We develop the K-theory of a C∗–algebra Oλ which represents the leaf space of measured foliations studied by Novikov, Masur, Thurston and Veech. The K-theory construction is based on the coding of geodesic lines due to Koebe and Morse. This method allows to calculate the range of Elliott group (K0, K + 0 , [1]) of Oλ, to establish a condition of strict ergodicity of the interval exchange transf...

Incidence relations among the cells of a regular CW complex produce a 2category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching on (the cells of) such a complex corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized ...

Tilting perverse sheaves arise in geometric representation theory as natural bases for many categories. (See [1] or Section 3 below for definitions and some background results.) For example, for category O of a complex reductive group G, tilting modules correspond to tilting sheaves on the Schubert stratification S of the flag variety B. In this setting, there are many characterizations of tilt...

Morse theory relates the set of critical points of a smooth functional defined on a Hilbert manifold to the topology of the manifold itself. Morse himself gave the first application of his theory to Riemannian geometry (cf. [6, 11, 12]), proving two very nice and famous results. In order to recall them, consider a Riemannian manifold (M, 〈 · , · 〉x) with Riemannian structure 〈 · , · 〉x. A curve...

Morse theory describes the relationship between a function's critical points and the homotopy type of the function's domain. The theorems of Morse theory were developed speci cally for functions on a manifold. This work adapts these theorems for use with parameterized families of implicit surfaces in computer graphics. The result is a theoretical basis for the determination of the global topolo...

Let (M, f,G) be a manifold, a function and a Riemann metric on the manifold. Topologists would use these data in order to analyze the manifold by means of Morse theory, that is by studying the dynamical system ẋ = ±∇f . Many recent applications of physics to topology are based on another point of view suggested in E. Witten’s paper Supersymmetry and Morse theory J. Diff. Geom. (1982). Given the...

Spectral flow is a general formula or computing the Fredholm index of an operator d ds ` Apsq : LpR,Hq Ñ LpR,Hq for a family As of (almost) self-adjoint operators on a Hilbert space H. The simplest version of such an operator is one where H “ R. This appears in Morse theory when studying the spaces of gradient trajectories. Thus Morse theory is a natural setting to consider such operators, with...

Morse theory has been considered to be a powerful tool in its applications to computational topology, computer graphics and geometric modeling. Forman introduced a discrete version of it, which is purely combinatorial. This opens Morse theory applications to a much larger scope. The main objective of this work is to illustrate Forman’s theory. We intend to use some of Forman’s concepts to visua...

Morse theory provides a powerful framework to study the topology of a manifold from a function de ned on it, but discrete constructions have remained elusive due to the di culty of translating smooth concepts to the discrete setting. Consider the problem of approximating the Morse-Smale (MS) complex of a Morse function from a point cloud and an associated nearest neighbor graph (NNG). While fol...

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities” relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman’s Sphere Theorem. The main corollaries of our work are: ...