arXiv research

New spaces help connect manifold structures on equivariant Poincaré spaces.

Uniform Poincaré inequalities established for various metric spaces.

Extends Poincaré-Lefschetz duality to pairs of ∞-categories.

The paper shows plentiful non-homotopy finite Poincaré duality spaces.

Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.

A special knot in the Poincaré sphere leads to a unique connected sum of lens spaces.

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

We describe classes of potential structures (covector fields) on Minkowski space that admit subgroups of the Poincaré group. We describe also seven classes of Maxwell spaces that admit subgroups of the Poincaré group.

Study Poincaré inequality in metric spaces via separating sets.

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space $X$ that admits Poincaré inequalities for a continuum of mutually singular measures.

Enhanced loop space decomposition for specific Poincaré complexes.

The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.

Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…

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.

Proves energy expression on Poincaré-Einstein spaces.

We investigate the possibility of improving the $p$-Poincaré inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge Λ_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $Λ_p:=[(N-1)/p]^{p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is …

We obtain multirelative connectivity statements about spaces of Poincare embeddings, as precursors to analogous statements about spaces of smooth embeddings. The latter are the key to convergence results in the functor calculus approach to spaces of embeddings.

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

Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.

We exhibit an infinite family of knots in the Poincare homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. In the appendix, it is shown that hyperbolic knots in the Poincare homology sphere with a lens space…

We establish a number of foundational results on Poincaré spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there exists a finite CW pair $(X,Y)$ satisfying relative Poincaré duality in dimension …

Proves existence and uniqueness of weighted metrics for smooth spaces.

We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is quasi-isomorphic to the Deligne sheaf.

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Timár. In this paper we focus on properties of the Poincaré profiles of groups with …

This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficien…

Characterizes when almost smooth spaces become RCD spaces.

In a flat space, the global topology of comoving space can induce a weak acceleration effect similar to dark energy. Does a similar effect occur in the case of the Poincare dodecahedral space S^3/I^*? Does the effect distinguish the Poincare space from other well-proportioned spaces? The residual acceleration effect in…

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…

A space is created to realize a specific cohomology module, showing PL structure but not smoothability.

Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.

The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.

The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.

The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.

In this paper we address the relation between the orbifold fundamental group and the topology of the underlying space. In particular, under the assumption that the orbifold fundamental group is equal to the fundamental group of the underlying space, we prove Poincaré Duality for orbifolds of dimension 4 and 5.

The paper finds manifold structures on complex spaces.

New Poincaré inequality for differential forms on manifolds.

A combination of Bestvina--Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented Q-Poincaré duality group which is not the fundamental group of an aspherical closed ANR Q-homology manifold. The acyclic construction suggests asking which Q-Poincaré duality groups act freely …

The paper extends stabilization methods to Poincaré Duality complexes.

Motivated by decompositions of spaces that arise in continuous and discrete Morse theory, we describe a so called fibrous decomposition Z = X_0(Y_1)X_1 ... X_{n-1}(Y_n)X_n of a space Z. Among the applications is a succinct formula for the Euler-Poincare characteristic of Z, e(Z) = e(X_0) - e(Y_1) + e(X_1) - ... + e(X_{…

Study on functional inequalities on simple edge spaces.

Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.

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

Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincaré inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio a…

The presentation of supergravity theories of our previous paper "Super-Poincare' algebras, space-times and supergravities (I)" is re-formulated in the language of Berezin-Leites-Kostant theory of supermanifolds. It is also shown that the equations of Cremmer, Julia and Scherk's theory of 11D-supergravity are equivalent…

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

This work presents a reformulation of the recently proposed Wasserstein autoencoder framework on a non-Euclidean manifold, the Poincaré ball model of the hyperbolic space. By assuming the latent space to be hyperbolic, we can use its intrinsic hierarchy to impose structure on the learned latent space representations. W…

In this paper we prove $L^{\infty}$ type gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to a broad class of complete manifolds satisfying weighted Poincaré inequalities. In particular, we obtain a gap theorem on the Euclidean space wi…