In mathematics, the Kervaire semi-characteristic, introduced by Michel Kervaire (1956), is an invariant of closed manifolds M of dimension taking values in , given by
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 .[1]
The difference is the deRham invariant of .[2]
References[edit]
- 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[edit]
- ^ 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.
- ^ Lusztig, George; Milnor, John; Peterson, Franklin P. (1969). Semi-characteristics and cobordism. Topology. Vol. 8. Topology. p. 357–359.