In mathematics, the Kervaire semi-characteristic, introduced by , is an invariant of

closed manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example ...

s ''M'' of dimension

4n+1

taking values in

\Z/2\Z

, given by :

k(M) = \sum_^ \dim H^(M,\R)\bmod 2

. showed that the Kervaire semi-characteristic of a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

is given by the

index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...

of a skew-adjoint

elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which i ...

. Assuming ''M'' is

oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...

, the Atiyah vanishing theorem states that if ''M'' has two linearly independent vector fields, then

k(M) = 0

References

* * *

Notes

{{reflist Differential topology