Intersection homology

In topology, a branch of mathematics, intersection homology is a special way to study shapes that have sharp points or corners, called singular spaces. It is very useful because regular methods for studying shapes sometimes fail on these tricky spaces.

Intersection homology was discovered by two mathematicians, Mark Goresky and Robert MacPherson, in the fall of 1974. They worked on it for several years to make it a powerful tool in mathematics.

This idea helped prove important guesses called the Kazhdan–Lusztig conjectures and understand a deep connection called the Riemann–Hilbert correspondence. It is also closely related to another advanced idea called L² cohomology.

Goresky–MacPherson approach

The homology groups of a compact, oriented, connected space have a special property called Poincaré duality. This means that for any two homology groups, there is a perfect pairing between them.

When a space has singularities, or places that don’t look like normal space, this duality breaks down. Goresky and MacPherson introduced a new way to study these spaces using "allowable" cycles. They showed that the intersection of these cycles gives well-defined homology classes, extending the idea of Poincaré duality to more complex spaces.

Small resolutions

A resolution of singularities is a way to make a complex shape smoother by mapping it to another shape. A small resolution happens when, for every size we look at, the tricky parts of the original shape are very small and don’t affect the overall structure much. This means that the homology — a way to study the shape — of the smoother shape matches the homology of the original shape.

There exists a special example where two different small resolutions have different structures in their cohomology. This shows that intersection homology usually doesn’t have a natural ring structure.

Sheaf theory

Deligne's formula for intersection cohomology describes a special way to study shapes with sharp points or corners. It connects two different ideas in mathematics: intersection homology and a tool called "sheaves."

This formula helps mathematicians understand complicated spaces by breaking them into simpler parts and studying how those parts fit together. One example is looking at a special kind of shape called an elliptic curve, which has a tiny point where it looks sharp. By studying this point carefully, mathematicians can learn more about the whole shape.

Properties of the complex IC(X)

The complex IC_p(X) has special properties that make it useful in mathematics. On most of the space, except for a very small part, the homology groups behave in a simple way. These properties help mathematicians understand the shape and structure of complicated spaces.

The complex is uniquely defined by these properties, meaning there is only one way it can look up to certain mathematical transformations. This also means that the properties do not change based on how the space is divided into simpler pieces.