Gregory Lupton

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.

Topological Complexity Of H-Spaces

Let X be a (not-necessarily homotopy-associative) H-space. We show that TCn+1(X) = cat(X-n), for n >= 1, where TCn+1(-) denotes the so-called higher topological complexity introduced by Rudyak, and cat(-) denotes the Lusternik-Schnirelmann category. We also generalize this equality to an inequality, which gives an upper bound for TCn+1(X), in the setting of a space Y acting on X.

Banach algebras and rational homotopy theory

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GLn(A), the group of invertible n × n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lcn(A) denote the space of “last columns” of GLn(A). We construct a natural isomorphism Ȟ(Max(A);Q) ∼= π2n−1−s(Lcn(A))⊗ Q for n > 1 2 s+1 which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F (X,G) for G a Lie group or, more generally, a rational H-space.

Free Torus Actions and Two-Stage Spaces

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

Variations on a conjecture of Halperin

Halperin has conjectured that the Serre spectral sequence of any fibration that has fibre space a certain kind of elliptic space should collapse at the E2-term. In this paper we obtain an equivalent phrasing of this conjecture, in terms of formality relations between base and total spaces in such a fibration (Theorem 3.4). Also, we obtain results on relations between various numerical invariants of the base, total and fibre spaces in these fibrations. Some of our results give weak versions of Halperin’s conjecture (Remark 4.4 and Corollary 4.5). We go on to establish some of these weakened forms of the conjecture (Theorem 4.7). In the last section, we discuss extensions of our results and suggest some possibilities for future work.

Equivalence classes of homotopy-associative comultiplications of finite complexes

Let X be a finite, 1-connected CW-complex which admits a homotopy-associative comultiplication. Then X has the rational homology of a wedge of spheres, Sn1 + 1 V … V Snr + 1. Two comultiplications of X are equivalent if there is a self-homotopy equivalence of X which carries one to the other. Let ba(X), respectively bac(X), denote the set of equivalence classes of homotopy classes of homotopy-associative, respectively, homotopy-associative and homotopycommutative, comultiplications of X. We prove the following basic finiteness result: Theorem 6.1 (1) If for each i, (a) ni ≠ nj + nk for every j, k with j < k and (b) ni ≠ 2nj for every j with nj even, then ba(X) is finite. (2) bac(X) is always finite. The methods of proof are algebraic and consist of a detailed examination of comultiplications of the free Lie algebra π#(ΩX) ⊗ Q. These algebraic methods and results appear to be of interest in their own right. For example, they provide dual versions of well-known results about Hopf algebras. In an appendix we show the group of self-homotopy equivalences that induce the identity on all homology groups is finitely generated.

On the Nilpotency of Subgroups of Self-Homotopy Equivalences

If X is a topological space, we denote by e(X) the set of homotopy classes of self-homotopy equivalences of X. Then e(X) is a group with group operation given by composition of homotopy classes. The group e(X) is a natural object in homotopy theory and has been studied extensively—see [Ar] for a survey of known results and applications of e(X). In this paper we continue our investigation of e #(X), the subgroup of e(X) consisting of homotopy classes which induce the identity on homotopy groups, and, to a lesser extent, of e *#(X), the subgroup of e #(X) consisting of homotopy classes which also induce the identity on homology groups (see §2 for precise definitions), which was begun in [A-L]. These groups are nilpotent and we focus primarily on the nilpotency class of e #(X). The determination of this nilpotency class appears in the list of problems on e(X) in [Ka, Problem 10]. For rational spaces we obtain both general results on the nilpotency class and a complete determination of the nilpotency class in specific cases. This leads to a lower bound for the nilpotency class of the groups e #(X) for certain finite complexes X by using derationalization techniques.

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