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In algebraic topology, the intersection number appears as the Poincaré dual of the cup product. Specifically, if two manifolds, X and Y, intersect transversely in a manifold M, the homology class of the intersection is the Poincaré dual of the cup product ⌣ of the Poincaré duals of X and Y.
A key example of self-intersection numbers is the exceptional curve of a blow-up, which is a central operation in birational geometry. Given an algebraic surface S, blowing up at a point creates a curve C. This curve C is recognisable by its genus, which is 0, and its self-intersection number, which is −1. (This is not obvious.)
In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a smooth structure .
The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. [2]
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
Intersection homology was originally defined on suitable spaces with a stratification, though the groups often turn out to be independent of the choice of stratification. There are many different definitions of stratified spaces. A convenient one for intersection homology is an n-dimensional topological pseudomanifold.
In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty.
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group (), is divisible by 16.