De Rham invariant

In geometric topology, the de Rham invariant is a mod 2 invariant of a -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, and the Kervaire invariant, a -dimensional quadratic invariant
It is named for Swiss mathematician Georges de Rham, and used in surgery theory.

Definition

The de Rham invariant of a -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 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.