Misplaced Pages

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (May 2020) (Learn how and when to remove this message)

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 ${\displaystyle k(M)=0}$.

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

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. 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.

Notes

1. Zhang, Weiping (2001-09-21). Lectures on Chern–Weil theory and Witten deformations. Nankai Tracts in Mathematics. Vol. 4. River Edge, NJ: World Scientific. p. 105. ISBN 9789814490627. MR 1864735. Retrieved 6 July 2018.
2. Lusztig, George; Milnor, John; Peterson, Franklin P. (1969). Semi-characteristics and cobordism. Topology. Vol. 8. Topology. p. 357–359.

• Differential topology