The degree of maps of free G-manifolds and the average

Suppose that G is a compact Lie group, M and N are orientable, free G-manifolds and f : M → N is an equivariant map. We show that the degree of f satisfies a formula involving data given by the classifying maps of the orbit spaces M/G and N/G. In particular, if the generator of the top dimensional cohomology of M/G with integer coefficients is in the image of the cohomology map induced by the classifying map for M, then the degree is one. The condition that the map be equivariant can be relaxed: it is enough to require that it be “nearly equivariant”, up to a positive constant. We will also discuss the G-average construction and show that the requirement that the map be equivariant can be replaced by a somewhat weaker condition involving the average of the map. These results are applied to maps into real, complex and quaternionic Stiefel manifolds. In particular, we show that a nearly equivariant map of a complex or quaternionic Stiefel manifold into itself has degree one. Dedicated to the memory of Jean Leray     

Uniqueness and transport density in Monge's mass transportation problem

Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures on , find the measure preserving map s(x) between them which minimizes the average distance transported. Here distance can be induced by the Euclidean norm, or any other uniformly convex and smooth norm on . Although the solution is never unique, we give a geometrical monotonicity condition singling out a particular optimal map s(x). Furthermore, a local definition is given for the transport cost density associated to each optimal map. All optimal maps are then shown to lead to the same transport density . Received: 18 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Observable concentration of mm-spaces into spaces with doubling measures

The property of measure concentration is that an arbitrary 1-Lipschitz function on an mm-space X is almost close to a constant function. In this paper, we prove that if such a concentration phenomenon arise, then any 1-Lipschitz map f from X to a space Y with a doubling measure also concentrates to a constant map. As a corollary, we get any 1-Lipschitz map to a Riemannian manifold with a lower Ricci curvature bounds also concentrates to a constant map.      

Non-linear maps preserving solvability

Let M_n be the algebra of all n×n complex matrices and let L be the general linear Lie algebra gl(n,C) or the special linear Lie algebra sl(n,C). A bijective (not necessarily linear) map preserves solvability in both directions if both and ⁻¹ map every solvable Lie subalgebra of L into some solvable Lie subalgebra. If n3 then every such map is either a composition of a bijective lattice preserving map with a similarity transformation and a map [a_ij][f(a_ij)] induced by a field automorphism , or a map of this type composed with the transposition. We also describe the general form of such maps in the case when n=2. Using Lie's theorem we will reduce the proof of this statement to the problem of characterizing bijective maps on M_n preserving triangularizability of matrix pairs in both directions. As a byproduct we will characterize bijective maps on M_n that preserve inclusion for lattices of invariant subspaces in both directions.     

Equivariant evaluation on free loop spaces

There is a natural evaluation map on the free loop space which sends a loop to its values at the kth roots of unity. This map is equivariant with respect to the action of the cyclic group on k elements . We study the induced map in -equivariant cohomology with mod two coefficients in the cases where for . Received: 17 February 2000; in final form: 28 February 2001 / Published online: 18 January 2002     

Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

Let X be a closed oriented connected topological manifold of dimension n ≥ 5 . The structure group is the abelian group of equivalence classes of all pairs (f, M) such that M is a closed oriented manifold and f : M → X is an orientation‐preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant map defines a group homomorphism from the topological structure group of X to the analytic structure group of X . Here X is the universal cover of X , Γ = π₁X is the fundamental group of X , and is a certain C^* ‐algebra. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected topological manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected homology manifold. As an application, we use the additivity of the higher rho invariant map to study nonrigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the algebraically reduced structure group of X by the number of torsion elements in π₁X . Here the algebraically reduced structure group of X is the quotient of modulo a certain action of self‐homotopy equivalences of X . We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds. © 2020 Wiley Periodicals LLC     

Discs in Stein manifolds containing given discrete sets

Denote by the open unit disc in . We prove that given a discrete subset S of a connected Stein manifold M there is a proper holomorphic map such that ; if the map f can be chosen to be an embedding. In addition we prove that we can prescribe higher order contacts of with given one dimensional submanifolds in M. Received: 19 June 2000; in final form: 29 November 2000 / Published online: 19 October 2001     

An effective algorithm for the cohomology ring of symplectic reductions

Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of is generated by a small number of classes satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two , quotiented by the diagonal 2-torus action. Submitted: September 2001, Revised: December 2001, Revised: February 2002.     

