In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of – either 0 or 1. It can be thought of as the simply-connected symmetric L-group and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, ), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant
It is named for Swiss mathematician Georges de Rham, and used in surgery theory.John W. Morgan, A product formula for surgery obstructions, 1978
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 as an integer mod 2; the Stiefel–Whitney number ; the (squared) Wu number, where is the Wu class of the normal bundle of and is the Steenrod square; formally, as with all characteristic numbers, this is evaluated on the fundamental class: ; in terms of a semicharacteristic.
References
Chess, Daniel, A Poincaré-Hopf type theorem for the de Rham invariant, 1980