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Arf–Kervaire Invariant
In mathematics, the Kervaire invariant is an invariant of a parallelizable manifold, framed (4k+2)-dimensional manifold that measures whether the manifold could be surgery theory, surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf. The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected ''quadratic'' L-theory, L-group L_, and thus analogous to the other invariants from L-theory: the signature (topology), signature, a 4k-dimensional invariant (either symmetric or quadratic, L^ \cong L_), and the De Rham invariant, a (4k+1)-dimensional ''symmetric'' invariant L^. In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelizable Manifold
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist Smooth function, smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a Basis of a vector space, basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of frame bundle, linear frames has a global section on M. A particular choice of such a basis of vector fields on M is called a Parallelization (mathematics), parallelization (or an absolute parallelism) of M. Examples *An example with n = 1 is the circle: we can take ''V''1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension n is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take n = 2, and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |