geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originat ...

, the de Rham invariant is a mod 2 invariant of a (4''k''+1)-dimensional manifold, that is, an element of

\mathbf/2

– either 0 or 1. It can be thought of as the simply-connected ''symmetric'' L-group

L^,

and thus analogous to the other invariants from L-theory: the

signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...

, a 4''k''-dimensional invariant (either symmetric or quadratic,

L^ \cong L_

), and the

Kervaire invariant In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a spher ...

, a (4''k''+2)-dimensional ''quadratic'' invariant

L_.

It is named for Swiss mathematician

Georges de Rham Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. Biography Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in ...

, and used in

surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while A ...

Definition

The de Rham invariant of a (4''k''+1)-dimensional manifold can be defined in various equivalent ways: * the rank of the 2-torsion in

H_(M),

as an integer mod 2; * the Stiefel–Whitney number

w_2w_

; * the (squared) Wu number,

v_Sq^1v_,

where

v_ \in H^(M;Z_2)

is the

Wu class Wu may refer to: States and regions on modern China's territory *Wu (state) (; och, *, italic=yes, links=no), a kingdom during the Spring and Autumn Period 771–476 BCE ** Suzhou or Wu (), its eponymous capital ** Wu County (), a former county i ...

of the normal bundle of

M

and

Sq^1

is the

Steenrod square In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, ...

; formally, as with all

characteristic number In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...

s, this is evaluated on the

fundamental class In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fund ...

(v_Sq^1v_,

; * in terms of a semicharacteristic.

References

* * Chess, Daniel,
A Poincaré-Hopf type theorem for the de Rham invariant
'' 1980 {{refend Geometric topology Surgery theory