John Oprea

Lusternik-Schnirelmann Category

Introduction to LS-category Lower bounds for LS-category Upper bounds for category Localization and category Rational homotopy and category Hopf invariants Category and critical points Category and symplectic topology Examples, computations and extensions Topology and analysis Basic homotopy Bibliography Index.

Cohomologically symplectic spaces: toral actions and the Gottlieb group

Aspects of symplectic geometry are explored from a homotopical viewpoint. In particular, the question of whether or not a given toral action is Hamiltonian is shown to be independent of geometry. Rather, a new homotopical obstruction is described which detects when an action is Hamiltonian. This new entity, the Ae,-invariant, allows many results of symplectic geometry to be generalized to manifolds which are only cohomologically symplectic in the sense that there is a degree 2 cohomology class which cups to a top class. Furthermore, new results in symplectic geometry also arise from this homotopical approach.

Symplectic manifolds and formality

Abstract We study some questions about symplectic manifolds, using techniques of rational homotopy theory. Our questions and results focus around formality properties of symplectic manifolds. We assume the presence of a symplectic structure on a manifold, and establish extra conditions sufficient to imply formality. The conditions are phrased in terms of the minimal model. In addition we study the question of whether or not a manifold can admit a Kahler structure. We use our results to give examples of non-aspherical symplectic manifolds that do not admit a Kahler structure.

On the Lusternik–Schnirelmann category of symplectic manifolds and the Arnold conjecture

In [Arn, Appendix 9] Arnold proposed a beautiful conjecture concerning the relation between the number of fixed points of certain (i.e., exact or Hamiltonian) selfdiffeomorphisms of a closed symplectic manifold (M,ω) and the minimum number of critical points of any smooth (= C∞) function on M . The first author succeeded in proving this form of the Arnold conjecture [R2] under the hypothesis that ω and c1 vanish on all spherical homology classes and that there is equality between the Lusternik–Schnirelmann category of M and the dimension of M . In this paper, we use a fundamental property of category weight to show that, for any closed symplectic manifold whose symplectic form vanishes on the image of the Hurewicz map, the required equality holds. Thus, we show that the original form of the Arnold Conjecture holds for all symplectic manifolds having ω|π2(M) = 0 = c1|π2(M).

Rational homotopy of gauge groups

In this brief paper, we observe that basic results from rational homotopy theory provide formulas for the rational homotopy groups of gauge groups of principal bundles K -> P -> B in terms of the rational homotopy groups of K and cohomology groups of B alone.

Quotient maps, group actions and Lusternik–Schnirelmann category

Abstract This note explores connections between Lusternik–Schnirelmann category, quotient maps and group actions. Category is used to restrict group actions in general and Hamiltonian actions in particular.

Higher Lefschetz Traces and Spherical Euler Characteristics

Higher analogs of the Euler characteristic and Lefschetz number are introduced. It is shown that they possess a variety of properties generalizing known features of those classical invariants. Applications are then given. In particular, it is shown that the higher Euler characteristics are obstructions to homotopy properties such as the TNCZ condition, and to a manifold being homologically Kähler. The Lefschetz number of a self-map f : X → X of a space X with finitely generated homology,

Geometry and the Foucault Pendulum

(1995). Geometry and the Foucault Pendulum. The American Mathematical Monthly: Vol. 102, No. 6, pp. 515-522.

Koszul–Sullivan Models and the Cohomology of Certain Solvmanifolds

In this paper we develop a technique of working with graded differential algebra models of solvmanifolds, overcoming the main difficulty arising from the ‘non-nilpotency’ of the corresponding Mostow fibrations. A graded differential model for solvmanifolds of the form G/Γ with G=R⋊ϕN is presented (N is a nilpotent Lie group, G is a semi-direct product). As an application, we prove the Benson–Gordon conjecture in dimension four.

A mapping theorem for topological complexity

We give new lower bounds for the (higher) topological complexity of a space in terms of the Lusternik‐Schnirelmann category of a certain auxiliary space. We also give new lower bounds for the rational topological complexity of a space, and more generally for the rational sectional category of a map, in terms of the rational category of a certain auxiliary space. We use our results to deduce consequences for the global (rational) homotopy structure of simply connected hyperbolic finite complexes. 55M30, 55P62; 55S40, 55Q15