![]() Classify the different smooth structures on a smoothable manifold. ![]() ![]() Which topological manifolds are smoothable?.This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of many problems is already known to be intractable.įor manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way, so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds.Ī major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones.Īs the topological ones are known, this breaks up into two parts: As there is no algorithm to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial) there is no algorithm to tell if two 4-manifolds have the same fundamental group. If the fundamental group is too large (for example, a free group on 2 generators), then Freedman's techniques seem to fail and very little is known about such manifolds.įor any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. A famous theorem of Michael Freedman ( 1982) implies that the homeomorphism type of the manifold only depends on this intersection form, and on a Z / 2 Z. The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).Ĥ-manifolds are important in physics because in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. A smooth 4-manifold is a 4-manifold with a smooth structure. In mathematics, a 4-manifold is a 4-dimensional topological manifold.
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |