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 $\mathbb {Z} /2\mathbb {Z}$ , given by

k(M)=\sum _{i=0}^{2n}\dim H^{2i}(M,\mathbb {R} ){\bmod {2}}

.

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 $k(M)=0$ .[1]

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

Zhang, Weiping (2001-09-21). Lectures on Chern–Weil theory and Witten deformations. Nankai Tracts in Mathematics. 4. River Edge, NJ: World Scientific. p. 105. ISBN 9789814490627. MR 1864735. Retrieved 6 July 2018.

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[1] Zhang, Weiping (2001-09-21). Lectures on Chern–Weil theory and Witten deformations. Nankai Tracts in Mathematics. 4. River Edge, NJ: World Scientific. p. 105. ISBN 9789814490627. MR 1864735. Retrieved 6 July 2018.