Essential manifolds with extra structures

We consider classes of algebraic manifolds $\mathcal{A}$, of symplectic manifolds $\mathcal{S}$, of symplectic manifolds with the hard Lefschetz property $\mathcal{HS}$ and the class of cohomologically symplectic manifolds $\mathcal{CS}$. For every class of manifolds $\mathcal{C}$ we denote by $\mathcal{EC}(\pi,n)$ a subclass of $n$-dimensional essential manifolds with fundamental group $\pi$. In this paper we prove that for all the above classes with symplectically aspherical form the condition $\mathcal{EC}(\pi,2n)\ne \emptyset$ implies that $\mathcal{EC}(\pi,2n-2)\ne\emptyset$ for every $n>2$. Also we prove that all the inclusions $\mathcal{EA}\subset\mathcal{EHS}\subset\mathcal{ES}\subset\mathcal{ECS}$ are proper.Comment: 13 pages, no figures, v2, typos correcte

Asymptotic topology

We establish some basic theorems in dimension theory and absolute extensor theory in the coarse category of metric spaces. Some of the statements in this category can be translated in general topology language by applying the Higson corona functor. The relation of problems and results of this `Asymptotic Topology' to Novikov and similar conjectures is discussed.Comment: 38 pages, AMSTe

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

One of the fundamental problems in Riemannian geometry is to understand the relation of locally defined curvature invariants and global properties of smooth manifolds. This workshop was centered around the investigation of scalar curvature, addressing questions in global analysis, geometric topology, relativity and minimal surface theory

Geometric Topology

Geometric topology has seen significant advances in the understanding and application of infinite symmetries and of the principles behind them. On the one hand, for advances in (geometric) group theory, tools from algebraic topology are applied and extended; on the other hand, spectacular results in topology (e.g., the proofs of new cases of the Novikov conjecture or the Atiyah conjecture) were only possible through a combination of methods of homotopy theory and new insights in the geometry of groups. This workshop focused on the rich interplay between algebraic topology and geometric group theory

Periodic symplectic cohomologies and obstructions to exact Lagrangian immersions

Given a Liouville manifold, symplectic cohomology is defined as the Hamiltonian Floer homology for the symplectic action functional on the free loop space. In this thesis, we propose two versions of periodic S^1-equivariant homology or S^1-equivariant Tate homology for the natural S^1-action on the free loop space. The first version is called periodic symplectic cohomology. We prove that there is a localization theorem or a fix point property for periodic symplectic cohomology. The second version is called the completed periodic symplectic cohomology which satisfies Goodwillie's excision isomorphism. Inspired by the work of Seidel and Solomon on the existence of dilations on symplectic cohomology, we formulate the notion of Liouville manifolds admitting higher dilations using Goodwillie's excision isomorphism on the completed periodic symplectic cohomology. As an application, we derive obstructions to existence of certain exact Lagrangian immersions in Liouville manifolds admitting higher dilations