Kervaire semi-characteristic

Template:More footnotes In mathematics, the Kervaire semi-characteristic, introduced by Template:Harvs, is an invariant of closed manifolds M of dimension $4n+1$ taking values in $\mathbb{Z}/2\mathbb{Z}$, given by

$k_F(M)=\sum _{i=0}^{2n}\dim H^{2i}(M,F)mod2$

where F is a field.

Template:Harvs 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 $k_{\mathbb{Q}}(M)-k_{\mathbb{Z}/2}(M)$ is the de Rham invariant of $M$.

References

• Template:Cite journal
• Template:Cite journal
• Template:Cite journal