De Rham invariant

Template: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 $𝐙 / 2$ – either 0 or 1. It can be thought of as the simply-connected symmetric L-group $L^{4 k + 1},$ and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, $L^{4 k} ≅ L_{4 k}$ ), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant $L_{4 k + 2} .$

It is named for Swiss mathematician Georges de Rham, and used in surgery theory.^[1]^[2]

Definition

The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:^[3]

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