Let G be a Lie groupoid over M such that the target-source map from G to M × M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction G|U of G to some neighborhood U of O in M is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid G|O on the normal bundle NO. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of finite type.

Abstract. The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing major results on paracompactness (Stone Theorem and Tamano Theorem), a result which serves as a connection to Ascoli Theorem. A new calculus of partitions of unity is introduced with applications to dimension theory and metric simplicial complexes. The geometric interpretation of this calculus is the barycentric subdivision of simplicial complexes. Also, joins of partitions of unity are often used; they are an algebraic version of joins of simplicial complexes. §0. Introduction. The explosion of research in topology makes it imperative that one ought to look at its foundations and decide what topics should be included in its mainstream. One of the primary criteria is interconnectedness and potential applications to many branches of topology and mathematics. The author believes that the gems of basic topology are: normality,

. We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in R 3 , and the problem of continuing two periodic orbits which are not homotopic as loops. Classifications: Primary 34C35; Secondary 58F25, 34A34 Keywords: isolated invariant set, Conley index, continuation, fiberwise pointed space 1. Introduction The Conley index is a useful tool in the study of flows. To obtain the index of an isolated invariant set S, one takes an index pair (P 1 ; P 2 ) for S and collapses P 2 to a point. (Recall that P 1 is an isolating neighborhood for S, and P 2 is an subset of P 1 such that each trajectory which leaves P 1 , passes through P 2 .) The homotopy type of the resulting quotient space is independent of the choice of index pair. Indeed, give...

Abstract. Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of S 1-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which have clear functorial properties with respect to diffeomorphism of 4-manifolds. For 4-manifolds with b1 = 0 and b+> 1 our invariants are equivalent to the Bauer-Furuta invariants, but they are finer in general. We study fundamental properties of the new invariants in a very general framework. In particular we prove a universal cohomotopy invariant jump formula and a multiplicative property. The formalism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromov-Witten invariants.

this paper we give a homotopy theoretic proof of Williams's metastable Poincar'e embedding theorem [Wi1], which was originally proven geometrically.

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We prove that every n-dimensional permutive cellular automaton is conjugate to a one-sided shift with compact set of states. This is a generalization of a theorem of R. Gilman.

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

This work is dedicated to mobile robot navigation. It presents a solution to the navigation problem using a simple world model based on a new approach to behaviours' mapping. This new theory of behaviours' mapping proposes a general solution not only to mobile robots' navigation but to the control of a wide range of autonomous (intelligent) behaviour based systems. Based on the phenomena of bifurcation and non-continuity of a behaviour or a group of behaviours that consist an autonomous behaviour based system, the new behaviours' space mapping theory enables direct mapping of the system's existing space to a graph. The latter mapping creates a world model that is made exclusively in terms of behaviours. We believe that an intelligent system require a world model to plan and to predict the result of its actions. We also believe that behaviours' based systems cope more successfully with the dynamics and changes of the real world. Combining the two poses a real problem in the creation of ...

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