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We study the structure of the formal groups associated to the Morava –theories of integral Eilenberg–Mac Lane spaces. The main result is that every formal group in the collection for a fixed enters in it together with its Serre dual, an analogue of a principal polarization on an abelian variety. We also identify the isogeny class of each of these formal groups over an algebraically closed field. These results are obtained with the help of the Dieudonné correspondence between bicommutative Hopf algebras and Dieudonné modules. We extend P Goerss’ results on the bilinear products of such Hopf algebras and corresponding Dieudonné modules.
Given a noncompact quasi-Fuchsian surface in a finite volume hyperbolic 3–manifold, we introduce a new invariant called the cusp thickness, that measures how far the surface is from being totally geodesic. We relate this new invariant to the width of a surface, which allows us to extend and generalize results known for totally geodesic surfaces. We also show that checkerboard surfaces provide examples of such surfaces in alternating knot complements and give examples of how the bounds apply to particular classes of knots. We then utilize the results to generate closed immersed essential surfaces.
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and nonterminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the virtual unknotting number of a knot, and show that any knot with nontrivial Jones polynomial has virtual unknotting number at least 2.
A Heegaard splitting of an open 3–manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard splittings. The main result is a classification of Heegaard splittings of those open 3–manifolds obtained by removing boundary components (not all of which are 2–spheres) from a compact 3–manifold. Also studied is the relationship between exhaustions and Heegaard splittings of eventually end-irreducible 3–manifolds. It is shown that Heegaard splittings of end-irreducible 3–manifolds are formed by amalgamating Heegaard splittings of boundary-irreducible compact submanifolds.
In this paper, we describe a canopolis (ie categorified planar algebra) formalism for Khovanov and Rozansky’s link homology theory. We show how this allows us to organize simplifications in the matrix factorizations appearing in their theory. In particular, it will put the equivalence of the original definition of Khovanov–Rozansky homology and the definition using Soergel bimodules in a more general context, allow us to give a new proof of the invariance of triply graded homology and give a new analysis of the behavior of triply graded homology under the Reidemeister IIb move.
We study contact structures compatible with genus one open book decompositions with one boundary component. Any monodromy for such an open book can be written as a product of Dehn twists around dual nonseparating curves in the once-punctured torus. Given such a product, we supply an algorithm to determine whether the corresponding contact structure is tight or overtwisted for all but a small family of reducible monodromies. We rely on Ozsváth–Szabó Heegaard Floer homology in our construction and, in particular, we completely identify the –spaces with genus one, one boundary component, pseudo-Anosov open book decompositions. Lastly, we reveal a new infinite family of hyperbolic three-manifolds with no co-orientable taut foliations, extending the family discovered by Roberts, Shareshian, and Stein in [J. Amer. Math. Soc. 16 (2003) 639–679]
Consider the space of pairwise commuting –tuples of elements in a compact Lie group . This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of , which allows us to derive formulas for its ordinary and equivariant cohomology in terms of the Lie algebra of a maximal torus in and the action of the Weyl group. This is an application of a general theorem concerning –spaces for which every element is fixed by a maximal torus.
Given a 3–manifold the second author defined functions , generalizing McMullen’s Alexander norm, which give lower bounds on the Thurston norm. We reformulate these invariants in terms of Reidemeister torsion over a non-commutative multivariable Laurent polynomial ring. This allows us to show that these functions are semi-norms.
We give a counterexample to a conjecture of D H Gottlieb and prove a strengthened version of it.
The conjecture says that a map from a finite CW–complex to an aspherical CW–complex with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of is trivial.
As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from to are contractible.
We use –Betti numbers and homological algebra over von Neumann algebras to prove the modified conjecture.
We study connective –theory for the classifying space of a finite cyclic –group and compute the –cohomology groups completely. We also compute the –homology groups, with the exception of a finite range of dimensions.
