Kervaire semi-characteristic

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

${\displaystyle k_{F}(M)=\sum _{i=0}^{2n}\dim H^{2i}(M,F){\bmod {2}}}$

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 ${\displaystyle k(M)=0}$.^[1]

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

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. JSTOR 1970757.
• Kervaire, Michel (1956). "Courbure intégrale généralisée et homotopie". Mathematische Annalen. 131: 219–252. doi:10.1007/BF01342961. ISSN 0025-5831. MR 0086302.
• Lee, Ronnie (1973). "Semicharacteristic classes". Topology. 12 (2): 183–199. doi:10.1016/0040-9383(73)90006-2. MR 0362367.