arXiv research

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

In this paper, using exclusively homotopy theoretical methods, we study degrees of maps between $(n-2)$-connected $(2n-1)$-dimensional Poincar\' e complexes which have torsion free integral homology. Necessary and sufficient algebraic conditions for the existence of map degrees between such Poincar\' e complexes are es…

The Poincaré map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincaré map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of limit cycles (…

Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.

Model for assembly map of bordism-invariant functors.

We prove two kinds of fibering theorems for maps X --> P, where X and P are Poincare spaces. The special case of P = S^1 yields a Poincare duality analogue of the fibering theorem of Browder and Levine.

We study a generalization of the familiar Poincaré map, first implicitely introduced by N.N. Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant tori.

Classifies mapping tori of specific groups, generalizing known results.

The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.

For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N;f) of continuous maps homotopic to f:M--> N.We show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self coincidence…

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…

We consider two types of minimal Poincaré $4$-complexes. One is defined with respect to the degree $1$-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincaré $4$-complexes were introduced by Hambleton, Kreck and Teichner. It…

On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and bound…

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Ca…

The normal map of curves is analyzed as a vector field on a cylinder.

We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scatte…

A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski…

For a Poincare duality space X and a map X -> B, consider the homotopy fiber product X x^B X. If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X x^B X has the structure of a graded associative algebra. When X -> B is the diagonal…

In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).

Study embeddings of manifolds via acyclic maps and surgery.

Introduces Epstein-Poincaré surfaces for G-oper, generalizing classical construction.

We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere holomorphic harmonic map from an open simply connected Riem…

We show that there are no tight nonholomorphic maps from irreducible domains into exceptional codomains, the only exception being the already known tight nonholomorphic maps from the Poincare disc. This follows up on previous work by the first author where this was shown for classical codomains.

New proof of chain duality for simplicial complexes.

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltot…

Classifies knots in the Poincaré sphere, using fixed points and folding automata.

Study of Fubini-Study forms on surfaces with punctures.

Establishing criteria for top cell inertness in complexes.

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact …

Let G be either a finite cyclic group of prime order or S^1. We find new relations between cohomology of a manifold (or a Poincare duality space) M with a G-action on it and cohomology of the fixed point set, M^G. Our main tool is the notion of Poincare duality on the Leray spectral sequence of the map M_G -> BG. We ap…

Proves simplicity of Lyapunov exponents for specific Anosov flows.

We prove a categorified version of the Poincaré lemma. The natural setting for our result is that of $\infty$-local systems. More precisely, we show that any smooth homotopy between maps $f$ and $g$ induces an $\mathsf{A}_\infty$-natural transformation between the corresponding pullback functors. This transformation is…

Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.

This paper explores topological aspects of index theory for infinite-dimensional manifolds.

We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in $\mathcal{E}_n$-algebra.

Study local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_…

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were in…

We present in modern language the contents of the famous note published by Henri Poincaré in 1901 "Sur une forme nouvelle des équations de la Mécanique", in which he proves that, when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the well known Euler-Lagrange equa…

We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property th…

The variational auto-encoder (VAE) is a popular method for learning a generative model and embeddings of the data. Many real datasets are hierarchically structured. However, traditional VAEs map data in a Euclidean latent space which cannot efficiently embed tree-like structures. Hyperbolic spaces with negative curvatu…

We prove that to every inclusion $A\hookrightarrow L$ of Lie algebroids over the same base manifold $M$ corresponds a Kapranov dg-manifold structure on $A[1]\oplus L/A$, which is canonical up to isomorphism. As a consequence, $Γ(Λ^\bullet A^\vee\otimes L/A)$ carries a canonical $L_\infty[1]$ algebra structure whose una…

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then t…

In this paper, we are interested in the construction of quasiconformal mappings between domains of the Heisenberg group H that minimise a mean distortion functional. We propose to construct such mappings by considering a corresponding problem between domains of Poincaré half-plane $\mathbb H$. The first map we construc…

Study rationality of meromorphic functions between real algebraic sets in the plane.

Let f: P-->W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C:=closure(W-T) be its complement. Then W is the homotopy push-out of a diagram C<--dT-->P. This homotopy push-out square is an example of what is called a Poincare embedding. We study…

We classify pro-$p$ Poincaré duality pairs in dimension two. We then use this classification to build a pro-$p$ analogue of the curve complex and establish its basic properties. We conclude with some statements concerning separability properties of the mapping class group.

A natural class of coloring complexes $X$ on closed manifold $M^n$ is investigated that gives a holonomy map $\mbox{Hol}_X: π_1(M) \to S_{n+1}$. By a $k$-multilayer complex construction the holonomy map may be defined to any finite permutation group $\mbox{Hol}_X: π_1(M) \to S_{n+k}$, $k>0$. Under isotopy of $X$ and su…