Wall theorems

Template:Short description In differential topology in mathematics, the Wall theorems are four results, which connect the smooth structure, auto-diffeomorphisms and the intersection form of a smooth 4-manifold with h-cobordisms, stable diffeomorphisms and induced form automorphisms. The theorems are named after C. T. C. Wall, who proved them in 1964.

Wall's theorem on h-cobordisms

Wall's theorem on h-cobordisms claims, that simply connected closed smooth 4-manifolds with isomorphic intersection form are even h-cobordant.

Wall's theorem on stabilization

Wall's theorem on stabilization claims, that for h-cobordant simply connected closed smooth 4-manifolds $M$ and $N$, there exists a natural number $n\in \mathbb{N}$ and a diffeomorphism $M\mathrm{\#}n(S^2\times S^2)\cong N\mathrm{\#}n(S^2\times S^2)$. Becoming diffeomorphic after connected sums with $S^2\times S^2$ is also called stably diffeomorphic. A combination with Wall's theorem on h-cobordisms yields that simply connected smooth 4-manifolds with isomorphic intersection form are even stably diffeomorphic.

Since both the classification of smooth structures on 4-manifolds and the existence of exotic 4-spheres are important open problems in differential topology, it seems that a connection could exist using connected sums with exotic 4-spheres. However, if $\mathrm{\Sigma }$ is an exotic 4-sphere, then Wall's theorem on stabilization guarantees the existence of a natural number $n\in \mathbb{N}$ and a diffeomorphism:

$\mathrm{\Sigma }\mathrm{\#}n(S^2\times S^2)\cong S^4\mathrm{\#}n(S^2\times S^2)\cong n(S^2\times S^2),$

meaning that the connected sum with $\mathrm{\Sigma }$ causes no change in the smooth structure.

Wall's theorem on automorphisms

Wall's theorem on automorphisms claims, that for a symmetric unimodular bilinear form $Q:{\mathbb{Z}}^n\times {\mathbb{Z}}^n\to \mathbb{Z}$ with $\mathrm{rk}(Q)-|\sigma (Q)|\ge 4$ and two elements $x,y\in \mathbb{Z}$ with same divisibility, self-intersection $Q(x,x)=Q(y,y)$ and type, there exists an automorphism $f:{\mathbb{Z}}^n\to {\mathbb{Z}}^n$ with $f(x)=y$.

Divisibility is the largest integer able to divide the element. Type refers to whether the element is characteristic or not. $\mathrm{rk}(Q)-|\sigma (Q)|$ is always even, hence the above condition only excludes definite forms with $\mathrm{rk}(Q)-|\sigma (Q)|=0$ and near-definite forms with $\mathrm{rk}(Q)-|\sigma (Q)|=2$. According to Serre's classification, this includes only the hyperbolic form $H=Q_{S^2\times S^2}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ as well as $[+1]\oplus n[-1]$ and $n[+1]\oplus [-1]$.

Wall's theorem on diffeomorphisms

Wall's theorem on diffeomorphisms claims, that for a simply connected smooth 4-manifold $M$ with indefinite intersection form $Q_M$, every automorphism of $Q_{M\mathrm{\#}(S^2\times S^2)}\cong Q_M\oplus H$ (with the hyperbolic form $H=Q_{S^2\times S^2}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$) comes from a self-diffeomorphism on $M\mathrm{\#}(S^2\times S^2)$. This shows an important difference between smooth and topological 4-manifolds since the latter require no stabilization. For a simply connected topological 4-manifold $M$ with indefinite intersection form $Q_M$, every automorphism of $Q_M$ comes from a self-homeomorphism on $M$.

Literature

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External links

• Wall theorems on nLab