Intersection homology

In topology, a branch of mathematics, intersection homology is a special way to study shapes with sharp points or corners. These shapes can be hard to study with regular methods.

Intersection homology was discovered by two mathematicians, Mark Goresky and Robert MacPherson, in 1974. They worked on it for several years to make it useful 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 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, they pair up perfectly.

When a space has singularities, or places that don’t look like normal space, this pairing stops working. Goresky and MacPherson created a new way to study these spaces using "allowable" cycles. They showed that the intersection of these cycles gives clear homology classes, extending 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 the tricky parts of the original shape are very small and don’t change the overall structure much. This means the homology of the smoother shape matches the homology of the original shape.

There is an example where two different small resolutions have different structures. This shows that intersection homology usually doesn’t have a natural ring structure.

Sheaf theory

Deligne's formula for intersection cohomology is a special way to study shapes that have sharp points or corners. It connects two ideas in mathematics: intersection homology and tools called "sheaves."

This formula helps mathematicians understand complicated spaces. They break the spaces into simpler parts and study how those parts fit together. One example is an elliptic curve, which has a tiny point that looks sharp. By studying this point, mathematicians can learn more about the whole shape.

Properties of the complex IC(X)

The complex IC_p(X) has special features that make it useful in math. In most of the space, except for a very tiny part, the homology groups act in a simple way. These features help mathematicians learn about the shape and structure of complicated spaces.

The complex is defined by these features in a unique way. This means there is only one way it can look, up to some math changes. It also means the features do not change based on how the space is split into simpler parts.