nLab Freedman classification

Idea

Freedman’s classification is a deep connection between topology and algebra, providing the full classification of simply connected closed topological 4-manifolds by their intersection form and their Kirby-Siebenmann invariant. It allows any symmetric unimodular bilinear form to be represented by such a manifold and translates isomorphism of these forms into homeomorphisms of such manifolds (for equal Kirby-Siebenmann invariant when it’s odd). It in particular expanded the topological h-cobordism theorem to four dimensions and solved the topological Poincaré conjecture in four dimensions.

Freedman’s classification was published by Michael Freedman in 1982 and earned him a Fields Medal in 1986. His proof was expanded in the collaborative textbook The Disc Embedding Theorem, on which multiple mathematicians began working on in 2013 and which was eventually published in 2021.

Statement

Every symmetric unimodular bilinear form can be realized as the intersection form of a simply connected closed topological 4-manifold.

• If the form is even, then there exists exactly one such manifold up to homeomorphism.

• If the form is odd, then there exist exactly two such manifolds up to homeomorphism, which have different binary Kirby-Siebenmann invariant, in particular implying that at most one of them is smoothable.

(Scorpan 05, p. 240)

Often, these results are rephrased as:

• If two simply connected closed topological 4-manifolds have the same even intersection form, they’re homeomorphic.

• If two simply connected closed topological 4-manifolds have the same odd intersection form and Kirby-Siebenmann invariant, they’re homeomorphic. In particular, if both are smoothable, they’re homeomorphic.

Examples

Freedman’s classification allows to translate purely algebraic isomorphisms of symmetric unimodular bilinear forms into homeomorphisms of closed simply connected topological 4-manifolds.

For example, $S^2\times S^2$ and $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$ have intersection forms of same signature $0$ and rank $2$:

$$Q_{S^2\times S^2} =\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix};$$

$$Q_{\mathbb{C}P^2\#\overline{\mathbb{C}P^2}} =\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$$

But since the former form is even, while the latter is odd, they cannot be isomorphic and hence there cannot be a homeomorphism $S^2\times S^2\cong\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$. But an expansion of the form or equivalently a connected sum can untangle the form and yield a homeomorphism:

$$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\oplus[-1] \cong[+1]\oplus 2[-1] \Rightarrow \left(S^2\times S^2\right)\#\overline{\mathbb{C}P^2} \cong\mathbb{C}P^2\# 2\overline{\mathbb{C}P^2}$$

$$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\oplus[+1] \cong 2[+1]\oplus[-1] \Rightarrow \left(S^2\times S^2\right)\#\mathbb{C}P^2 \cong 2\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$$

There is an isomorphism of forms:

$$E_8\oplus -E_8 \cong 8\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}.$$

$M_{E_8}\#\overline{M_{E_8}}$ and $# 8(S^2\times S^2)$ both have this as their even intersection form and are both smoothable, so Freedman’s classifciation gives a homeomorphism:

$$M_{E_8}\#\overline{M_{E_8}} \cong# 8(S^2\times S^2)$$

(Scorpan 05, p. 240)

There is an isomorphism of forms:

$$E_8\oplus[-1] \cong 8[+1]\oplus[-1].$$

$M_{E_8}\#\overline{\mathbb{C}P^2}$ and $8\mathbb{C}P^2 \#\overline{\mathbb{C}P^2}$ both have this as their odd intersection form, but different Kirby-Siebenmann invariant. A way around is using the fake second complex projective space to flip it and obtain a homeomorphism:

$$M_{E_8}\# *\overline{\mathbb{C}P^2} \cong 8\mathbb{C}P^2\#\overline{\mathbb{C}P^2}.$$

There is an isomorphism of forms:

$$-E_8\oplus[+1] \cong [+1]\oplus 8[-1].$$

$\overline{M_{E_8}}\#\mathbb{C}P^2$ and $\mathbb{C}P^2 \# 8\overline{\mathbb{C}P^2}$ both have this as their odd intersection form, but different Kirby-Siebenmann invariant. Again using the fake second complex projective space yields a homeomorphism:

$$\overline{M_{E_8}}\# *\mathbb{C}P^2 \cong\mathbb{C}P^2 \# 8\overline{\mathbb{C}P^2}.$$

(Scorpan 05, p. 242-243)

