Estimating the rational LS-category of elliptic spaces

An elliptic space is one whose rational homotopy and rational cohomology are both finite dimensional. We prove, for Toomer's invariant, two improvements of the estimate of the Mapping theorem relying on data from the homotopy Lie algebra of the space. In particular, we show that if is elliptic,

where is the rational homotopy Lie algebra of and its centre.

Several interesting examples are presented to illustrate our results.

     

LS-category of compact Hausdorff foliations

The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author, is an invariant of foliated homotopy type with values in . A foliation with all leaves compact and Hausdorff leaf space is called compact Hausdorff. The transverse saturated category of a compact Hausdorff foliation is always finite.

In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category in terms of the geometry of and the Epstein filtration of the exceptional set . The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that

We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.

     

The equivariant LS-category of polar actions

We will provide a lower bound for arbitrary proper actions in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant LS-category of its generalized Weyl group. As an application we reprove a theorem of Singhof that determines the classical Lusternik-Schnirelmann category for U(n) and SU(n).     

The rational LS-category of -trivial fibrations

We provide new upper and lower bounds for the rational LS-category of a rational fibration of simply connected spaces that depend on a measure of the triviality of which is strictly finer than the vanishing of the higher holonomy actions. In particular, we prove that if is -trivial for some and enjoys Poincaré duality, then

     

The six Grothendieck operations on o-minimal sheaves

In this note, we report on our work on the formalism of the Grothendieck six operations on o-minimal sheaves. As an application to the theory of definable groups, we see that the cohomology of a definably compact group with coefficients in a field is a connected, bounded, Hopf algebra of finite type.     

Weighted o-minimal hybrid systems

We consider weighted o-minimal hybrid systems, which extend classical o-minimal hybrid systems with cost functions. These cost functions are “observer variables” which increase while the system evolves but do not constrain the behaviour of the system. In this paper, we prove two main results: (i) optimal o-minimal hybrid games are decidable; (ii) the model-checking of WCTL, an extension of CTL which can constrain the cost variables, is decidable over that model. This has to be compared with the same problems in the framework of timed automata where both problems are undecidable in general, while they are decidable for the restricted class of one-clock timed automata.     

Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which admits local definable cell decomposition.     

Definable functions continuous on curves in o-minimal structures

We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve has a definable extension to a closed set. This situation is translated into a question about types: What are the conditions on an n-type such that, for any bounded definable function, the germ of the function on the type has a definable continuous global extension? Certain categories of definable types have this property, and we give the precise conditions that are equivalent to existence of the global extension.     

A trichotomy theorem for o-minimal structures

Let M = M, <, ... be alinearly ordered structure. We defineM to be o-minimal if every definable subset of M is a finiteunion of intervals. Classical examples are ordered divisibleabelian groups and real closed fields. We prove a trichotomytheorem for the structure that an arbitraryo-minimal M can induceon a neighbourhood of any a in M. Roughly said, one of the followingholds:

(i) a is trivial (technical term), or

(ii) a has aconvex neighbourhood on which M induces the structureof anordered vector space, or

(iii) a is contained in an open intervalon which M inducesthe structure of an expansion of a real closedfield.

The proof uses ‘geometric calculus’ whichallows one to recover a differentiable structure by purely geometricmethods. 1991 Mathematics Subject Classification: primary 03C45;secondary 03C52, 12J15, 14P10.     

Wild theories with o-minimal open core

Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let $T^{\prime }$ be a consistent theory. Then there is a complete theory $T^{?}$ extending T such that T is an open core of $T^{?}$, but every model of $T^{?}$ interprets a model of $T^{\prime }$. If $T^{\prime }$ is NIP, $T^{?}$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.     

Definably simple groups in o-minimal structures

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.

     

An o-minimal structure without mild parameterization

We prove, by explicit construction, that not all sets definable in polynomially bounded o-minimal structures have mild parameterization. Our methods do not depend on the bounds particular to the definition of mildness and therefore our construction is also valid for a generalized form of parameterization, which we call G-mild. Moreover, we present a cell decomposition result for certain o-minimal structures which may be of independent interest. This allows us to show how our construction can produce polynomially bounded, model complete expansions of the real ordered field which, in addition to lacking G-mild parameterization, nonetheless still have analytic cell decomposition.     

Cochain model for thickenings and its application to rational LS-category

A thickening of a finite CW-complex X is by definition a compact manifold M of the same simple homotopy type as X. We give a model for the cochain complex of the boundary of that manifold, C ^*(δM), as a module over the cochain algebra C ^*(X). We also show how to construct an algebraic model of the rational homotopy type of δC ^*(M) from a model of X. Using this rational model, we prove a new formula for the rational Lusternik–Schnirelmann category of X. Received: 24 September 1999