Harmonic cohomology of symplectic manifolds

Mathieu (Math. Helv. 70 (1995) 1) introduced a canonic filtration in the de Rham cohomology of a symplectic manifold and proved, that the middle filtration space is the space of harmonic cohomology classes. We give an interpretation of the other filtration spaces, prove a Künneth theorem for harmonic cohomology, prove Poincaré duality for harmonic cohomology and show how surjectivity of certain Lefschetz type mappings is related to properties of the filtration. For a closed symplectic manifold M we also introduce symplectic invariants , and show if M is of dimension 2n with n even.     

Moment polytopes for symplectic manifolds with monodromy

A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such “almost-toric 4-manifolds” which admits a Hamiltonian S¹-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application, we derive a Duistermaat-Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.     

Free circle actions with contractible orbits on symplectic manifolds

We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on the constructions of FGM.     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

Examples of cell-like decompositions of the infinite-dimensional manifolds σ and Σ

Denote by σ the subspace of Hilbert space {(x_i)?l₂:x_i=0 for all but finitely many i}. Examples of cell-like decompositions of σ are constructed that have decomposition spaces that are not homeomorphic to σ. At one extreme is a cell-like decomposition G of σ produced using ghastly finite dimensional examples such that the decomposition space σ?G contains no embedded 2-cell but (σ?G)×$\text{R}$ is homeomorphic to σ. At the other extreme is a cell-like decomposition G of σ satisfying: (a) the nondegeneracy set N_G={g?G:g≠point} consists of countably many arcs (necessarily tame); (b) the nondegeneracy set N_G is a closed subset of the decomposition space σ?G; (c) each map f:B²→σ?G of a 2-cell into σ?G can be approximated arbitrarily closely by an embedding; (d) σ?G is not homeomorphic to σ but (σ?G)×$\text{R}$ is homeomorphic to σ. The fact that both conditions (a) and (b) can be satisfied (and have (d) hold) is directly attributable to σ’s incompleteness as a topological space.     

Lie solutions of Riemannian transport equations on compact manifolds

Given a couple of smooth positive measures of same total mass on a compact Riemannian manifold, the associated optimal transport equation admits a symplectic Monge-Ampère structure, hence Lie solutions (in a restricted sense, though, still expressing measure-transport). Properties of such solutions are recorded; a structure result is obtained for regular ones (each consisting of a closed 1-form composed with a diffeomorphism) and a quadratic cost-functional proposed for them.     

Topology of manifolds modeled on countable direct limits of Menger compacta

We construct n-dimensional counterparts of manifolds modeled on the space ?² equipped by the bounded weak topology (-manifolds). For -manifolds we prove the characterization, triangulation and classification theorems. In addition, a universal map of onto Q^∞ (the countable direct limit of Hilbert cubes and Z-embeddings) is constructed and characterized.     

Moduli spaces over manifolds with involutions

Most of the work was carried out under the support of K.C. Wong Education Foundation Ltd.     

Characterizing Hilbert cube manifolds by their homological structure

A hierarchy of disjoint ?ech carriers properties is introduced; and each is shown to be characteristic of ANR's whose products with 2-cells are Hilbert cube manifolds.     

The missing boundary problem for smooth manifolds of dimension greater than or equal to six

Let M be a non-compact differentiable manifold of dimension ?6. Suppose both M and ?M are 1-ended spaces. We give necessary and sufficient conditions for M to be diffeomorphic to the complement of a compact subset of the boundary of a compact manifold. There are four conditions: two geometric conditions and two algebraic obstructions. We give examples to show that these obstructions are not always trivial. In particular, an example of a manifold is constructed which does not have a completion but any tubular neighborhood of codimension ?3 has a completion. We also classify the different ways to complete a given manifold.     

A property that characterizes Euler characteristic among invariants of combinatorial manifolds

If a real valued invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension in the manifold, then the invariant is completely determined by the Euler characteristic of the manifold and its boundary. So essentially, the Euler characteristic is the unique invariant of this type.     

Stochastic differential equations on product loop manifolds

We study infinite systems of stochastic differential equations in spaces of loops with values in compact Riemannian manifolds. We prove existence of solutions with deterministic initial conditions and continuity of the corresponding paths.     

A note on a global invertibility of mappings on Rn

We provide sufficient conditions for a mapping f : Rⁿ → Rⁿ to be a global di?eomorphism in case f need not be continuously di?erentiable. Instead it is assumed to be strictly (Hadamard-like) and Fréchet di?erentiable. We use classical local invertibility conditions together with the non-smooth critical point theory.     

Residue formulation of the Chern character on smooth manifolds

The Chern character of a complex vector bundle is most conveniently defined as the exponential of a curvature of a connection. It is well known that its cohomology class does not depend on the particular connection chosen. It has been shown by Quillen that a connection may be perturbed by an endomorphism of the vector bundle, such as a symbol of some elliptic differential operator. This point of view, as we intend to show, allows one to relate Chern character to a noncommutative sibling formulated by Connes and Moscovici.     

Restrictions on symplectic fibrations

Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E₂. In the appendix, we prove nontriviality of certain Gromov-Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of  along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.     

Boundaries and complements of infinite- dimensional manifolds in the model space

Let M be a manifold modeled on a locally convex linear metric space E≌E^ω (or ≌E^ω_f and N a Z-submanifold of M. Then N is collared in M. In this paper, we study the following problem [1, 3]: Under what conditions can M be embedded in E so that N is the topological boundary of M in E? We gain a more mild sufficient condition than the previous papers [7, 8] and a necessary and sufficient condition in the case M has the homotopy type of Sⁿ (and each component of N is simply connected if n?2) and in the case N has the homotopy type of Sⁿ (n?2). Also we obtain a necessary and sufficient condition under which M can be embedded in E so that bd M = N and cl(E\M) has the homotopy type of Sⁿ (we assume that M and N are simply connected if n ? 2).     

Transversality for equivariant exact 1-forms and gauge theory on 3-manifolds

An equivariant jet transversality framework is developed for the study of critical sets of invariant functions on G manifolds. Techniques are developed to extend transversality results to the infinite dimensional Fredholm setting. As an application, the generic structure of the SU(4) perturbed flat moduli space of an integral homology three-sphere is described, as well as the generic structure of the parameterized moduli space for a path of perturbations. A similar analysis of the U(3) moduli space for rational homology three-spheres is also carried out.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

On a theorem of Shchepin and Repovš concerning the smoothness of compacta

We show that a theorem of Shchepin and Repovš concerning the smoothness of compacta follows from the theory of semicontinuous relations.