Kervaire semi-characteristic

In mathematics, the Kervaire semi-characteristic, introduced by Michel Kervaire (1956), is an invariant of closed manifolds M of dimension $4n+1$ taking values in $\mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}}$, given by

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

where F is a field.

Michael Atiyah and Isadore Singer (1971) 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 $k_{\mathbb{\mathbb{Q}}}(M)-k_{\mathbb{\mathbb{Z}}/2}(M)$ is the deRham invariant of $M$.

References

• Atiyah, Michael F.; Singer, Isadore M. (1971). "The Index of Elliptic Operators V". Annals of Mathematics. Second Series 93 (1): 139–149. doi:10.2307/1970757. 
• Kervaire, Michel (1956). "Courbure intégrale généralisée et homotopie". Mathematische Annalen 131: 219–252. doi:10.1007/BF01342961. ISSN 0025-5831. 
• Lee, Ronnie (1973). "Semicharacteristic classes". Topology 12 (2): 183–199. doi:10.1016/0040-9383(73)90006-2.