Dirichlet units and critical points of closed 1-forms

S.P. Novikov developed an analog of the Morse theory for closed 1-forms. In this paper we suggest an analog of the Lusternik-Schnirelman theory for closed 1-forms. For any cohomology class ξ ε H¹(X,R) we define an integer cl(ξ) (the cup-length associated with ξ); we prove that any closed 1-form representing ξ has at least cl(ξ) - 1 critical points. The number cl(ξ) is defined using cup-products in cohomology of some flat line bundles, such that their monodromy is described by complex numbers, which are not Dirichlet units.     

Novikov homology,jump loci and Massey products

Let X be a finite CW complex, and ρ: π ₁(X) → GL(l, ?) a representation. Any cohomology class α ∈ H ¹(X, ?) gives rise to a deformation γ _t of ρ defined by γ _t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ _gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple. If α ∈ H ¹(X, ?) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ _gen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H ¹(X, ?).     

Equivalence of closed 1-forms on surfaces with edge

We investigate closed 1-forms with isolated zeros on surfaces with edge. A criterion for the topological equivalence of closed 1-forms is proved.     

Topological and ergodic properties of closed 1-forms with incommensurable periods

V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 1–12, April–June, 1991.     

Finite semifields and nonsingular tensors

In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455–464, 1981).     

Lepage forms,closed 2-forms and second-order ordinary differential equations

Lepage 2-forms appear in the variational sequence as representatives of the classes of 2-forms. In the theory of ordinary differential equations on jet bundles they are used to construct exterior differential systems associated with the equations and to study solutions, and help to solve the inverse problem of the calculus of variations: since variational equations are characterized by Lepage 2-forms that are closed. In this paper, a general setting for Lepage forms in the variational sequence is presented, and Lepage 2-forms in the theory of second-order differential equations in general and of variational equations in particular, are investigated in detail. The text was submitted by the authors in English.     

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L ²(, ). Denote by T=T(t) _t0and S=S(t) _t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.     

A local limit theorem for closed geodesics and homology

In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies.

     

Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, ∂-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Necessary and sufficient conditions the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found. The vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper-Kähler manifold is shown.     

Every nonsingular C1 flow on a closed manifold of dimension greater than two has a global transverse disk

We prove three results about global cross sections which are disks, henceforth called global transverse disks. First we prove that every nonsingular (fixed point free) C¹ flow on a closed (compact, no boundary) connected manifold of dimension greater than 2 has a global transverse disk. Next we prove that for any such flow, if the directed graph G_h has a loop then the flow does not have a closed manifold which is a global cross section. This property of G_h is easy to read off from the first return map for the global transverse disk. Lastly, we give criteria for an “M-cellwise continuous” (a special case of piecewise continuous) map $\text{h}\text{:D}{}^2\text{→D}{}^2$ that determines whether h is the first return map for some global transverse disk of some flow ϕ. In such a case, we call ϕ the suspension of h.     

Finite axiomatizations for existentially closed posets and semilattices

In this paper we exhibit axiomatizations for the theories of existentially closed posets and existentially closed semilattices. We do this by considering an infinite axiomatization which characterizes these structures in terms of embeddings of finite substructures, an axiomatization which exists for any locally finite universal class with a finite language and with the joint embedding and amalgamation properties. We then find particular finite subsets of these axioms which suffice to axiomatize both classes. Research supported by an NSERC Postdoctoral Fellowship. Research supported by NSERC Grant No. A7256.     

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.     

Continuous families of Riemannian manifolds, Isospectral on functions but not on 1-forms

The purpose of this paper is to present the first continuous families of Riemannian manifolds that are isospectral on functions but not on 1-forms, and, simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. Examples of isospectral manifolds that are not isospectral on forms are sparse, as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral. The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method, a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method, and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson method. To the memory of Hubert Pesce, a valued friend and colleague.     

Matchings,path covers and domination

We show that if $G$ is a graph with minimum degree at least three, then ${\gamma }_t\left(G\right)\le {\alpha }^{\prime }\left(G\right)+(\mathrm{pc}\left(G\right)?1)∕2$ and this bound is tight, where ${\gamma }_t\left(G\right)$ is the total domination number of $G$, ${\alpha }^{\prime }\left(G\right)$ the matching number of $G$ and $\mathrm{pc}\left(G\right)$ the path covering number of $G$ which is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover. We show that if $G$ is a connected graph on at least six vertices, then ${\gamma }_{\mathrm{nt}}\left(G\right)\le {\alpha }^{\prime }\left(G\right)+\mathrm{pc}\left(G\right)∕2$ and this bound is tight, where ${\gamma }_{\mathrm{nt}}\left(G\right)$ denotes the neighborhood total domination number of $G$. We observe that every graph $G$ of order $n$ satisfies ${\alpha }^{\prime }\left(G\right)+\mathrm{pc}\left(G\right)∕2\ge n∕2$, and we characterize the trees achieving equality in this bound.     

Period quotient maps of meromorphic 1-forms and minimal surfaces on Tori

Consider on a complex 1-dimensional torus T_λ an abelian differential of the second kind α_λ. Assign to each λ the period quotient of α_λ for two independent cycles on T_λ. For appropriate choices of α_λ, the image of the modular sphere under this map is simple to describe, extending the classical case of holomorphic α_λ Using this information for different α_λ simultaneously, we discuss applications to existence and uniqueness questions for the Chen-Gackstatter surface, the Costa surface, and the translation invariant periodic helicoid with handles. In particular, there is now a conceptual and essentially non-computational proof for the uniqueness of the Chen-Gackstatter surface.     

Coherent measures and the existence of smooth Lyapunov 1-forms for flows

Let a smooth vector field V on a smooth closed manifold M be given and let Z⊂M be an isolated invariant set for the flow of V. In this situation, we give a necessary and sufficient condition for the existence of a Lyapunov 1-form for (V,Z) in terms of the relative asymptotic cycles associated with certain invariant measures of the flow.