Related papers: Simplicial intersection homology revisited

Suppose M is a complex manifold of dimension $n+1$ and K is a hypersurface in M. By Poincar\'e duality we define a residue morphism $res:H^{k+1}(M\setminus K)\longrightarrow H_{2n-k}(K)$ which generalizes the classical Leray residue…

We say that a complex analytic space, $X$, is an intersection cohomology manifold if and only if the shifted constant sheaf on $X$ is isomorphic to intersection cohomology; this is quickly seen to be equivalent to $X$ being a homology…

Taking the l^1-completion and the topological dual of the singular chain complex gives rise to l^1-homology and bounded cohomology respectively. In contrast to l^1-homology, major structural properties of bounded cohomology are well…

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity,…

Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, in which case intersection cohomology may depend on the…

We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological…

This expository article presents a self-contained introduction to simplicial homology for finite simplicial complexes, emphasizing concrete computation and geometric intuition. Beginning with orientations of simplices and the construction…

M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for…

The dynamics of large complex systems are predominately modeled through pairwise interactions, the principle underlying structure being a network of the form of a digraph or quiver. Significant success has been obtained in applying the…

We show the intersection of a compact almost complex subvariety of dimension $4$ and a compact almost complex submanifold of codimension $2$ is a $J$-holomorphic curve. This is a generalization of positivity of intersections for…

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…

This paper is a contribution to piecewise linear (PL) symplectic topology. We define the notion of PL symplectic manifold as being a combinatorial manifold endowed with a piecewise constant Whitney symplectic form and investigate possible…

For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation of the `coarse' underlying space together with the local isotropy groups of M. We prove that, for any local system on M, this complex S(M) has…

In this paper, we review a method for computing and parameterizing the set of homotopy classes of chain maps between two chain complexes. This is then applied to finding topologically meaningful maps between simplicial complexes, which in…

This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection…

We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce…

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse…

Given a diagram for a trisection of a 4-manifold $X$, we describe the homology and the intersection form of $X$ in terms of the three subgroups of $H_1(\Sigma;\mathbb{Z})$ generated by the three sets of curves and the intersection pairing…

The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell…

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005)…