Concept
Intersection homology
Related concepts (5)
Perverse sheaf
The mathematical term perverse sheaves refers to a certain associated to a topological space X, which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces (intersection homology of Mark Goresky and Robert MacPherson) and the algebraic theory of differential equations (microlocal calculus and holonomic D-modules of Joseph Bernstein, Masaki Kashiwara and Takahiro Kawai).
Local system
In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from point to point. Local coefficient systems were introduced by Norman Steenrod in 1943. The category of perverse sheaves on a manifold is equivalent to the category of local systems on the manifold. Let X be a topological space.
Singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string.
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the ()th homology group of M, for all integers k Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.