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We prove that for any compact orientable connected –manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open –disk removed admits a Lagrangian embedding into the standard symplectic –space. Moreover, the minimal Maslov number of the Lagrangian embedding is equal to .
We construct multiplicative twisted versions of differential cohomology theories for all highly structured ring spectra and twists. We prove existence and give a full classification of differential refinements of twists under mild assumptions. Various concrete examples are discussed and related to earlier approaches.
For every group , we define the set of hyperbolic structures on , denoted by , which consists of equivalence classes of (possibly infinite) generating sets of such that the corresponding Cayley graph is hyperbolic; two generating sets of are equivalent if the corresponding word metrics on are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded –actions on hyperbolic spaces. We are especially interested in the subset of acylindricallyhyperbolic structures on , ie hyperbolic structures corresponding to acylindrical actions. Elements of can be ordered in a natural way according to the amount of information they provide about the group . The main goal of this paper is to initiate the study of the posets and for various groups . We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of , and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.
Connected sums of lens spaces which smoothly bound a rational homology ball are classified by P Lisca. In the classification, there is a phenomenon that a connected sum of a pair of lens spaces appears in one of the typical cases of rational homology cobordisms. We consider smooth negative-definite cobordisms among several disjoint union of lens spaces and a rational homology –sphere to give a topological condition for the cobordism to admit the above “pairing” phenomenon. By using Donaldson theory, we show that if has a certain minimality condition concerning the Chern–Simons invariants of the boundary components, then any must have a counterpart in negative-definite cobordisms with a certain condition only on homology. In addition, we show an existence of a reducible flat connection through which the pair is related over the cobordism. As an application, we give a sufficient condition for a closed smooth negative-definite –orbifold with two isolated singular points whose neighborhoods are homeomorphic to the cones over lens spaces and to admit a finite uniformization.
We construct a sequence of smooth concordance invariants defined using truncated Heegaard Floer homology. The invariants generalize the concordance invariants of Ozsváth and Szabó and of Hom and Wu. We exhibit an example in which the gap between two consecutive elements in the sequence can be arbitrarily large. We also prove that the sequence contains more concordance information than , , , and .
We show that the semidirect product construction for –operads and the levelwise Borel construction for –cooperads are intertwined by the topological operadic bar construction. En route we give a generalization of the bar construction of M Ching from reduced to certain nonreduced topological operads.
We study the symplectic embedding capacity function for ellipsoids into dilates of polydisks as both and vary through . For , Frenkel and Müller showed that has an infinite staircase accumulating at , while for integer , Cristofaro-Gardiner, Frenkel and Schlenk found that no infinite staircase arises. We show that for arbitrary , the restriction of to is determined entirely by the obstructions from Frenkel and Müller’s work, leading on this interval to have a finite staircase with the number of steps tending to as . On the other hand, in contrast to the results of Cristofaro-Gardiner, Frenkel and Schlenk, for a certain doubly indexed sequence of irrational numbers we find that has an infinite staircase; these include both numbers that are arbitrarily large and numbers that are arbitrarily close to , with the corresponding accumulation points respectively arbitrarily large and arbitrarily close to .
We study the topological invariant reflecting the complexity of algorithms for autonomous robot motion. Here, stands for the configuration space of a system and is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in . We focus on the case when the space is aspherical; then the number depends only on the fundamental group and we denote it by . We prove that can be characterised as the smallest integer such that the canonical –equivariant map of classifying spaces
can be equivariantly deformed into the –dimensional skeleton of . The symbol denotes the classifying space for free actions and denotes the classifying space for actions with isotropy in the family of subgroups of which are conjugate to the diagonal subgroup. Using this result we show how one can estimate in terms of the equivariant Bredon cohomology theory. We prove that , where denotes the cohomological dimension of with respect to the family of subgroups . We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion-free hyperbolic groups as well as all torsion-free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher (2017) are exactly the classes having Bredon cohomology extensions with respect to the family .
The zero locus of a generic section of a vector bundle over a manifold defines a submanifold. A classical problem in geometry asks to realise a specified submanifold in this way. We study two cases: a point in a generalised flag manifold and the diagonal in the direct product of two copies of a generalised flag manifold. These cases are particularly interesting since they are related to ordinary and equivariant Schubert polynomials, respectively.
A –structure on a group was introduced by Bestvina in order to extend the notion of a group boundary beyond the realm of CAT(0) and hyperbolic groups. A refinement of this notion, introduced by Farrell and Lafont, includes a –equivariance requirement, and is known as an –structure. The general questions of which groups admit – or –structures remain open. Here we show that all Baumslag–Solitar groups admit –structures and all generalized Baumslag–Solitar groups admit –structures.
This is an exposition of homotopical results on the geometric realisation of semisimplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen’s Theorems A and B for nonunital topological categories; the effect on classifying spaces of changing the topology on the space of objects; the group-completion theorem.
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