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 1964 (On simply-connected 4-manifolds), Thrm. 2Scorpan 05, p. 155
Wall's theorem on stabilization Wall's theorem on stabilization claims, that for h-cobordant simply connected closed smooth 4-manifolds and , there exists a natural number and a diffeomorphism .Wall 1964 (On simply-connected 4-manifolds), Thrm. 3Scorpan 05, p. 149 Becoming diffeomorphic after connected sums with 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 is an exotic 4-sphere, then Wall's theorem on stabilization guarantees the existence of a natural number and a diffeomorphism:
meaning that the connected sum with 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 with and two elements with same divisibility, self-intersection and type, there exists an automorphism with .Scorpan 05, p. 152
Divisibility is the largest integer able to divide the element. Type refers to whether the element is characteristic or not. is always even, hence the above condition only excludes definite forms with and near-definite forms with . According to Serre's classification, this includes only the hyperbolic form as well as and .
Wall's theorem on diffeomorphisms Wall's theorem on diffeomorphisms claims, that for a simply connected smooth 4-manifold with indefinite intersection form , every automorphism of (with the hyperbolic form ) comes from a self-diffeomorphism on .Scorpan 05, p. 153 This shows an important difference between smooth and topological 4-manifolds since the latter require no stabilization. For a simply connected topological 4-manifold with indefinite intersection form , every automorphism of comes from a self-homeomorphism on .
Literature
References
External links
Wall theorems on nLab