Kervaire manifold
Encyclopedia
In mathematics
, specifically in differential topology
, a Kervaire manifold K4n+2 is a piecewise-linear manifold of dimension 4n+2 constructed by by plumbing together the tangent bundle
s of two 2n+1-spheres, and then gluing a ball
to the result. In 10 dimensions this gives a piecewise-linear manifold with no smooth structure
.
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...
, specifically in differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, a Kervaire manifold K4n+2 is a piecewise-linear manifold of dimension 4n+2 constructed by by plumbing together the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
s of two 2n+1-spheres, and then gluing a ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
to the result. In 10 dimensions this gives a piecewise-linear manifold with no smooth structure
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....
.