Manifolds and cell complexes Four-manifolds Theorem

Freedman's classification of simply connected 4-manifolds

Builds on 4-manifold Topological manifold Fundamental group Intersection form Symmetric bilinear form Kirby–Siebenmann class Homeomorphism

Statement

A unimodular form is a symmetric bilinear form $Q \colon \Z^n \times \Z^n \to \Z$ whose matrix has determinant $\pm 1$ . It is even if $Q(x, x)$ is even for all $x \in \Z^n$ , and odd otherwise.

For every unimodular form $Q$ there is a closed, simply connected, topological 4-manifold whose intersection form is isomorphic to $Q$ .
Two closed, simply connected, topological $4$ -manifolds are homeomorphic if and only if their intersection forms are isomorphic and their Kirby–Siebenmann invariants in $\Z/2$ are equal.
If $Q$ is even, there is exactly one such manifold with form $Q$ up to homeomorphism. If $Q$ is odd, there are exactly two, one with each value of the Kirby–Siebenmann invariant.

Remarks

Michael Freedman proved the theorem in 1982. The main step is his proof that Casson handles are homeomorphic to standard open handles, which makes the topological h-cobordism theorem hold for simply connected $4$ -manifolds. For an even form, the Kirby–Siebenmann invariant equals the signature divided by $8$ , modulo $2$ , so it is determined by $Q$ . The manifold with odd form and Kirby–Siebenmann invariant $1$ admits no smooth structure. For example, the form $(1)$ is realized by the complex projective plane $\C P^2$ and by a fake $\C P^2$ that is not homeomorphic to it. The $E_8$ manifold realizes the $E_8$ form. The case $Q = 0$ of rank $0$ is the four-dimensional topological Poincaré conjecture: a homotopy $4$ -sphere is homeomorphic to $S^4$ .

Going deeper

For a student

The theorem reduces the topological classification of closed simply connected $4$ -manifolds to the algebra of unimodular forms over $\Z$ . Indefinite forms are classified by rank, signature and type, so these manifolds are well understood. The smooth picture is very different: Donaldson's theorem and Rokhlin's theorem show that many of Freedman's manifolds have no smooth structure. Together the two results give exotic $\R^4$ .

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4-manifold concept
Fundamental group concept
Homeomorphism concept
Intersection form concept
Kirby–Siebenmann class concept
Symmetric bilinear form concept
Topological manifold concept
Freedman's classification of simply connected 4-manifolds statement

Tag	`HSD6FKCYRC`
URI	`m2://HSD6FKCYRC`
Kind	statement, theorem
Writing	W0 stub
MSC	57K40, 57N65
Status	This copy of the site has no build data. Run `m2 status HSD6FKCYRC`.
About	4-manifold, Intersection form
Uses	4-manifold, Topological manifold, Fundamental group, Intersection form, Symmetric bilinear form, Kirby–Siebenmann class, Homeomorphism
Changed	2 November 2026

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