An Elementary Proof of the Fundamental Theorem of Projective Geometry (Dedicated to Alfred Frölicher)

The following version of the fundamental theorem is proved: Let V, W be vector spaces and g: P(V)\E P(W) a morphism between the associated projective spaces. If the image of g is not contained in a line, then there exists a semilinear map f: V W which induces g. The difficulty lies in the fact that the homomorphism of division rings associated to the map f can be nonsurjective. As an application, a short proof of Wigner's theorem is also proposed.     

Roots of mappings on nonorientable surfaces¶and equations in free groups

Let f:M ₁→M ₂ be a continuous map and c:M ₁→M ₂ a constant map between closed (not necessarily orientable) surfaces. By definition the pair (f,c) has the Wecken property if f can be deformed into a map f' such that the number of coincidence points of (f',c) is the same as the number of essential coincidence classes of (f,c) and, hence, every essential coincidence class consists of exactly one point. When both surfaces are orientable the problem to determine all maps which have the Wecken property was solved in [14]. Let A(f) denote the absolute degree as defined in [6] or [15] and . Here we show that a map f has the Wecken property iff either the Euler characteristic or . In free groups there are solved certain quadratic equations closely related to the root problem. Received: Received: 18 January 2001 / Revised version: 27 November 2001     

The geometric Hopf invariant and double points

The geometric Hopf invariant of a stable map F is a stable _boxclose/2{{\mathbb Z}/2} -equivariant map h(F) such that the stable \mathbb Z/2{{\mathbb Z}/2} -equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper, we express the geometric Hopf invariant of the Umkehr map F of an immersion f : M^m \looparrowright Nⁿ{f : M^m \looparrowright N^n} in terms of the double point set of f. We interpret the Smale–Hirsch–Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m <  2n – 1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.     

Bounds on Supremum Norms for Hecke Eigenfunctions of Quantized Cat Maps

We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are ≪_ε N^ε for all ε > 0 in the case of N square free. Submitted: March 6, 2006; Accepted: April 30, 2006     

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C ¹ (Theorem 3). Then we consider the case of the C ⁰ integrable symplectic twist maps and we prove that for such a map, there exists a dense G _δ subset of the set of its invariant curves such that every curve of this G _δ subset is C ¹ (Theorem 4).     

Schottky uniformization¶and vector bundles over Riemann surfaces

We study a natural map from representations of a free group of rank g in GL(n,ℂ), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient condition for this map to be a submersion, when restricted to representations producing stable bundles. Using a generalized version of Riemann's bilinear relations, this condition is shown to be true on the subspace of unitary Schottky representations. Received: 13 June 2000 / Revised version: 29 December 2000     

A Result on Localization of Equilibria

In this paper, we deal with a general equilibrium problem where a bimap F: A×B X×Y2 ^Z is involved. This problem contains the scalar equilibrium problem as a very special case. The general equilibrium is considered via the properties of the map G: B2 ^A naturally associated to the problem. The main result shows that, to have solutions on every convex subsets B ₁ of B, localized via a map T: B2 ^A , a necessary and sufficient condition is the KKM property of the map G with respect to T. The assumptions require that T satisfies a regularity condition with respect to G, and it is proved that this condition is quite sharp, providing a suitable counterexample.     

An approximation theorem for sections of reflexive sheaves

 We show that, if X is a Stein manifold and D ? X an open set (not necessarily Stein) such that the restriction map has dense image, then, for any reflexive coherent analytic sheaf ℱ on X, the map has dense image, too. We also characterize the reflexivity of a torsion-free coherent sheaf on complex manifolds in terms of absolute gap sheaves or Kontinuit?tssatz. Received: 14 September 2001 / Revised version: 29 January 2002