nLab Wall theorems

Idea

Wall’s theorems are four related results connecting the smooth structure, auto-diffeomorphisms and the intersection form of a smooth 4-manifolds with h-cobordisms, stably diffeomorphisms and induced form automorphisms.

Wall’s theorem on h-cobordisms

Proposition

(Wall’s theorem on h-cobordisms) Simply connected smooth 4-manifolds with isomorphic intersection forms are h-cobordant.

(Scorpan 05, p. 155)

Wall’s theorem on stabilization

Proposition

(Wall’s theorem on stabilization) For h-cobordant simply connected smooth 4-manifolds $M$ and $N$, there exists a natural number $n\in\mathbb{N}$ and a diffeomorphism:

$$M\#n\left(S^2\times S^2\right) \cong N\#n\left(S^2\times S^2\right).$$

(Scorpan 05, p. 149)

Becoming diffeomorphic after connected sums with $S^2\times S^2$ is also called stably diffeomorphic. Combining Wall’s theorem on h-cobordisms with Wall’s theorem on stabilization directly yields:

Proposition

Simply connected smooth 4-manifolds with isomorphic intersection forms are stably diffeomorphic.

Wall’s theorem on automorphisms

Proposition

(Wall’s theorem on automorphisms) For a symmetric unimodular bilinear form $Q\colon\mathbb{Z}^n\times\mathbb{Z}^n\rightarrow\mathbb{Z}$ with $rk(Q)-|\sigma(Q)|\geq 4$ and for two elements $x,y\in\mathbb{Z}^n$ with same divilibility, self-intersection $Q(x,x)=Q(y,y)$ and type, there exists an automorphism $f\colon\mathbb{Z}^n\rightarrow\mathbb{Z}^n$ with $f(x)=y$.

(Scorpan 05, p. 152)

Divisibility is the largest integer able to divide the element. Type refers to whether the element is characteristic or not.

$rk(Q)-|\sigma(Q)|$ is always even, hence the above condition excludes only definite forms with $rk(Q)-|\sigma(Q)|=0$ and “near-definite” forms with $rk(Q)-|\sigma(Q)|=2$. According to Serre’s classification theorem, the “near-definite” forms only include $H=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ as well as $[+1]\oplus n[-1]$ and $n[+1]\oplus[-1]$.

Wall’s theorem on diffeomorphisms

Proposition

(Wall’s theorem on diffeomorphisms) For a simply connected smooth 4-manifolds $M$ with indefinite intersection form $Q_M$, every automorphism of $Q_{M\#(S^2\times S^2)}\cong Q_M\oplus H$ (with the hyperbolic form $H=Q_{S^2\times S^2}=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$) comes from a self-diffeomorphism on $M\#(S^2\times S^2)$.

(Scorpan 05, p. 153)

For topological manifolds, no stabilization is necessary:

Proposition

For a simply connected topological 4-manifolds $M$ with indefinite intersection form $Q_M$, every automorphism of $Q_M$ comes from a self-homeomorphism on $M$, which is unique up to isotopy.

(Scorpan 05, p. 153)

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

Named after:

• C. T. C. Wall: Diffeomorphisms of 4-Manifolds Journal of London Mathematical Society s1–39 1 (1964) 131–140 [doi:10.1112/jlms/s1-39.1.131]

• C. T. C. Wall: On simply-connected 4-manifolds, Journal of the London Mathematical Society s1–39 1 (1964) 141–-149 [doi:10.1112/jlms/s1-39.1.141, pdf]

See also:

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