
We determine a set of generators for the Brunnian braids on a general surface for or . For the case or , a set of generators for the Brunnian braids on is given by our generating set together with the homotopy groups of a –sphere.
We prove fibrewise versions of classical theorems of Hopf and Leray–Samelson. Our results imply the fibrewise H-triviality after rationalization of a certain class of fibrewise H-spaces. They apply, in particular, to universal adjoint bundles. From this, we may retrieve a result of Crabb and Sutherland [Proc. London Math. Soc. 81 (2000) 747–768], which is used there as a crucial step in establishing their main finiteness result.
A few examples of –groups are presented whose Morava K–theory is determined by representation theory. By contrast, a –primary example shows that in general relations arising from representation theory do not suffice to calculate the Chern subring of .
Given a group acting on a graph quasi-isometric to a tree, we give sufficient conditions for a pseudocharacter to be bushy. We relate this with the conditions studied by Bestvina and Fujiwara on their work on bounded cohomology and obtain some results on the space of pseudocharacters.
Higher Dimensional Automata (HDA) are topological models for the study of concurrency phenomena. The state space for an HDA is given as a pre-cubical complex in which a set of directed paths (d-paths) is singled out. The aim of this paper is to describe a general method that determines the space of directed paths with given end points in a pre-cubical complex as the nerve of a particular category.
The paper generalizes the results from Raussen [Algebr. Geom. Topol. 10 (2010) 1683–1714; Appl. Algebra Engrg. Comm. Comput. 23 (2012) 59–84] in which we had to assume that the HDA in question arises from a semaphore model. In particular, important for applications, it allows for models in which directed loops occur in the processes involved.
Generalizing Block and Weinberger’s characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of on its Stone–Čech compactification we obtain a homological characterization of exactness of the group.
We prove that the Todd genus of a compact complex manifold of complex dimension with vanishing odd degree cohomology is one if the automorphism group of contains a compact –dimensional torus as a subgroup. This implies that if a quasitoric manifold admits an invariant complex structure, then it is equivariantly homeomorphic to a compact smooth toric variety, which gives a negative answer to a problem posed by Buchstaber and Panov.
We give a new upper bound for Farber’s topological complexity for rational spaces in terms of Sullivan models. We use it to determine the topological complexity in some new cases, and to prove a Ganea-type formula in these and other cases.
In analogy with the vector bundle theory we define universal and strongly universal Lefschetz fibrations over bounded surfaces. After giving a characterization of these fibrations we construct very special strongly universal Lefschetz fibrations when the fiber is the torus or an orientable surface with connected boundary and the base surface is the disk. As a by-product we also get some immersion results for –dimensional –handlebodies.
We consider the extension of classical –dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul and we identify the dual dg operad governing –algebras with involution in terms of Möbius graphs which are a generalisation of ribbon graphs. We then generalise open topological conformal field theories to open Klein topological conformal field theories and give a generators and relations description of the open KTCFT operad. We deduce an analogue of the ribbon graph decomposition of the moduli spaces of Riemann surfaces: a Möbius graph decomposition of the moduli spaces of Klein surfaces (real algebraic curves). The Möbius graph complex then computes the homology of these moduli spaces. We also obtain a different graph complex computing the homology of the moduli spaces of admissible stable symmetric Riemann surfaces which are partial compactifications of the moduli spaces of Klein surfaces.