We consider symplectic manifolds with Hamiltonian torus actions which are “almost but not quite completely integrable”: the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are “centered” (see below) and the moment map is proper. In particular, this classifies the moment map preimages of all sufficientl...

Given a (not necessarily discrete) proper metric space M with bounded geometry, we define a groupoid G(M). We show that the coarse Baum–Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M) is an isomorphism, is hereditary by taking closed subspaces.

We address the problem of computing the fundamental group of a symplectic S-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian S-action. Several examples are presented to illustrate our main results.

We address the problem of computing the fundamental group of a symplectic S1-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian S1-action. Several examples are presented to illustrate our main results.

This paper gives three noncompact variants of Lefschetz–Nielsen fixed-point theory parallel to developments that have occurred in surgery theory. Thus, we study when a proper map can be properly homotoped, or boundedly homotoped, or even homotoped in a C bounded fashion to a fixedpoint free map. As an example, the universal cover of a compact aspherical manifold always has a fixed-point free se...

A model of a topologically organized neural network of a Hopfield type with nonlinear analog neurons is shown to be very effective for path planning and obstacle avoidance. This deterministic system can rapidly provide a proper path, from any arbitrary start position to any target position, avoiding both static and moving obstacles of arbitrary shape. The model assumes that an (external) input ...

Let T × M → M be a Hamiltonian torus action on a connected symplectic manifold M for which the associated moment map Φ : M → t∗ is proper as a map into a convex open set ρ ⊆ t∗. We consider a closed submanifold Q of M and show that under certain local conditions on Q one has Φ(Q) = Φ(M). We apply this result in the special case that Q arises as the fixed point set of some involution σ on M whic...

We define a moment map for a torus action on a smooth manifold without a two-form. Cobordisms of such structures are meaningful even if the manifolds are non compact, as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant the normal bundle of its fixed point set. Two formulas follow easily: Guillemin’s topological version of the...

We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph C∗-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an AF algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph w...

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a ‘system of open neighborhoods at infinity’ while an exterior map is a continuous map which is ‘continuous at infinity’. The category of spaces and proper maps is a subcategory of the category of exterior spaces. In this paper we show that the category of e...