Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local Ck diffeomorphisms for k∈N∪{∞,ω} which admit embedding vector fields with some smoothness. Finally we prove that for any k∈N∪{∞} under the coefficient topology the subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck diffeomorphisms.     

Localizability of the embedding problem with symplectic kernel

In this paper we consider the question of how much information is supplied by local solutions to a global embedding problem for the special case in which the normal subgroup belonging to the given group extension is the projective symplectic group PSp(2m, q). It is proved that for suitable Galois extensions K of a given number field k (which constitute part of the data of the embedding problem), the local solutions in a sense determine whether or not an extension K ? K, Galois over k, with $\text{G(}\text{L}\text{K}\text{) ≈ PSp(2m, q)}$, represents a global solution to the embedding problem.     

Embedding of a universally consistent 2-extension in a universally consistent extension

A Galois extension is considered universally consistent with the period q if for any problem of embedding of this extension with an abelian kernel of the period q the consistency condition holds. Let K be a universally consistent extension of the period 2n + 1 of an algebraic number field k, such that 2 completely splits in K, and let (K/k, φ) be an embedding problem with the cyclic kernel of order 2. It is proven that (under some group-theoretical restrictions) there exists a solution of this embedding problem that is universally consistent with the period 2n.     

The Hopf invariant of a Haefliger knot

We show that for any given differentiable embedding of the three-sphere in six-space there exists a Seifert surface (in six-space) with arbitrarily prescribed signature. This implies, according to our previous paper, that given such a (6,3)-knot endowed with normal one-field, we can construct a Seifert surface so that the outward normal field along its boundary coincides with the given normal one-field. This aspect enables us to understand the resemblance between Ekholm–Szűcs’ formula for the Smale invariant and a formula in our previous paper for differentiable (6,3)-knots. As a consequence, we show that an immersion of the three-sphere in five-space can be regularly homotoped to the projection of an embedding in six-space if and only if its Smale invariant is even. We also correct a sign error in our previous paper: “A geometric formula for Haefliger knots” [Topology 43: 1425–1447 2004].      

The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. To this end, we associate with such a current an equivariant cyclic cohomology class of Connes' C^∗-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points.     

Geometric formulas for Smale invariants of codimension two immersions

We give three formulas expressing the Smale invariant of an immersion f of a (4k−1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k−1)-space.The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k−1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k−1)-space. Moreover, any generic homotopy in (6k−1)-space connecting f to g must have at least a_k(2k−1)! cusps, where a_k=2 if k is odd and a_k=1 if k is even.     

Capturing k-ary existential second order logic with k-ary inclusion–exclusion logic

In this paper we analyze k-ary inclusion–exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k].We also introduce several useful operators that can be expressed in INEX[k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.     

On finite simple groups acting on homology spheres

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each dimension n there are only finitely many finite simple groups which admit a faithful, smooth action on some homology sphere of dimension n, and in particular on the n-sphere. We discuss also the finite simple groups which admit an action on a homology sphere of dimension 3, 4 or 5.     

Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is smooth except at a finite set of conical singularities. This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen (Sharp eigenvalue bounds and minimal surfaces in the ball, 2013). Then we get existence results among all metrics on surfaces of a given genus, leading to the existence of minimal isometric immersions of smooth compact Riemannian manifold (M, g) of dimension 2 into some k-sphere by first eigenfunctions. At last, we also answer a conjecture of Friedlander and Nadirashvili (Int Math Res Not 17:939–952, 1999) which asserts that the supremum of the first eigenvalue of the Laplacian on a conformal class can be taken as close as we want of its value on the sphere on any orientable surface.     

Embeddings in the 3/4 range

We give a complete obstruction to turning an immersion f:M^m→Rⁿ into an embedding when 3n?4m+5. It is a secondary obstruction, and exists only when the primary obstruction, due to André Haefliger, vanishes. The obstruction lives in a twisted cobordism group, and its vanishing implies the existence of an embedding in the regular homotopy class of f in the range indicated. We use Tom Goodwillie's calculus of functors, following Michael Weiss, to help organize and prove the result.     

Haefliger’s codimension-one singular foliations, open books and twisted open books in dimension 3

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by André Haefliger as being ??₁-structures. We prove that under the obvious linear embedding condition, they are ??₁-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C ^r , r ?? 1. In particular, in dimension 3, this gives a very simple proof of Thurston??s 1976 regularization theorem without using Mather??s homology equivalence.     

On smooth foliations with Morse singularities

Let M be a smooth manifold and let F be a codimension one, C^∞ foliation on M, with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb (1946) [11] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper (1962) [4] classifies manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F, we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S¹, we are able to extend Haefliger?s theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.     

The Schwartz space of a smooth semi-algebraic stack

Schwartz functions, or measures, are defined on any smooth semi-algebraic (“Nash”) manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are defined as geometric stacks in the category of Nash manifolds. Moreover, when those are obtained from algebraic quotient stacks of the form X/G, with X a smooth affine variety and G a reductive group defined over a number field k, we define, whenever possible, an “evaluation map” at each semisimple k-point of the stack, without using truncation methods. This corresponds to a regularization of the sum of those orbital integrals whose semisimple part corresponds to the chosen k-point. These evaluation maps produce, in principle, a distribution which generalizes the Arthur–Selberg trace formula and Jacquet’s relative trace formula, although the former, and many instances of the latter, cannot actually be defined by the purely geometric methods of this paper. In any case, the stack-theoretic point of view provides an explanation for the pure inner forms that appear in many versions of the Langlands, and relative Langlands, conjectures.     

Global existence and blow-up for harmonic map heat flow

For degree-one equivariant maps on bounded domains, the question of finite-time blow-up vs. global existence of solutions to the harmonic map heat flow has been well studied. In this paper we study the Cauchy problem for degree-m equivariant harmonic map heat flow from (2+1)-dimensional space-time into the 2-sphere with initial energy close to the energy of harmonic maps. It is proved that solutions are globally smooth for m?4, whereas for m=1, we show that finite-time singularities can form for this class of data.     

Morphisms between complete Riemannian pseudogroups

We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a closure and its maps are C^∞ along the orbit closures. Here, completeness and closure are versions for morphisms of concepts introduced by Haefliger for pseudogroups. This result is applied to approximate foliated maps by smooth ones in the case of transversely complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. This generalizes the topological invariance of their basic cohomology, shown by El Kacimi-Alaoui-Nicolau. A different proof of the spectral sequence invariance was also given by the second author.     

Embedding ordered fields in formal power series fields

In the paper it is shown how an embedding of an ordered field F into a formal power series field can be extended canonically to an embedding of any simple extension F(y) of F. Properties of the extended embedding are studied in detail. Several applications are given.     

The Rank of Abelian Varieties over Infinite Galois Extensions

G. Frey and M. Jarden (1974, Proc. London Math. Soc.28, 112-128) asked if every Abelian variety A defined over a number field k with dim A>0 has infinite rank over the maximal Abelian extension k^ab of k. We verify this for the Jacobians of cyclic covers of P¹, with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d -dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2).