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We prove a generalization of Bennequin’s inequality for Legendrian knots in a 3-dimensional contact manifold , under the assumption that is the boundary of a 4-dimensional manifold and the version of Seiberg-Witten invariants introduced by Kronheimer and Mrowka [Invent. Math. 130 (1997) 209–255] is nonvanishing. The proof requires an excision result for Seiberg-Witten moduli spaces; then the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside .
We give a new proof of a theorem of Kleiner–Leeb: that any quasi-isometrically embedded Euclidean space in a product of symmetric spaces and Euclidean buildings is contained in a metric neighborhood of finitely many flats, as long as the rank of the Euclidean space is not less than the rank of the target. A bound on the size of the neighborhood and on the number of flats is determined by the size of the quasi-isometry constants.
Without using asymptotic cones, our proof focuses on the intrinsic geometry of symmetric spaces and Euclidean buildings by extending the proof of Eskin–Farb’s quasiflat with holes theorem for symmetric spaces with no Euclidean factors.
We construct an explicit semifree model for the fiber join of two fibrations and from semifree models of and . Using this model, we introduce a lower bound of the sectional category of a fibration which can be calculated from any Sullivan model of and which is closer to the sectional category of than the classical cohomological lower bound given by the nilpotency of the kernel of . In the special case of the evaluation fibration we obtain a computable lower bound of Farber’s topological complexity . We show that the difference between this lower bound and the classical cohomological lower bound can be arbitrarily large.
Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots with 11 crossings.
Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley’s criterion does not apply.
We show that if two 3–manifolds with toroidal boundary are glued via a “sufficiently complicated" map then every Heegaard splitting of the resulting 3–manifold is weakly reducible. Additionally, suppose is a manifold obtained by gluing and , two connected small manifolds with incompressible boundary, along a closed surface . Then the following inequality on genera is obtained:
Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.
The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in this paper. It is shown that the assignment is functorial for fusion-preserving homomorphisms in a way which extends the assignment of stable –completed classifying spaces to finite groups, and admits a transfer theory analogous to that for finite groups. Furthermore the group of homotopy classes of maps between classifying spectra is described, and in particular it is shown that a fusion system can be reconstructed from its classifying spectrum regarded as an object under the stable classifying space of the underlying –group.
In this paper, we examine the “derived completion” of the representation ring of a pro- group with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg–MacLane spectrum , and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor from groups to ring spectra, and show that the map becomes an equivalence after completion when is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the –adic Heisenberg group.
We provide a lower bound for the coherence of the homotopy commutativity of the Brown–Peterson spectrum, , at a given prime and prove that it is at least –homotopy commutative. We give a proof based on Dyer–Lashof operations that cannot be a Thom spectrum associated to –fold loop maps to for at and at odd primes. Other examples where we obtain estimates for coherence are the Johnson–Wilson spectra, localized away from the maximal ideal and unlocalized. We close with a negative result on Morava-–theory.
Let be the space of base-point-preserving maps from a connected finite CW complex to a connected space . Consider a CW complex of the form and a space whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map is greater than the Whitehead length of , then has the rational homotopy type of the product space . This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex are greater than and the connectivity of is greater than or equal to , then the mapping space can be decomposed rationally as the product of iterated loop spaces.
The usual construction of link invariants from quantum groups applied to the superalgebra is shown to be trivial. One can modify this construction to get a two variable invariant. Unusually, this invariant is additive with respect to connected sum or disjoint union. This invariant contains an infinity of Vassiliev invariants that are not seen by the quantum invariants coming from Lie algebras (so neither by the colored HOMFLY-PT nor by the colored Kauffman polynomials).
We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3–manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if has an irreducible, horizontal Heegaard splitting, has a base orbifold of positive genus, and is not a circle bundle. This characterizes precisely which Seifert fibered spaces satisfy the converse of Waldhausen’s conjecture.
For a knot in we construct according to Casson—or more precisely taking into account Lin and Heusener’s further works—a volume form on the –representation space of the group of . We prove that this volume form is a topological knot invariant and explore some of its properties.
In this work we construct Calabi quasi-morphisms on the universal cover of the group of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of .
Let be a complete finite-volume hyperbolic 3–manifold with compact non-empty geodesic boundary and toric cusps, and let be a geometric partially truncated triangulation of . We show that the variety of solutions of consistency equations for is a smooth manifold or real dimension near the point representing the unique complete structure on . As a consequence, the relation between deformations of triangulations and deformations of representations is completely understood, at least in a neighbourhood of the complete structure. This allows us to prove, for example, that small deformations of the complete triangulation affect the compact tetrahedra and the hyperbolic structure on the geodesic boundary only at the second order.
We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima.
We consider properties of the total curvature functional on the space of 2–sphere immersions into 3–space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most and we establish a non-injectivity result for local minima.