Kervaire semi-characteristic

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

[math]\displaystyle{ k_F(M) = \sum_{i=0}^{2n} \dim H^{2i}(M,F)\bmod 2 }[/math]

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 [math]\displaystyle{ k(M) = 0 }[/math].

The difference [math]\displaystyle{ k_\Q(M)-k_{\Z/2}(M) }[/math] is the deRham invariant of [math]\displaystyle{ M }[/math].

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.