In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns module over the Novikov ring to compact subsets of closed manifolds. We show existence restriction maps, and prove some basic properties. Our main contribution is identify natural geometric situation in cohomology two satisfy Mayer-Vietoris property. This tailored work und...

The Brown representability theorem gives a list of conditions for the representablity of a set-valued contravariant functor which is defined on the classical stable homotopy category. It has had many uses through the years, and has long been part of the canon of Algebraic Topology. It is entirely reasonable to ask for a more general version of the theorem, which would give conditions on a close...

It is shown that the hyperspace of non-empty finite subsets of a space X is an ANR (an AR) for stratifiable spaces if and only if X is a 2-hyper-locally-connected (and connected) stratifiable space. 0. Introduction. For a space X, let F(X) denote the hyperspace of non-empty finite subsets of X with the Vietoris topology, i.e., the topology generated by the sets 〈U1, . . . , Un〉 = {A ∈ F(X) | A ...

Computational topology is a set of algorithmic methods developed to understand topological invariants such as loops and holes in high-dimensional data sets. In particular, a method know as persistent homology has been used to understand such shapes and their persistence in point clouds and networks. It has only been applied to neuronal networks in recent years. While most tools from network sci...

In Domain Theory quasicontinuous domains pop up from time to time generalizing slightly the powerful notion of a continuous domain. It is the aim of this paper to show that quasicontinuous domains occur in a natural way in relation to the powerdomains of finitely generated and compact saturated subsets. Properties of quasicontinuous domains seem to be best understood from that point of view. Th...