Functoriality and duality in Morse-Conley-Floer homology

In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known~\cite{abbondandoloschwarz, aizenbudzapolski,audindamian, kronheimermrowka, schwarz}, but the proofs use isomorphisms to other homology theories. We give direct proofs by analyzing appropriate moduli spaces. The notions of isolating map and flow map allows the results to generalize to local Morse homology and Morse-Conley-Floer homology. We prove Poincar\'e type duality statements for local Morse homology and Morse-Conley-Floer homology.Comment: To appear in the Journal of Fixed Point theory and its Application

Morse-Conley-Floer Homology

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figure

Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation

Continuation of algebraic structures in families of dynamical systems is described using category theory, sheaves, and lattice algebras. Well-known concepts in dynamics, such as attractors or invariant sets, are formulated as functors on appropriate categories of dynamical systems mapping to categories of lattices, posets, rings or abelian groups. Sheaves are constructed from such functors, which encode data about the continuation of structure as system parameters vary. Similarly, morphisms for the sheaves in question arise from natural transformations. This framework is applied to a variety of lattice algebras and ring structures associated to dynamical systems, whose algebraic properties carry over to their respective sheaves. Furthermore, the cohomology of these sheaves are algebraic invariants which contain information about bifurcations of the parametrized systems

The Drosophila homologue of Rootletin is required for mechanosensory function and ciliary rootlet formation in chordotonal sensory neurons

BACKGROUND: In vertebrates, rootletin is the major structural component of the ciliary rootlet and is also part of the tether linking the centrioles of the centrosome. Various functions have been ascribed to the rootlet, including maintenance of ciliary integrity through anchoring and facilitation of transport to the cilium or at the base of the cilium. In Drosophila, Rootletin function has not been explored. RESULTS: In the Drosophila embryo, Rootletin is expressed exclusively in cell lineages of type I sensory neurons, the only somatic cells bearing a cilium. Expression is strongest in mechanosensory chordotonal neurons. Knock-down of Rootletin results in loss of ciliary rootlet in these neurons and severe disruption of their sensory function. However, the sensory cilium appears largely normal in structure and in localisation of proteins suggesting no strong defect in ciliogenesis. No evidence was found for a defect in cell division. CONCLUSIONS: The role of Rootletin as a component of the ciliary rootlet is conserved in Drosophila. In contrast, lack of a general role in cell division is consistent with lack of centriole tethering during the centrosome cycle in Drosophila. Although our evidence is consistent with an anchoring role for the rootlet, severe loss of mechanosensory function of chordotonal (Ch) neurons upon Rootletin knock-down may suggest a direct role for the rootlet in the mechanotransduction mechanism itself