In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a 5-manifold is a 5-dimensional topological manifold

Topological manifold

In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

, possibly with a piecewise linear or smooth structure

Differential structure

In mathematics, an n-dimensional differential structure on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold...

.

Non-simply connected

Simply connected space

In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...

 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups

Word problem for groups

In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...

. Simply connected compact 5-manifolds were first classified by Dennis Barden

Dennis Barden

Dennis Barden is a mathematician at the University of Cambridge, who proved the s-cobordism theorem and classified the simply connected compact 5-manifolds.-Publications:...

 and another proof was later given by A. V. Zhubr. Rather surprisingly, this turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Poincaré conjecture

Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

, and the 4-dimensional case

4-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...

 was solved by Freedman

Michael Freedman

Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...

 (1982) in the topological case, but is a very hard unsolved problem in the smooth case.

Indeed, in dimension 5 smooth classification is governed by classical algebraic topology, namely, two simply connected

Simply connected space

In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...

 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving linking form and the second Stiefel–Whitney class

Stiefel–Whitney class

In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney class is an example of a \mathbb Z_2characteristic class associated to real vector bundles.-General presentation:...

. Moreover any such isomorphism is induced by some diffeomorphism.

External links

• 1-connected 5-manifolds at the Manifold Atlas.