In mathematics, a 5-manifold is a 5-dimensional

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...

, possibly with a piecewise linear or

smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...

. Non- simply connected 5-manifolds are impossible to classify, as this is harder than solving the

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 el ...

.. Simply connected compact 5-manifolds were first classified by Stephen Smale and then in full generality by

Dennis Barden Dennis Barden is a mathematician at the University of Cambridge working in the fields of geometry and topology. He is known for his classification of the simply connected compact 5-manifolds and, together with Barry Mazur and John R. Stalling ...

, while another proof was later given by Aleksey V. Zhubr. This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by

Michael Freedman Michael Hartley Freedman (born April 21, 1951) is an American 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 4-dimensional gene ...

(1982) in the topological case, but is a very hard unsolved problem in the smooth case. In dimension 5, the smooth classification of simply connected manifolds is governed by classical

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

. Namely, two simply connected, smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving the linking form and the second

Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...

. Moreover, any such isomorphism in second homology is induced by some diffeomorphism. It is undecidable if a given 5-manifold is homeomorphic to

S^5

, the 5-sphere.

Examples

Here are some examples of smooth, closed, simply connected 5-manifolds: *

S^5

, the 5-sphere. *

S^2\times S^3

, the product of a 2-sphere with a 3-sphere. *

S^2\widetilde S^3

, the total space of the non-trivial

S^3

-bundle over

S^2

. *

\operatorname(3)/\operatorname(3)

, the homogeneous space obtained as the quotient of the

special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...

SU(3) by the rotation subgroup SO(3).

References

External links

* * Geometric topology Manifolds {{topology-stub