Kervaire semi-characteristic

In mathematics, the Kervaire semi-characteristic, introduced by, is an invariant of closed manifolds M of dimension $$4n+1$$ taking values in $$\Z/2\Z$$, given by


 * $$k_F(M) = \sum_{i=0}^{2n} \dim H^{2i}(M,F)\bmod 2$$

where F is a field.

showed that the Kervaire semi-characteristic of a differentiable manifold is given by the index of a skew-adjoint elliptic operator.

Assuming M is oriented, the Atiyah vanishing theorem states that if M has two linearly independent vector fields, then $$k(M) = 0$$.

The difference $$k_\Q(M)-k_{\Z/2}(M)$$ is the de Rham invariant of $$M$$.