Corollaries

A strong corollary of Freedman’s classification is that it fully describes the splittings of simply connected closed topological 4-manifold into connected sums, which breaks down for smooth 4-manifolds:

Proposition

If the intersection form of a simply connected closed topological 4-manifold $M$ splits:

$$Q_M \cong Q_1\oplus Q_2,$$

then there are simply connected closed topological 4-manifolds $N_1$ and $N_2$ with them as intersection forms and a corresponding connected sum:

$$M \cong N_1#N_2 \cong *N_1#*N_2.$$

Another strong corollary of Freedman’s classification is that it allows to untangle the entire topological structure of a simply connected closed topological 4-manifold: A connected sum with either $\mathbb{C}P^2$ or $\overline{\mathbb{C}P^2}$ always makes the intersection form odd. A connected sum with $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$ furthermore always makes it indefinite. In this case, Serre’s classification theorem or the Hesse-Minkowski theorem give an isomorphism to a direct sum of $[+1]$ and $[-1]$, which can be also be realized by a connected sum of $\mathbb{C}P^2$ and $\overline{\mathbb{C}P^2}$.

Proposition

For a simply connected closed topological 4-manifold $M$, there is a homeomorphism:

$$M\#\mathbb{C}P^2\#\overline{\mathbb{C}P^2} \cong\left(b_2^+(M)+1\right)\mathbb{C}P^2\#\left(b_2^-(M)+1\right)\overline{\mathbb{C}P^2}.$$

(Scorpan 05, p. 238)

Connection with Donaldson’s theorem

Freedman’s classification can be combined with Donaldson's theorem, which causes both to be restricted: Freedman’s classification requires simple connectedness and Donaldson's theorem requires smoothness, both of which are not required for the respective other result. (Initially, simple connectedness was required for Donaldson's theorem when first published in 1983, but it was improved to work without in 1987.)

Proposition

A simply connected compact oriented smooth 4-manifold $M$ with definite intersection form is homeomorphic to $\# b_2^+(M)\mathbb{C}P^2$ if it’s positive definite or homeomorphic to $\# b_2^-(M)\overline{\mathbb{C}P^2}$ if it’s negative definite.

Proof

According to Donaldson's theorem, $M$ has a diagonal intersection form, which is $\oplus b_2^+(M)(+1)$ if it’s positive definite or $\oplus b_2^-(M)(-1)$ if it’s negative definite. $\# b_2^+(M)\mathbb{C}P^2$ with $Q_{\mathbb{C}P^2}=(+1)$ and $\# b_2^-(M)\overline{\mathbb{C}P^2}$ with $Q_{\overline{\mathbb{C}P^2}}=(-1)$ also have these respective intersection forms and are simply connected due to $\pi_1(\mathbb{C}P^2)\cong\pi_1(\overline{\mathbb{C}P^2})\cong 1$, so are homeomorphic according to Freedman’s classification. (Required here is that for two topological 4-manifolds $M$ and $N$, one has $Q_{M# N}\cong Q_M\oplus Q_N$ and $\pi_1(M# N)\cong\pi_1(M)*\pi_1(N)$.)

Related entries

Articles on geometry and topology of 4-manifolds:

• Basic concepts:

  • Donaldson theory and related articles
  • Seiberg-Witten theory and related articles
  • intersection form
  • Kirby-Siebenmann invariant
  • signature

• Important examples

  • Dolgachev surface$\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$
  • E₈ manifold$M_{E_8}$
  • exotic R^4
  • fake second complex projective space$\mathbb{C}P^2$

• Central results:

  • Donaldson theorem (Fields Medal 1986)
  • Freedman classification (Fields Medal 1986)
  • Furuta theorem (10/8 theorem)
  • Hirzebruch signature theorem
  • Hitchin-Thorpe inequality
  • Quinn theorem
  • Rokhlin theorem
  • van der Blij lemma
  • Wall theorems
  • Wu formula

• Open problems:

  • 3/2 conjecture
  • 11/8 conjecture
  • simple type conjecture
  • Poincaré conjecture

References

• Michael Freedman, The topology of 4-manifolds, J. Differential Geometry 17 3 (1982), pp. 357–453 [doi:10.4310/jdg/1214437136 ISSN 0022-040X MR 0679066]

• Michael Freedman, Frank Quinn, Topology of 4-Manifolds (1990) [doi:10.1515/9781400861064]

• Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society (2005) [ISBN 978-1470468613]