Английская Википедия:De Rham invariant

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Шаблон:Short description In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of <math>\mathbf{Z}/2</math> – either 0 or 1. It can be thought of as the simply-connected symmetric L-group <math>L^{4k+1},</math> and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, <math>L^{4k} \cong L_{4k}</math>), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant <math>L_{4k+2}.</math>

It is named for Swiss mathematician Georges de Rham, and used in surgery theory.

Definition

The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:

• the rank of the 2-torsion in <math>H_{2k}(M),</math> as an integer mod 2;
• the Stiefel–Whitney number <math>w_2w_{4k-1}</math>;
• the (squared) Wu number, <math>v_{2k}Sq^1v_{2k},</math> where <math>v_{2k} \in H^{2k}(M;Z_2)</math> is the Wu class of the normal bundle of <math>M</math> and <math>Sq^1</math> is the Steenrod square; formally, as with all characteristic numbers, this is evaluated on the fundamental class: <math>(v_{2k}Sq^1v_{2k},[M])</math>;
• in terms of a semicharacteristic.

References

• Шаблон:Citation
• Chess, Daniel, A Poincaré-Hopf type theorem for the de Rham invariant, 1980