Theorem

Donaldson's Theorem

Topology

Donaldson's Theorem, in differential topology and gauge theory, states that if a closed, oriented, smooth four-dimensional manifold has a definite intersection form, that form must be diagonalizable, reducing to the identity matrix over the integers in the positive definite case. Proved by Simon Donaldson using instanton gauge theory, the theorem was originally stated for simply connected manifolds and was later extended to four-manifolds with any fundamental group. It showed that many topological four-manifolds admit no smooth structure at all, revealing a sharp divide between topology and smooth geometry unique to dimension four.

Facts

Statement

A definite intersection form of a closed, oriented, smooth manifold of dimension 4 is diagonalizable. 1

Proof Year

1983 1

Classification

Statement Form

Classification Theorem 1

Connections

Has Statement Form

Classification Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Differential Topology

Source Donaldson's theorem, Wikipedia

Sources

1. Donaldson's theorem, Wikipedia

Introduction
a definite intersection form of a closed, oriented, smooth manifold of dimension 4 is diagonalizable.
References
Donaldson, S. K. (1983-01-01).
In Branch: Differential Topology, Lead sentence
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection for

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Donaldson's Theorem

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