In this paper we use the singularity method of Koschorke [Lecture Notes in Math. 847 (1981)] to study the question of how many different nonstable homotopy classes of monomorphisms of vector bundles lie in a stable class and the percentage of stable monomorphisms which are not homotopic to stabilized nonstable monomorphisms. Particular attention is paid to tangent vector fields. This work complements some results of Koschorke [Lecture Notes in Math. 1350, 1988, Topology Appl. 75 (1997)], Libardi–Rossini [Proc. of the XI Brazil. Top. Meeting 2000] and Libardi–do Nascimento–Rossini [Revesita de Mátematica e Estatística 21 (2003)].
We construct a new family, indexed by odd integers , of –dimensional quantum field theories that we call quantum hyperbolic field theories (QHFT), and we study its main structural properties. The QHFT are defined for marked –bordisms supported by compact oriented –manifolds with a properly embedded framed tangle and an arbitrary –character of (covering, for example, the case of hyperbolic cone manifolds). The marking of QHFT bordisms includes a specific set of parameters for the space of pleated hyperbolic structures on punctured surfaces. Each QHFT associates in a constructive way to any triple with marked boundary components a tensor built on the matrix dilogarithms, which is holomorphic in the boundary parameters. When the QHFT tensors are scalar-valued, and coincide with the Cheeger–Chern–Simons invariants of –characters on closed manifolds or cusped hyperbolic manifolds. We establish surgery formulas for QHFT partitions functions and describe their relations with the quantumhyperbolic invariants of Baseilhac and Benedetti (either defined for unframed links in closed manifolds and characters trivial at the link meridians, or cusped hyperbolic –manifolds). For every –character of a punctured surface, we produce new families of conjugacy classes of “moderately projective" representations of the mapping class groups.
We study the relationship between two sets and of Coxeter generators of a finitely generated Coxeter group by proving a series of theorems that identify common features of and . We describe an algorithm for constructing from any set of Coxeter generators of a set of Coxeter generators of maximum rank for .
A subset of is called complete if any two elements of generate a finite group. We prove that if and have maximum rank, then there is a bijection between the complete subsets of and the complete subsets of so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of and have the same multiset of entries.
Suppose is a closed irreducible orientable 3–manifold, is a knot in , and are bridge surfaces for and is not removable with respect to . We show that either is equivalent to or . If is not a 2–bridge knot, then the result holds even if is removable with respect to . As a corollary we show that if a knot in has high distance with respect to some bridge sphere and low bridge number, then the knot has a unique minimal bridge position.
In this paper we give a sheaf theory interpretation of the twisted cohomology of manifolds. To this end we develop a sheaf theory on smooth stacks. The derived push-forward of the constant sheaf with value along the structure map of a gerbe over a smooth manifold is an object of the derived category of sheaves on . Our main result shows that it is isomorphic in this derived category to a sheaf of twisted de Rham complexes.
Let be a geodesic metric space with uniformly generated. If has asymptotic dimension one then is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriented surface with genus and one boundary component is at least two.
We prove that , . Here is a commutative algebra over the prime field of characteristic and is considered as a bimodule, where the left multiplication is the usual one, while the right multiplication is given via Frobenius endomorphism and denotes the Hochschild homology over . This result has implications in Mac Lane homology theory. Among other results, we prove that , provided is an algebra over a field of characteristic and is a strict homogeneous polynomial functor of degree with .
A well-known formula of R J Herbert’s relates the various homology classes represented by the self-intersection immersions of a self-transverse immersion. We prove a geometrical version of Herbert’s formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert’s formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise the formula to other homology theories. The proof is based on Herbert’s but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions.
We give concrete constructions of discrete and faithful representations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we find a general criterion for when discrete and faithful representations exist, and show that the criterion is satisfied in particular cases. There are direct applications towards constructing representations of surface groups into higher-rank Lie groups, and, in particular, into lattices in higher-rank Lie groups.
We give another proof of a theorem of Scharlemann and Tomova and of a theorem of Hartshorn. The two theorems together say the following. Let be a compact orientable irreducible 3–manifold and a Heegaard surface of . Suppose is either an incompressible surface or a strongly irreducible Heegaard surface in . Then either the Hempel distance or is isotopic to . This theorem can be naturally extended to bicompressible but weakly incompressible surfaces.