Kervaire semi-characteristic

In mathematics, the Kervaire semi-characteristic, introduced by Kervaire (1956), is an invariant of manifolds M of dimension 4n+1 taking values in Z/2Z, given by

k(M) =

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

Atiyah & Singer (1971) showed that it 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, The Annals of Mathematics, Vol. 93, No. 1, 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

Notes

↑ Weiping, Zhang (2001-09-21). Lectures On Chern-weil Theory And Witten Deformations. World Scientific. p. 105. ISBN 9789814490627. Retrieved 6 July 2018.

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[1] Weiping, Zhang (2001-09-21). Lectures On Chern-weil Theory And Witten Deformations. World Scientific. p. 105. ISBN 9789814490627. Retrieved 6 July 2018.