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

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

Investment & Strategy at Morse Cutting Tool

[Excerpt] Local #277 of the United Electrical, Radio and Machine Workers of America contracted with the ICA for a preliminary assessment of the long term viability of Morse Cutting Tools. The union had become alarmed by declining employment at Morse and by Morse management\u27s statements regarding the company\u27s inadequate profitability and shrinking market share. Of even greater concern was the threat that the conglomerate which owns Morse, Gulf+Western (G+W), might close the New Bedford plant. We were asked to examine Morse\u27s position in the cutting tool business with particular attention to the adequacy of G+W\u27s investment in plant and equipment and of Morse\u27s management strategy

Rigidity and gluing for Morse and Novikov complexes

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M,\omega)$ with $c_{1}|_{\pi_{2}(M)}=[\omega]|_{\pi_{2}(M)}=0$. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently $C^{0}$ close generic function/hamiltonian. The gluing result is a type of Mayer-Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.Comment: 46 pages, LATEX file with XYPIC diagrams, and one .EPS file. Final version, accepted for publication by the Journal of the European Mathematical Societ

C^0-topology in Morse theory

Let $f$ be a Morse function on a closed manifold $M$, and $v$ be a Riemannian gradient of $f$ satisfying the transversality condition. The classical construction (due to Morse, Smale, Thom, Witten), based on the counting of flow lines joining critical points of the function $f$ associates to these data the Morse complex $M_*(f,v)$. In the present paper we introduce a new class of vector fields ($f$-gradients) associated to a Morse function $f$. This class is wider than the class of Riemannian gradients and provides a natural framework for the study of the Morse complex. Our construction of the Morse complex does not use the counting of the flow lines, but rather the fundamental classes of the stable manifolds, and this allows to replace the transversality condition required in the classical setting by a weaker condition on the $f$-gradient (almost transversality condition) which is $C^0$-stable. We prove then that the Morse complex is stable with respect to $C^0$-small perturbations of the $f$-gradient, and study the functorial properties of the Morse complex. The last two sections of the paper are devoted to the properties of functoriality and $C^0$-stability for the Novikov complex $N_*(f,v)$ where $f$ is a circle-valued Morse map and $v$ is an almost transverse $f$-gradient.Comment: 22 pages, Latex file, one typo correcte

Morse Inequalities for Orbifold Cohomology

This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show that a generic function on an orbifold is Morse. In obtaining these results we develop for differentiable Deligne-Mumford stacks those tools of differential geometry and topology -- flows of vector fields, the strong topology -- that are essential to the development of Morse theory on manifolds