Search

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic forms of compact K\"ahler manifolds. Some topological implications are deduced., Comment: 7 pages, to appear in Proc. AMS

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of $d$-harmonic forms of pure bidegree. There is also a generalization of Hodge Index Theorem for compact almost Kähler 4-manifolds. In particular, these provide topological bounds on the dimension of the space of d-harmonic forms of pure bidegree, as well as several new obstructions to the existence of a symplectic form compatible with a given almost complex structure.

For a topological flow $(V,\phi)$ - i.e., $V$ is a linearly compact vector space and $\phi$ a continuous endomorphism of $V$ - we gain a deep understanding of the relationship between $(V,\phi)$ and the Bernoulli shift: a topological flow $(V,\phi)$ is essentially a product of one-dimensional left Bernoulli shifts as many as $\mathrm{ent}^*(V,\phi)$ counts. This novel comprehension brings us to introduce a notion of corank for topological flows designed for coinciding with the value of the topological entropy of $(V,\phi)$. As an application, we provide an alternative proof of the so-called Bridge Theorem for locally linearly compact vector spaces connecting the topological entropy to the algebraic entropy by means of Lefschetz duality., Comment: the organisation of the paper has been substantially changed

A theorem of Lambrechts and Stanley is used to find the rational cohomology of the complement of an embedding S4n−1 → S2n ×S m as a module and demonstrate that it is not necessarily determined by the map induced on cohomology by the embedding, nor is it a trivial extension. This demonstrates that the theorem is an improvement on the classical Lefschetz duality.

We use the mapping cone for the relative deRham cohomology of a manifold with boundary in order to show that the Chern–Gauss–Bonnet Theorem for oriented Riemannian vector bundles over such manifolds is a manifestation of Lefschetz Duality in any of the two embodiments of the latter. We explain how Thom isomorphism fits into this picture, complementing thus the classical results about Thom forms with compact support. When the rank is odd, we construct, by using secondary transgression forms introduced here, a new closed pair of forms on the disk bundle associated to a vector bundle, pair which is Lefschetz dual to the zero section.

Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyaga's paper (1974), where Moser's theorem is proven for manifolds with boundary. A cohomological interpretation of Banyaga's operator is given, which allows a proof of Lefschetz duality using differential forms., 9 pages; mistakes corrected, final accepted version

By analogy with the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a so-called Bridge Theorem.

International audience; We prove the Lefschetz duality for intersection (co)homology in the framework of ∂-pesudomanifolds. We work with general perversities and without restriction on the coefficient ring.

We prove a Poincare-Alexander-Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen-Macaulayness of relative equivariant cohomology modules arising from the orbit filtration., Comment: 28 pages. This is a substantially expanded version of Section 6 of arXiv:1111.0957v1. v2: new result (Prop. 2.7) about equivariant homology in the case of freely acting subgroups; minor changes. v3: mistake in the proof of Prop. 2.7 corrected

The moduli space of Fuchsian projective connections on a closed Riemann surface admits a Poisson structure. The moduli space of projective monodromy representations on the punctured Riemann surface also admits a Poisson structure which arises from the Poincare-Lefschetz duality for cohomology. We shall show that the former Poisson structure coincides with the pull-back of the latter by the projective monodromy map. This result explains intrinsically why a Hamiltonian structure arises in the monodromy preserving deformation

The Conley index for continuous dynamical systems is defined for (one-sided) semiflows. For (two-sided) flows, there are two indices defined: one for the forward flow; and one for the reverse flow. In general, the two indices give different information about the flow; but for flows on orientable manifolds, there is a duality isomorphism between the homology Conley indices of the forward and reverse flows. This duality preserves the algebraic structure of many of the constructions of the Conley index theory: sums and products; continuation; attractor-repeller sequences and connection matrices.

Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F(M,k). We prove that our model it is at least a Sigma_k-equivariant differential graded model. We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold., Comment: Minor revision

A general result which produces product evaluations of determinants of certain raising operators for sl(2) representations is obtained. The most combinatorially interesting cases occur for self-dual raising operators of Peck posets. Applications include the following: A nice product expression is found for the determinant of the Lefschetz duality linear transformation on the cohomology of a Grassmannian. Known product expressions for the cardinalities of two sets of plane partitions are re-derived. The appearance of rising factorials for the hooks in one of these product expressions is “explained” by the appearance of rising factorials in sl(2) determinants. A higher dimensional generalization in a certain sense of MacMahon's famous product enumeration result for Ferrers diagrams contained in a box is stated in the context of nonintersecting lattice paths.

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 CP. This homotopy push-out square is an example of what is called a Poincare embedding. We study how to construct algebraic models, in particular in the sense of Sullivan, of that homotopy push-out from a model of the map f. When the codimension is high enough this allows us to completely determine the rational homotopy type of the complement C = W-f(P). Moreover we construct examples to show that our restriction on the codimension is sharp. Without restriction on the codimension we also give differentiable modules models of Poincare embeddings and we deduce a refinement of the classical Lefschetz duality theorem, giving information on the algebra structure of the cohomology of the complement., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-9.abs.html

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ3. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ3, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincare duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds.

We prove a Lefschetz duality theorem for intersection homology. Usually, this result applies to pseudomanifolds with boundary which are assumed to have a “collared neighborhood of their boundary”. Our duality does not need this assumption and is a generalization of the classical one.

We seek an analogue for topological manifolds of closed tubular neighbourhoods (for smooth imbeddings) and closed regular neighbourhoods (for piecewise linear imbeddings). We succeed when the dimension of the ambient manifold is at least six. The proof uses topological handle theory, the results of Siebenmann's thesis, and a strong version of the Lefschetz Duality Theorem which yields a duality formula for Wall's finiteness obstruction. 0. Introduction. Let DIFF, PL, TOP denote the categories of smooth, piece- wise linear and topological manifolds and, respectively, smooth, piecewise linear and continuous maps. In each of DIFF and PL, given an imbedding i. X —> Y, there is a natural class C (i), of closed neighbourhoods of i(X) in Y, namely, tubular neighbourhoods in DIFF and regular neighbourhoods in PL. C(z) has the following properties: (A) C(z') is a fundamental system of neighbourhoods of i(X) in Y. (B) If N £ c(z'), i(X) c_» N is a simple homotopy equivalence and, if codim X > 3, dN *—> N induces an isomorphism of fundamental groups. (C) There is a deformation retraction r : N —> i(X) such that Map (dN—'i(X)) = N (Map is the mapping cylinder construction).